Unit 5 Polynomials Exponents n Classification of Polynomials
![Unit 5 Polynomials Exponents n Classification of Polynomials n Multiplying and Dividing Polynomials n Unit 5 Polynomials Exponents n Classification of Polynomials n Multiplying and Dividing Polynomials n](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-1.jpg)
![Multiplying Monomials OBJECTIVE: Given five problems, the student will be able to multiply the Multiplying Monomials OBJECTIVE: Given five problems, the student will be able to multiply the](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-2.jpg)
![Check for Understanding n n n a 4 • a 7 a 4 b Check for Understanding n n n a 4 • a 7 a 4 b](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-3.jpg)
![Rules for Multiplying Variables with Exponents n What do you do when Multiplying variables Rules for Multiplying Variables with Exponents n What do you do when Multiplying variables](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-4.jpg)
![Rules for Multiplying Variables with Exponents 2 (a 3 b c)(a 2 b 5 Rules for Multiplying Variables with Exponents 2 (a 3 b c)(a 2 b 5](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-5.jpg)
![Examples (3 3 x (-5 4 y 3 a )(-7 2 x b )( Examples (3 3 x (-5 4 y 3 a )(-7 2 x b )(](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-6.jpg)
![n What is the n When taking the power of a power, you multiply n What is the n When taking the power of a power, you multiply](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-7.jpg)
![Examples (3 x 3 y 4)3 3 4 3 4 (3 x y )(3 Examples (3 x 3 y 4)3 3 4 3 4 (3 x y )(3](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-8.jpg)
![Find the Following Products: (3 g 7 h 3 j 2) (2 g 2 Find the Following Products: (3 g 7 h 3 j 2) (2 g 2](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-9.jpg)
![Find the Following Products: (3 g 7 h 3 j 2) (2 g 2 Find the Following Products: (3 g 7 h 3 j 2) (2 g 2](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-10.jpg)
![Check for Understanding n n n a 4 • a 7 n a 4 Check for Understanding n n n a 4 • a 7 n a 4](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-11.jpg)
![Summary Inspiration: What is the exponent rule for multiplying monomials? n How is the Summary Inspiration: What is the exponent rule for multiplying monomials? n How is the](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-12.jpg)
![Dividing Monomials OBJECTIVE: Given four division problems, the student will be able to divide Dividing Monomials OBJECTIVE: Given four division problems, the student will be able to divide](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-13.jpg)
![Check for Understanding x 11 3 x t 4 9 t 3 5 m Check for Understanding x 11 3 x t 4 9 t 3 5 m](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-14.jpg)
![What is the rule for dividing monomials? n If I add the exponents when What is the rule for dividing monomials? n If I add the exponents when](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-15.jpg)
![x 6 x 2 =x 6– 2 4 d 9 d 3 =d 9– x 6 x 2 =x 6– 2 4 d 9 d 3 =d 9–](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-16.jpg)
![What happens when the exponent is the same in the numerator and denominator? 5 What happens when the exponent is the same in the numerator and denominator? 5](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-17.jpg)
![x 3 y 4 = x 3 -2 y 4 -1 = x 1 x 3 y 4 = x 3 -2 y 4 -1 = x 1](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-18.jpg)
![What happens when the exponent is negative? 2 x 5 x =x 2– 5 What happens when the exponent is negative? 2 x 5 x =x 2– 5](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-19.jpg)
![d 3 d 7 =d 3– 7 -4 =d 1 4 d n 5 d 3 d 7 =d 3– 7 -4 =d 1 4 d n 5](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-20.jpg)
![m 6 n 2 p 8 11 2 7 m n p = m m 6 n 2 p 8 11 2 7 m n p = m](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-21.jpg)
![Check for Understanding x 11 = x 8 3 x t 4 = 1 Check for Understanding x 11 = x 8 3 x t 4 = 1](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-22.jpg)
![Polynomial Terminology OBJECTIVE: Given two polynomials, the student will be able to find the Polynomial Terminology OBJECTIVE: Given two polynomials, the student will be able to find the](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-23.jpg)
![Check for Understanding: Find the degree of each monomial in the polynomial term and Check for Understanding: Find the degree of each monomial in the polynomial term and](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-24.jpg)
![Types of Polynomials: n Monomial: n One or more numbers multiplied together n The Types of Polynomials: n Monomial: n One or more numbers multiplied together n The](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-25.jpg)
![Degree of the Polynomial Degree: Monomial n The monomial of the n Tally up Degree of the Polynomial Degree: Monomial n The monomial of the n Tally up](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-26.jpg)
![Find the Degree of the Polynomial 3 x – 5 First Degree -3 a Find the Degree of the Polynomial 3 x – 5 First Degree -3 a](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-27.jpg)
![Writing Polynomials in Descending Order n Write the terms so that the powers of Writing Polynomials in Descending Order n Write the terms so that the powers of](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-28.jpg)
![Writing Polynomials in Descending Order n Write the terms so that the powers of Writing Polynomials in Descending Order n Write the terms so that the powers of](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-29.jpg)
![Writing Polynomials in Descending Order n Write the terms so that the powers of Writing Polynomials in Descending Order n Write the terms so that the powers of](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-30.jpg)
![Check for Understanding: Find the degree of the polynomial. Then arrange them in descending Check for Understanding: Find the degree of the polynomial. Then arrange them in descending](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-31.jpg)
![Check for Understanding: Find the degree of the polynomial. Then arrange them in descending Check for Understanding: Find the degree of the polynomial. Then arrange them in descending](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-32.jpg)
![Adding and Subtracting Polynomials OBJECTIVE: Given four polynomial problems the student will be able Adding and Subtracting Polynomials OBJECTIVE: Given four polynomial problems the student will be able](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-33.jpg)
![Check for Understanding 1. (4 x 2 + 5 x) + (-7 x 2 Check for Understanding 1. (4 x 2 + 5 x) + (-7 x 2](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-34.jpg)
![What are the steps to combining polynomials? 1. “Stack” the two polynomials and align What are the steps to combining polynomials? 1. “Stack” the two polynomials and align](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-35.jpg)
![+ Find the sum of (3 + 2 a + a 2) and (5 + Find the sum of (3 + 2 a + a 2) and (5](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-36.jpg)
![Let’s Try a Few: (3 x 4 + 2 x 3 + x 2) Let’s Try a Few: (3 x 4 + 2 x 3 + x 2)](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-37.jpg)
![(3 x 4 + 2 x 3 + x 2)+(-5 x 4 + 8 (3 x 4 + 2 x 3 + x 2)+(-5 x 4 + 8](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-38.jpg)
![How do you find 1. Stack the two the difference of equations and align How do you find 1. Stack the two the difference of equations and align](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-39.jpg)
![(x 3 + 4 x 2 - 7 x - 2) - (2 x (x 3 + 4 x 2 - 7 x - 2) - (2 x](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-40.jpg)
![(4 x 3 + 8 x 2 - x + 21) - (3 x (4 x 3 + 8 x 2 - x + 21) - (3 x](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-41.jpg)
![(3 x 2 y + xy – 5) - (2 x 2 y – (3 x 2 y + xy – 5) - (2 x 2 y –](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-42.jpg)
![Check for Understanding 1. (4 x 2 + 5 x) + (-7 x 2 Check for Understanding 1. (4 x 2 + 5 x) + (-7 x 2](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-43.jpg)
![Summary Inspiration: n n What do you do when adding polynomials? What do you Summary Inspiration: n n What do you do when adding polynomials? What do you](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-44.jpg)
![Multiplying a Polynomial by a Monomial OBJECTIVE: Given three problems, the student will be Multiplying a Polynomial by a Monomial OBJECTIVE: Given three problems, the student will be](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-45.jpg)
![Check for Understanding 1. 4 a 2 (-8 a 3 c + c -11) Check for Understanding 1. 4 a 2 (-8 a 3 c + c -11)](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-46.jpg)
![Use Distributive Property What property do I use when multiplying a polynomial by a Use Distributive Property What property do I use when multiplying a polynomial by a](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-47.jpg)
![Use Distributive Property n Distribute the monomial on the outside of the parentheses to Use Distributive Property n Distribute the monomial on the outside of the parentheses to](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-48.jpg)
![2 x 2 (5 x 2 -7 x +8) 2 x 2(5 x 2) 2 x 2 (5 x 2 -7 x +8) 2 x 2(5 x 2)](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-49.jpg)
![Step It Up a Notch: Simplify 3 t(2 t -4) + 6(5 t 2 Step It Up a Notch: Simplify 3 t(2 t -4) + 6(5 t 2](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-50.jpg)
![Step It Up a Notch: Simplify Get rid of the parentheses 3 t(2 t Step It Up a Notch: Simplify Get rid of the parentheses 3 t(2 t](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-51.jpg)
![3(y 3 +8 y) + 4 y(2 y 3 -8 y 2 +2 y 3(y 3 +8 y) + 4 y(2 y 3 -8 y 2 +2 y](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-52.jpg)
![Get rid of the parentheses by distributing. 3(y 3 +8 y) + 4 y(2 Get rid of the parentheses by distributing. 3(y 3 +8 y) + 4 y(2](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-53.jpg)
![Get rid of the parentheses by distributing. 3(y 3 +8 y) + 4 y(2 Get rid of the parentheses by distributing. 3(y 3 +8 y) + 4 y(2](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-54.jpg)
![Check for Understanding 1. 4 a 2 (-8 a 3 c + c -11) Check for Understanding 1. 4 a 2 (-8 a 3 c + c -11)](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-55.jpg)
![Summary Inspiration: What property is used when multiplying a polynomial by a monomial? n Summary Inspiration: What property is used when multiplying a polynomial by a monomial? n](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-56.jpg)
![Multiplying Polynomials OBJECTIVE: Given 4 multiplication problems, the student will be able to find Multiplying Polynomials OBJECTIVE: Given 4 multiplication problems, the student will be able to find](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-57.jpg)
![Check for Understanding 1. (y – 7) (y + 3) 2. (2 m + Check for Understanding 1. (y – 7) (y + 3) 2. (2 m +](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-58.jpg)
![What is the result of multiplying two binomials? n When multiplying two binomials, the What is the result of multiplying two binomials? n When multiplying two binomials, the](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-59.jpg)
![Multiplying Two Binomials What is the mnemonic device to remember how to multiply two Multiplying Two Binomials What is the mnemonic device to remember how to multiply two](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-60.jpg)
![Multiplying Two Binomials: FOIL First Outside Inside Last (d + 2) (d + 8) Multiplying Two Binomials: FOIL First Outside Inside Last (d + 2) (d + 8)](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-61.jpg)
![FOIL First Outside Inside Last (d + 2) (d + 8) 2 d +8 FOIL First Outside Inside Last (d + 2) (d + 8) 2 d +8](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-62.jpg)
![FOIL First Outside Inside Last (d + 2) (d + 8) 2 d +8 FOIL First Outside Inside Last (d + 2) (d + 8) 2 d +8](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-63.jpg)
![FOIL First Outside Inside Last (d + 2) (d + 8) 2 d +8 FOIL First Outside Inside Last (d + 2) (d + 8) 2 d +8](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-64.jpg)
![Combine the Middle Terms 2 d +8 d +2 d +16 2 d +10 Combine the Middle Terms 2 d +8 d +2 d +16 2 d +10](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-65.jpg)
![Multiply: (2 m – 4) (m + 5) FOIL First Outside Inside Last (2 Multiply: (2 m – 4) (m + 5) FOIL First Outside Inside Last (2](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-66.jpg)
![Multiplying Two Binomials: FOIL First Outside Inside Last (2 m – 4) (m + Multiplying Two Binomials: FOIL First Outside Inside Last (2 m – 4) (m +](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-67.jpg)
![FOIL First Outside Inside Last (2 m – 4) (m + 5) 2 2 FOIL First Outside Inside Last (2 m – 4) (m + 5) 2 2](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-68.jpg)
![FOIL First Outside Inside Last (2 m – 4) (m + 5) 2 2 FOIL First Outside Inside Last (2 m – 4) (m + 5) 2 2](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-69.jpg)
![FOIL First Outside Inside Last (2 m – 4) (m + 5) 2 2 FOIL First Outside Inside Last (2 m – 4) (m + 5) 2 2](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-70.jpg)
![Combine the Middle Terms 2 2 m +10 m-4 m -20 2 2 m Combine the Middle Terms 2 2 m +10 m-4 m -20 2 2 m](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-71.jpg)
![Multiply: (2 x + 2) (5 x - 8) FOIL First Outside Inside Last Multiply: (2 x + 2) (5 x - 8) FOIL First Outside Inside Last](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-72.jpg)
![FOIL First Outside Inside Last (2 x + 2) (5 x - 8) First FOIL First Outside Inside Last (2 x + 2) (5 x - 8) First](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-73.jpg)
![(2 a +3 b) (5 a -2 b) First Outside Inside Last 2 2 (2 a +3 b) (5 a -2 b) First Outside Inside Last 2 2](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-74.jpg)
![Check for Understanding 1. (y – 7) (y + 3) y 2 – 4 Check for Understanding 1. (y – 7) (y + 3) y 2 – 4](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-75.jpg)
![Special Products OBJECTIVE: Same as 9 -7. Special Products OBJECTIVE: Same as 9 -7.](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-76.jpg)
![What is a difference of squares and how is it different than the other What is a difference of squares and how is it different than the other](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-77.jpg)
![n A difference (SUBTRACTION) of (PERFECT) squares is exactly that…the result (ANSWER) will be n A difference (SUBTRACTION) of (PERFECT) squares is exactly that…the result (ANSWER) will be](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-78.jpg)
![Difference of Squares: Example (3 g + 4) (3 g - 4) By Marianne Difference of Squares: Example (3 g + 4) (3 g - 4) By Marianne](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-79.jpg)
![Difference of Squares (3 g + 4) (3 g - 4) 2 9 g Difference of Squares (3 g + 4) (3 g - 4) 2 9 g](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-80.jpg)
![(3 x + 5) (3 x - 5) 9 x 2 + 15 x (3 x + 5) (3 x - 5) 9 x 2 + 15 x](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-81.jpg)
![Why can I distribute the power to a monomial, but not a binomial? (4 Why can I distribute the power to a monomial, but not a binomial? (4](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-82.jpg)
![Perfect Squares (3 x – 2 6) (3 x – 6)2 = (3 x Perfect Squares (3 x – 2 6) (3 x – 6)2 = (3 x](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-83.jpg)
![Perfect Squares (3 x + 6 y)2= (3 x +6 y) 9 x 2+18 Perfect Squares (3 x + 6 y)2= (3 x +6 y) 9 x 2+18](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-84.jpg)
![Summary Inspiration: n n What does FOIL stand for? Show the steps on the Summary Inspiration: n n What does FOIL stand for? Show the steps on the](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-85.jpg)
- Slides: 85
![Unit 5 Polynomials Exponents n Classification of Polynomials n Multiplying and Dividing Polynomials n Unit 5 Polynomials Exponents n Classification of Polynomials n Multiplying and Dividing Polynomials n](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-1.jpg)
Unit 5 Polynomials Exponents n Classification of Polynomials n Multiplying and Dividing Polynomials n Combining Polynomials n Special Products n By Marianne Ilmanen
![Multiplying Monomials OBJECTIVE Given five problems the student will be able to multiply the Multiplying Monomials OBJECTIVE: Given five problems, the student will be able to multiply the](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-2.jpg)
Multiplying Monomials OBJECTIVE: Given five problems, the student will be able to multiply the monomials with 80% accuracy. Ca. SS 10. 0: Students add, subtract, multiply and divided monomials.
![Check for Understanding n n n a 4 a 7 a 4 b Check for Understanding n n n a 4 • a 7 a 4 b](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-3.jpg)
Check for Understanding n n n a 4 • a 7 a 4 b 2 c • a 2 b c 3 (7 x 4 y 7) (-3 x 2 y-2 ) (a 4 b c 3 )4 (3 m n 5 p 2)3 By Marianne Ilmanen
![Rules for Multiplying Variables with Exponents n What do you do when Multiplying variables Rules for Multiplying Variables with Exponents n What do you do when Multiplying variables](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-4.jpg)
Rules for Multiplying Variables with Exponents n What do you do when Multiplying variables with exponents? n When multiplying, you multiply like variables within the expression. n When multiplying, you add the exponents. By Marianne Ilmanen
![Rules for Multiplying Variables with Exponents 2 a 3 b ca 2 b 5 Rules for Multiplying Variables with Exponents 2 (a 3 b c)(a 2 b 5](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-5.jpg)
Rules for Multiplying Variables with Exponents 2 (a 3 b c)(a 2 b 5 c) 2 (a 3 b 2+1 =a = 3 a By Marianne Ilmanen c) (a 2 b 3+2 b 1+5 c 5 b 6 c 5 c)
![Examples 3 3 x 5 4 y 3 a 7 2 x b Examples (3 3 x (-5 4 y 3 a )(-7 2 x b )(](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-6.jpg)
Examples (3 3 x (-5 4 y 3 a )(-7 2 x b )( - a 5 y) 2 b) -21 3+2 x -21 5 x 5 3+1 a 5 4 a By Marianne Ilmanen 4+5 y 9 y 1+2 b 3 b
![n What is the n When taking the power of a power you multiply n What is the n When taking the power of a power, you multiply](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-7.jpg)
n What is the n When taking the power of a power, you multiply the exponents. rule for taking a power of a power? (a 2 b 3 c)(a 2 b 3 c) (a 2 b 3 c)4 a 2+2+2+2 b 3+3+3+3 c 1+1+1+1 8 a Shortcut… 12 b 4 c = (a 2 b 3 c)4 a 2 • 4 b 3 • 4 c 1 • 4 By Marianne Ilmanen = 8 a 12 b 4 c
![Examples 3 x 3 y 43 3 4 3 4 3 x y 3 Examples (3 x 3 y 4)3 3 4 3 4 (3 x y )(3](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-8.jpg)
Examples (3 x 3 y 4)3 3 4 3 4 (3 x y )(3 x y ) 1 • 3 3 • 3 4 • 3 3 x y 9 12 27 x y (2 3 b 2 4 c) 24 b 3 • 4 c 2 • 4 16 b 12 c 8 By Marianne Ilmanen
![Find the Following Products 3 g 7 h 3 j 2 2 g 2 Find the Following Products: (3 g 7 h 3 j 2) (2 g 2](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-9.jpg)
Find the Following Products: (3 g 7 h 3 j 2) (2 g 2 h 3 j) m 7(m 3 n 2) (xy)(yz)(xz) 3(ax 3 y)2 By Marianne Ilmanen
![Find the Following Products 3 g 7 h 3 j 2 2 g 2 Find the Following Products: (3 g 7 h 3 j 2) (2 g 2](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-10.jpg)
Find the Following Products: (3 g 7 h 3 j 2) (2 g 2 h 3 j) 6 g 9 h 6 j 3 m 7(m 3 n 2) m 10 n 2 (xy)(yz)(xz) x 2 y 2 z 2 3(ax 3 y)2 3 a 2 x 6 y 2 By Marianne Ilmanen
![Check for Understanding n n n a 4 a 7 n a 4 Check for Understanding n n n a 4 • a 7 n a 4](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-11.jpg)
Check for Understanding n n n a 4 • a 7 n a 4 b 2 c • a 2 b c 3 n (7 x 4 y 7)(-3 x 2 y-2) n (a 4 b c 3)4 n (3 m n 5 p 2)3 n By Marianne Ilmanen a 11 a 6 b 3 c 4 -21 x 6 y 5 a 16 b 4 c 12 27 m 3 n 15 p 6
![Summary Inspiration What is the exponent rule for multiplying monomials n How is the Summary Inspiration: What is the exponent rule for multiplying monomials? n How is the](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-12.jpg)
Summary Inspiration: What is the exponent rule for multiplying monomials? n How is the rule different for coefficients in the monomials? n What is the exponent rule for raising a monomial to another power? n How does this rule affect the coefficients in the monomial? n By Marianne Ilmanen
![Dividing Monomials OBJECTIVE Given four division problems the student will be able to divide Dividing Monomials OBJECTIVE: Given four division problems, the student will be able to divide](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-13.jpg)
Dividing Monomials OBJECTIVE: Given four division problems, the student will be able to divide monomials with 75% accuracy. n Ca. SS 10. 0: Students add, subtract, multiply and divided monomials. n By Marianne Ilmanen
![Check for Understanding x 11 3 x t 4 9 t 3 5 m Check for Understanding x 11 3 x t 4 9 t 3 5 m](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-14.jpg)
Check for Understanding x 11 3 x t 4 9 t 3 5 m n m 2 n 3 6 12 5 a b c a 6 b 2 c 7 By Marianne Ilmanen
![What is the rule for dividing monomials n If I add the exponents when What is the rule for dividing monomials? n If I add the exponents when](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-15.jpg)
What is the rule for dividing monomials? n If I add the exponents when I multiply monomials, then I will SUBTRACT need to _____ the exponents when I divide the monomials. By Marianne Ilmanen
![x 6 x 2 x 6 2 4 d 9 d 3 d 9 x 6 x 2 =x 6– 2 4 d 9 d 3 =d 9–](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-16.jpg)
x 6 x 2 =x 6– 2 4 d 9 d 3 =d 9– 3 6 m 3 m m 3 3– 1 2 =m =m 1 m By Marianne Ilmanen =x =d
![What happens when the exponent is the same in the numerator and denominator 5 What happens when the exponent is the same in the numerator and denominator? 5](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-17.jpg)
What happens when the exponent is the same in the numerator and denominator? 5 x 1 5 x =x 1 5– 5 =x Anything divided by itself equals one !!! Therefore, 0 x =1 Anything to the zero power equals one !!! By Marianne Ilmanen 0
![x 3 y 4 x 3 2 y 4 1 x 1 x 3 y 4 = x 3 -2 y 4 -1 = x 1](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-18.jpg)
x 3 y 4 = x 3 -2 y 4 -1 = x 1 y 3 2 2 x y xy 3 a 7 b 3 = a 7 -2 b 3 -3 = a 5 b 0 2 3 a b a 5 By Marianne Ilmanen
![What happens when the exponent is negative 2 x 5 x x 2 5 What happens when the exponent is negative? 2 x 5 x =x 2– 5](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-19.jpg)
What happens when the exponent is negative? 2 x 5 x =x 2– 5 -3 =x x·x = x·x·x 1 1 1 x·x·x = 1 3 x So, a negative exponent means “put me on the other level…in this case, in the denominator. By Marianne Ilmanen
![d 3 d 7 d 3 7 4 d 1 4 d n 5 d 3 d 7 =d 3– 7 -4 =d 1 4 d n 5](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-20.jpg)
d 3 d 7 =d 3– 7 -4 =d 1 4 d n 5 n 17 =n 5– 17 -12 =n 1 By Marianne Ilmanen 12 n
![m 6 n 2 p 8 11 2 7 m n p m m 6 n 2 p 8 11 2 7 m n p = m](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-21.jpg)
m 6 n 2 p 8 11 2 7 m n p = m 6 -11 n 2 -2 p 8 -7 = -5 0 1 m n p By Marianne Ilmanen p 5 m
![Check for Understanding x 11 x 8 3 x t 4 1 Check for Understanding x 11 = x 8 3 x t 4 = 1](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-22.jpg)
Check for Understanding x 11 = x 8 3 x t 4 = 1 5 9 t t m 3 n 5 = mn 2 2 3 m n a 6 b 12 c 5 b 10 = 6 2 7 2 a b c c By Marianne Ilmanen
![Polynomial Terminology OBJECTIVE Given two polynomials the student will be able to find the Polynomial Terminology OBJECTIVE: Given two polynomials, the student will be able to find the](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-23.jpg)
Polynomial Terminology OBJECTIVE: Given two polynomials, the student will be able to find the degree and arrange the terms in descending order with 75% accuracy. Ca. SS 10. 0: Students add, subtract, multiply and divide polynomials.
![Check for Understanding Find the degree of each monomial in the polynomial term and Check for Understanding: Find the degree of each monomial in the polynomial term and](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-24.jpg)
Check for Understanding: Find the degree of each monomial in the polynomial term and arrange them in descending order with respect to X. abx 3 + bcx + 34 –x 7 7 x 3 + 5 yz - 3 xy 4 + x 4 yz – 13 x 2 z By Marianne Ilmanen
![Types of Polynomials n Monomial n One or more numbers multiplied together n The Types of Polynomials: n Monomial: n One or more numbers multiplied together n The](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-25.jpg)
Types of Polynomials: n Monomial: n One or more numbers multiplied together n The sum of two monomials n The sum of three monomials 2 x 5 y n Binomial: 2 x 5 + 3 y n Trinomial: 2 x 5 + 3 y - 7 By Marianne Ilmanen
![Degree of the Polynomial Degree Monomial n The monomial of the n Tally up Degree of the Polynomial Degree: Monomial n The monomial of the n Tally up](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-26.jpg)
Degree of the Polynomial Degree: Monomial n The monomial of the n Tally up the highest degree is the exponents: Winner: n EXAMPLES: n EXAMPLE: 8 y 3 degree of 3 8 y 3 + 2 x - xy 4 + xy – 7 42 qrs degree of 3 4 y 4 ab degree of 6 3 rd 1 st 5 th 2 nd 0 th Degree Polynomial 5 -14 degree of 0 By Marianne Ilmanen
![Find the Degree of the Polynomial 3 x 5 First Degree 3 a Find the Degree of the Polynomial 3 x – 5 First Degree -3 a](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-27.jpg)
Find the Degree of the Polynomial 3 x – 5 First Degree -3 a 2 + 7 a – 19 Second Degree 14 x + x 2 y – 37 x 4 Fourth Degree – 7 a 2 b 2 + 5 a 2 b 3 - 4 ab 3 Fifth Degree By Marianne Ilmanen
![Writing Polynomials in Descending Order n Write the terms so that the powers of Writing Polynomials in Descending Order n Write the terms so that the powers of](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-28.jpg)
Writing Polynomials in Descending Order n Write the terms so that the powers of y are in order from largest to smallest. 8 y 3 + 2 x - xy 4 + xy 2 – 7 y By Marianne Ilmanen
![Writing Polynomials in Descending Order n Write the terms so that the powers of Writing Polynomials in Descending Order n Write the terms so that the powers of](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-29.jpg)
Writing Polynomials in Descending Order n Write the terms so that the powers of y are in order from largest to smallest. 8 y 3 + 2 x - xy 4 + xy 2 – 7 y 3 rd 0 4 th 2 nd 1 st - xy 4 + 8 y 3 + xy 2 – 7 y + 2 x By Marianne Ilmanen
![Writing Polynomials in Descending Order n Write the terms so that the powers of Writing Polynomials in Descending Order n Write the terms so that the powers of](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-30.jpg)
Writing Polynomials in Descending Order n Write the terms so that the powers of a are in order from largest to smallest. 10 – a 2 b 3 + 7 ab – a 5 b 2 - 9 a 4 + b 3 0 2 nd 1 st 5 th 4 th 0 – a 5 b 2 - 9 a 4 – a 2 b 3 + 7 ab + b 3 + 10 By Marianne Ilmanen
![Check for Understanding Find the degree of the polynomial Then arrange them in descending Check for Understanding: Find the degree of the polynomial. Then arrange them in descending](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-31.jpg)
Check for Understanding: Find the degree of the polynomial. Then arrange them in descending order with respect to X. abx 3 + bcx + 34 –x 7 Seventh Degree -x 7 + abx 3 + bcx + 34 7 x 3 + 5 yz - 3 xy 4 + x 4 yz – 13 x 2 z x 4 yz + 7 x 3 – 13 x 2 z - 3 xy 4 + 5 yz Sixth Degree By Marianne Ilmanen
![Check for Understanding Find the degree of the polynomial Then arrange them in descending Check for Understanding: Find the degree of the polynomial. Then arrange them in descending](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-32.jpg)
Check for Understanding: Find the degree of the polynomial. Then arrange them in descending order with respect to X. 3 x 5 + 15 yz - 3 xy 4 + 7 x 4 yz 2 – 13 x 2 z 7 x 4 yz 2 + 3 x 5 – 13 x 2 z - 3 xy 4 + 15 yz Seventh Degree By Marianne Ilmanen
![Adding and Subtracting Polynomials OBJECTIVE Given four polynomial problems the student will be able Adding and Subtracting Polynomials OBJECTIVE: Given four polynomial problems the student will be able](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-33.jpg)
Adding and Subtracting Polynomials OBJECTIVE: Given four polynomial problems the student will be able to find the sum or difference with 75% accuracy.
![Check for Understanding 1 4 x 2 5 x 7 x 2 Check for Understanding 1. (4 x 2 + 5 x) + (-7 x 2](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-34.jpg)
Check for Understanding 1. (4 x 2 + 5 x) + (-7 x 2 + x) 2. (6 x 3 + 3 x 2 - 7) + (x 3 - 6 x 2 – 2 x) 3. (3 x 2 + 5 x - 6) - (7 x 2 - 9) 4. (5 b – 7 ab + 8 a) - (5 ab – 4 a) By Marianne Ilmanen
![What are the steps to combining polynomials 1 Stack the two polynomials and align What are the steps to combining polynomials? 1. “Stack” the two polynomials and align](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-35.jpg)
What are the steps to combining polynomials? 1. “Stack” the two polynomials and align the like terms. 2. Combine straight down. Remember: each term has variables that act as a “label” when you add, so the same “label” is carried into your answer. By Marianne Ilmanen
![Find the sum of 3 2 a a 2 and 5 + Find the sum of (3 + 2 a + a 2) and (5](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-36.jpg)
+ Find the sum of (3 + 2 a + a 2) and (5 - 8 a + a 2) 1. “Stack” the like terms. 2. Combine straight down. 3 + 2 a + a 2 5 - 8 a + a 2 8 – 6 a + 2 a 2 By Marianne Ilmanen
![Lets Try a Few 3 x 4 2 x 3 x 2 Let’s Try a Few: (3 x 4 + 2 x 3 + x 2)](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-37.jpg)
Let’s Try a Few: (3 x 4 + 2 x 3 + x 2) + (-5 x 4 + 8 x 3 + x 2) (5 x 2 – 4) + (3 x 2 + 8 x + 4) By Marianne Ilmanen
![3 x 4 2 x 3 x 25 x 4 8 (3 x 4 + 2 x 3 + x 2)+(-5 x 4 + 8](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-38.jpg)
(3 x 4 + 2 x 3 + x 2)+(-5 x 4 + 8 x 3 + x 2) 3 x 4 + 2 x 3 + x 2 (+) -5 x 4 + 8 x 3 + x 2 -2 x 4 +10 x 3 +2 x 2 (5 x 2 -4)+(3 x 2 + 8 x + 4) 5 x 2 + 0 x - 4 (+) 3 x 2 + 8 x + 4 8 x 2 + 8 x + 0 8 x 2 + 8 x By Marianne Ilmanen
![How do you find 1 Stack the two the difference of equations and align How do you find 1. Stack the two the difference of equations and align](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-39.jpg)
How do you find 1. Stack the two the difference of equations and align two polynomials? the like terms. 2. 3. Find the difference: (x 3 + 4 x 2 - 7 x - 2) - (2 x 2 - 9 x + 4) Distribute the negative through the bottom polynomial. Combine straight down. x 3 + 4 x 2 - 7 x - 2 (-) 0 x 3 + 2 x 2 - 9 x + 4 By Marianne Ilmanen
![x 3 4 x 2 7 x 2 2 x (x 3 + 4 x 2 - 7 x - 2) - (2 x](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-40.jpg)
(x 3 + 4 x 2 - 7 x - 2) - (2 x 2 - 9 x + 4) Stack. Distribute the negative through the bottom polynomial. -1(2 x 2 - 9 x + 4) = -2 x 2 + 9 x – 4 Combine straight down. By Marianne Ilmanen x 3 + 4 x 2 - 7 x - 2 0 x 3 - 2 x 2 + 9 x - 4 x 3 + 2 x 2 + 2 x - 6
![4 x 3 8 x 2 x 21 3 x (4 x 3 + 8 x 2 - x + 21) - (3 x](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-41.jpg)
(4 x 3 + 8 x 2 - x + 21) - (3 x 2 - 10 x) Stack. -1(3 x 2 - 10 x) = -3 x 2 + 10 x Distribute. Combine straight down. 4 x 3 + 8 x 2 - x + 21 - 3 x 2 + 10 x 4 x 3 + 5 x 2 + 9 x +21 By Marianne Ilmanen
![3 x 2 y xy 5 2 x 2 y (3 x 2 y + xy – 5) - (2 x 2 y –](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-42.jpg)
(3 x 2 y + xy – 5) - (2 x 2 y – 10 xy + 3) Stack. -1(2 x 2 y – 10 xy + 3) Distribute. = -2 x 2 y + 10 xy – 3 Combine straight down. By Marianne Ilmanen 3 x 2 y + xy – 5 -2 x 2 y + 10 xy – 3 1 x 2 y + 11 xy – 8
![Check for Understanding 1 4 x 2 5 x 7 x 2 Check for Understanding 1. (4 x 2 + 5 x) + (-7 x 2](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-43.jpg)
Check for Understanding 1. (4 x 2 + 5 x) + (-7 x 2 -3 x 2 + 6 x + x) 2. (6 x 3 + 3 x 2 – 7 x) + (x 3 - 6 x 2 – 2 x) 7 x 3 - 3 x 2 – 9 x 3. (3 x 2 + 5 x - 6) - (7 x 2 - 9) - 4 x 2 + 5 x + 3 4. (5 b – 7 ab + 8 a) - (5 ab – 4 a) By Marianne Ilmanen 5 b – 12 ab + 12 a
![Summary Inspiration n n What do you do when adding polynomials What do you Summary Inspiration: n n What do you do when adding polynomials? What do you](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-44.jpg)
Summary Inspiration: n n What do you do when adding polynomials? What do you do when subtracting polynomials? What is the rule for combining things with exponents…how is the exponent affected or not affected? What do you do if you have two polynomials to add, but there is a monomial within one of the polynomials that does not have a like term in the other polynomial? By Marianne Ilmanen
![Multiplying a Polynomial by a Monomial OBJECTIVE Given three problems the student will be Multiplying a Polynomial by a Monomial OBJECTIVE: Given three problems, the student will be](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-45.jpg)
Multiplying a Polynomial by a Monomial OBJECTIVE: Given three problems, the student will be able to find the product with 80% accuracy. Ca. SS 10. 0: Students add, subtract, multiply and divided polynomials.
![Check for Understanding 1 4 a 2 8 a 3 c c 11 Check for Understanding 1. 4 a 2 (-8 a 3 c + c -11)](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-46.jpg)
Check for Understanding 1. 4 a 2 (-8 a 3 c + c -11) 2. -3 x (-8 xy - 5 y +4) + 2 (x 2 y -2 xy + x) 3. 5 w 2 (-8 w 3 + 2 w) + 2 w (w 4 – 5 w 2 -2) By Marianne Ilmanen
![Use Distributive Property What property do I use when multiplying a polynomial by a Use Distributive Property What property do I use when multiplying a polynomial by a](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-47.jpg)
Use Distributive Property What property do I use when multiplying a polynomial by a monomial? n Distribute the monomial on the outside of the parentheses to each term of the polynomial inside the parentheses. Remember to add the exponents of the variables when multiplying!!!! By Marianne Ilmanen
![Use Distributive Property n Distribute the monomial on the outside of the parentheses to Use Distributive Property n Distribute the monomial on the outside of the parentheses to](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-48.jpg)
Use Distributive Property n Distribute the monomial on the outside of the parentheses to each term of the polynomial inside the parentheses. 3 d (4 d 2 -8 d -15) 3 d(4 d 2) + 3 d(-8 d) + 3 d(-15) 12 d 3 -24 d 2 -45 d By Marianne Ilmanen
![2 x 2 5 x 2 7 x 8 2 x 25 x 2 2 x 2 (5 x 2 -7 x +8) 2 x 2(5 x 2)](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-49.jpg)
2 x 2 (5 x 2 -7 x +8) 2 x 2(5 x 2) + 2 x 2(-7 x) +2 x 2(8) 10 x 4 -14 x 3 + 16 x 2 3 gh (5 g 2 -8 gh +2 h) 3 gh(5 g 2) +3 gh(-8 gh) +3 gh(2 h) 15 g 3 h-24 g 2 h 2 +6 gh 2 By Marianne Ilmanen
![Step It Up a Notch Simplify 3 t2 t 4 65 t 2 Step It Up a Notch: Simplify 3 t(2 t -4) + 6(5 t 2](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-50.jpg)
Step It Up a Notch: Simplify 3 t(2 t -4) + 6(5 t 2 +2 t -7) By Marianne Ilmanen
![Step It Up a Notch Simplify Get rid of the parentheses 3 t2 t Step It Up a Notch: Simplify Get rid of the parentheses 3 t(2 t](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-51.jpg)
Step It Up a Notch: Simplify Get rid of the parentheses 3 t(2 t -4) + 6(5 t 2 +2 t -7) by distributing. 6 t 2 -12 t + 30 t 2 +12 t -42 Combine like terms 36 t 2 + 0 t -42 36 t 2 - 42 By Marianne Ilmanen
![3y 3 8 y 4 y2 y 3 8 y 2 2 y 3(y 3 +8 y) + 4 y(2 y 3 -8 y 2 +2 y](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-52.jpg)
3(y 3 +8 y) + 4 y(2 y 3 -8 y 2 +2 y +9) By Marianne Ilmanen
![Get rid of the parentheses by distributing 3y 3 8 y 4 y2 Get rid of the parentheses by distributing. 3(y 3 +8 y) + 4 y(2](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-53.jpg)
Get rid of the parentheses by distributing. 3(y 3 +8 y) + 4 y(2 y 3 -8 y 2 +2 y +9) 3 y 3 +24 y + 8 y 4 -32 y 3 +8 y 2 +36 y By Marianne Ilmanen
![Get rid of the parentheses by distributing 3y 3 8 y 4 y2 Get rid of the parentheses by distributing. 3(y 3 +8 y) + 4 y(2](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-54.jpg)
Get rid of the parentheses by distributing. 3(y 3 +8 y) + 4 y(2 y 3 -8 y 2 +2 y +9) 3 y 3 +24 y + 8 y 4 -32 y 3 +8 y 2 +36 y 8 y 4 -29 y 3 +8 y 2 + 60 y By Marianne Ilmanen Combine like terms
![Check for Understanding 1 4 a 2 8 a 3 c c 11 Check for Understanding 1. 4 a 2 (-8 a 3 c + c -11)](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-55.jpg)
Check for Understanding 1. 4 a 2 (-8 a 3 c + c -11) -32 a 5 c + 4 a 2 c -44 a 2 2. -3 x (-8 xy - 5 y +4) + 2 (x 2 y -2 xy + x) 24 x 2 y +15 xy -12 x + 2 x 2 y -4 xy + 2 x 26 x 2 y +11 xy -10 x 3. 5 w 2 (-8 w 3 + 2 w) + 2 w (w 4 – 5 w 2 -2) -40 w 5 + 10 w 3 + 2 w 5 – 10 w 3 - 4 w -38 w 5 - 4 w By Marianne Ilmanen
![Summary Inspiration What property is used when multiplying a polynomial by a monomial n Summary Inspiration: What property is used when multiplying a polynomial by a monomial? n](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-56.jpg)
Summary Inspiration: What property is used when multiplying a polynomial by a monomial? n What do you do when multiplying the coefficients? n What do you do when multiplying the variables? n What is the last step if you have any like terms? n By Marianne Ilmanen
![Multiplying Polynomials OBJECTIVE Given 4 multiplication problems the student will be able to find Multiplying Polynomials OBJECTIVE: Given 4 multiplication problems, the student will be able to find](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-57.jpg)
Multiplying Polynomials OBJECTIVE: Given 4 multiplication problems, the student will be able to find the product with 75% accuracy. Ca. SS 10. 0: Students add, subtract, multiply and divided polynomials.
![Check for Understanding 1 y 7 y 3 2 2 m Check for Understanding 1. (y – 7) (y + 3) 2. (2 m +](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-58.jpg)
Check for Understanding 1. (y – 7) (y + 3) 2. (2 m + 5) (3 m - 8) 3. (3 p – 7) (3 p + 7) 4. (0. 5 y – 7) (0. 4 y + 3) By Marianne Ilmanen
![What is the result of multiplying two binomials n When multiplying two binomials the What is the result of multiplying two binomials? n When multiplying two binomials, the](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-59.jpg)
What is the result of multiplying two binomials? n When multiplying two binomials, the result is a binomial, four termed polynomial or (most commonly) a trinomial. (x – 7) (x + 3) x 2 – 4 x - 21 By Marianne Ilmanen
![Multiplying Two Binomials What is the mnemonic device to remember how to multiply two Multiplying Two Binomials What is the mnemonic device to remember how to multiply two](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-60.jpg)
Multiplying Two Binomials What is the mnemonic device to remember how to multiply two binomials? FOIL First Outside Inside Last (d + 2) (d + 8) By Marianne Ilmanen
![Multiplying Two Binomials FOIL First Outside Inside Last d 2 d 8 Multiplying Two Binomials: FOIL First Outside Inside Last (d + 2) (d + 8)](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-61.jpg)
Multiplying Two Binomials: FOIL First Outside Inside Last (d + 2) (d + 8) 2 d By Marianne Ilmanen
![FOIL First Outside Inside Last d 2 d 8 2 d 8 FOIL First Outside Inside Last (d + 2) (d + 8) 2 d +8](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-62.jpg)
FOIL First Outside Inside Last (d + 2) (d + 8) 2 d +8 d By Marianne Ilmanen
![FOIL First Outside Inside Last d 2 d 8 2 d 8 FOIL First Outside Inside Last (d + 2) (d + 8) 2 d +8](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-63.jpg)
FOIL First Outside Inside Last (d + 2) (d + 8) 2 d +8 d +2 d By Marianne Ilmanen
![FOIL First Outside Inside Last d 2 d 8 2 d 8 FOIL First Outside Inside Last (d + 2) (d + 8) 2 d +8](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-64.jpg)
FOIL First Outside Inside Last (d + 2) (d + 8) 2 d +8 d +2 d +16 By Marianne Ilmanen
![Combine the Middle Terms 2 d 8 d 2 d 16 2 d 10 Combine the Middle Terms 2 d +8 d +2 d +16 2 d +10](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-65.jpg)
Combine the Middle Terms 2 d +8 d +2 d +16 2 d +10 d +16 By Marianne Ilmanen
![Multiply 2 m 4 m 5 FOIL First Outside Inside Last 2 Multiply: (2 m – 4) (m + 5) FOIL First Outside Inside Last (2](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-66.jpg)
Multiply: (2 m – 4) (m + 5) FOIL First Outside Inside Last (2 m – 4) (m + 5) By Marianne Ilmanen
![Multiplying Two Binomials FOIL First Outside Inside Last 2 m 4 m Multiplying Two Binomials: FOIL First Outside Inside Last (2 m – 4) (m +](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-67.jpg)
Multiplying Two Binomials: FOIL First Outside Inside Last (2 m – 4) (m + 5) 2 2 m By Marianne Ilmanen
![FOIL First Outside Inside Last 2 m 4 m 5 2 2 FOIL First Outside Inside Last (2 m – 4) (m + 5) 2 2](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-68.jpg)
FOIL First Outside Inside Last (2 m – 4) (m + 5) 2 2 m +10 m By Marianne Ilmanen
![FOIL First Outside Inside Last 2 m 4 m 5 2 2 FOIL First Outside Inside Last (2 m – 4) (m + 5) 2 2](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-69.jpg)
FOIL First Outside Inside Last (2 m – 4) (m + 5) 2 2 m +10 m- 4 m By Marianne Ilmanen
![FOIL First Outside Inside Last 2 m 4 m 5 2 2 FOIL First Outside Inside Last (2 m – 4) (m + 5) 2 2](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-70.jpg)
FOIL First Outside Inside Last (2 m – 4) (m + 5) 2 2 m +10 m- 4 m - 20 By Marianne Ilmanen
![Combine the Middle Terms 2 2 m 10 m4 m 20 2 2 m Combine the Middle Terms 2 2 m +10 m-4 m -20 2 2 m](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-71.jpg)
Combine the Middle Terms 2 2 m +10 m-4 m -20 2 2 m +6 m -20 By Marianne Ilmanen
![Multiply 2 x 2 5 x 8 FOIL First Outside Inside Last Multiply: (2 x + 2) (5 x - 8) FOIL First Outside Inside Last](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-72.jpg)
Multiply: (2 x + 2) (5 x - 8) FOIL First Outside Inside Last (2 x + 2) (5 x - 8) By Marianne Ilmanen
![FOIL First Outside Inside Last 2 x 2 5 x 8 First FOIL First Outside Inside Last (2 x + 2) (5 x - 8) First](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-73.jpg)
FOIL First Outside Inside Last (2 x + 2) (5 x - 8) First Outside Inside Last 2 10 x -16 x +10 x -16 10 x 2 -6 x -16 By Marianne Ilmanen
![2 a 3 b 5 a 2 b First Outside Inside Last 2 2 (2 a +3 b) (5 a -2 b) First Outside Inside Last 2 2](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-74.jpg)
(2 a +3 b) (5 a -2 b) First Outside Inside Last 2 2 10 a -4 ab +15 ab -6 b 2 10 a +11 ab By Marianne Ilmanen 2 -6 b
![Check for Understanding 1 y 7 y 3 y 2 4 Check for Understanding 1. (y – 7) (y + 3) y 2 – 4](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-75.jpg)
Check for Understanding 1. (y – 7) (y + 3) y 2 – 4 y - 21 2 – m - 40 6 m 2. (2 m + 5) (3 m - 8) 3. (3 p – 7) (3 p + 7) 9 p 2 - 49 4. (5 y – 7 z) (4 y + 3 z) 20 y 2 – 13 yz – 21 z 2 By Marianne Ilmanen
![Special Products OBJECTIVE Same as 9 7 Special Products OBJECTIVE: Same as 9 -7.](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-76.jpg)
Special Products OBJECTIVE: Same as 9 -7.
![What is a difference of squares and how is it different than the other What is a difference of squares and how is it different than the other](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-77.jpg)
What is a difference of squares and how is it different than the other trinomial products we have talked about? n A difference (SUBTRACTION) of (PERFECT) squares is exactly that…the result (ANSWER) will be two perfect squares, one subtracted from the other. By Marianne Ilmanen
![n A difference SUBTRACTION of PERFECT squares is exactly thatthe result ANSWER will be n A difference (SUBTRACTION) of (PERFECT) squares is exactly that…the result (ANSWER) will be](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-78.jpg)
n A difference (SUBTRACTION) of (PERFECT) squares is exactly that…the result (ANSWER) will be two perfect squares, one subtracted from the other. (g + 8) (g - 8) FOIL it out… g 2 +0 g – 64 2 g – 64 It is different because the “OI” terms are additive inverses so they “zero out. ” There is no first degree term in a difference of squares. By Marianne Ilmanen
![Difference of Squares Example 3 g 4 3 g 4 By Marianne Difference of Squares: Example (3 g + 4) (3 g - 4) By Marianne](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-79.jpg)
Difference of Squares: Example (3 g + 4) (3 g - 4) By Marianne Ilmanen
![Difference of Squares 3 g 4 3 g 4 2 9 g Difference of Squares (3 g + 4) (3 g - 4) 2 9 g](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-80.jpg)
Difference of Squares (3 g + 4) (3 g - 4) 2 9 g +12 g – 12 g -16 0 9 g 2 – 16 By Marianne Ilmanen FL
![3 x 5 3 x 5 9 x 2 15 x (3 x + 5) (3 x - 5) 9 x 2 + 15 x](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-81.jpg)
(3 x + 5) (3 x - 5) 9 x 2 + 15 x – 25 9 x 2 - 25 (9 – z 9) (9 + z 9) 81 - 9 z 9 + 9 z 9 – z 18 81 – z 18 (5 b – 12 a) (5 b + 12 a) 25 b 2 - 60 ab + 60 ab – 144 a 2 25 b 2 – 144 a 2 (½x + ½y) (½x - ½y) ¼ x 2 + ¼xy – ¼ y 2 By Marianne Ilmanen ¼ x 2 – ¼ y 2
![Why can I distribute the power to a monomial but not a binomial 4 Why can I distribute the power to a monomial, but not a binomial? (4](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-82.jpg)
Why can I distribute the power to a monomial, but not a binomial? (4 xy)2 (4 xy) (42 x 2 y 2) (16 x 2 y 2) No FOIL (4 x + y)2 (4 x + y) 16 x 2+4 xy+y 2 16 x 2+8 xy+y 2 By Marianne Ilmanen FOIL
![Perfect Squares 3 x 2 6 3 x 62 3 x Perfect Squares (3 x – 2 6) (3 x – 6)2 = (3 x](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-83.jpg)
Perfect Squares (3 x – 2 6) (3 x – 6)2 = (3 x – 6) 9 x 2 – 18 x +36 -36 x 9 x 2 – 36 x + 36 (2 x + 5 y)2=(2 x + 5 y) 4 x 2 +10 x +25 y 2 By Marianne Ilmanen -36 x 9 x 2 – 36 x + 36
![Perfect Squares 3 x 6 y2 3 x 6 y 9 x 218 Perfect Squares (3 x + 6 y)2= (3 x +6 y) 9 x 2+18](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-84.jpg)
Perfect Squares (3 x + 6 y)2= (3 x +6 y) 9 x 2+18 xy+36 +36 xy 9 x 2 +36 xy + 36 (5 a – 12 b)2 (5 a– 12 b)2= (5 a – 12 b) 25 a 2 -60 ab– 60 ab +144 b 2 -120 ab 25 a 2 – 120 ab + 144 b 2 By Marianne Ilmanen
![Summary Inspiration n n What does FOIL stand for Show the steps on the Summary Inspiration: n n What does FOIL stand for? Show the steps on the](https://slidetodoc.com/presentation_image_h/8a63a026ffa62db611765ba64527d033/image-85.jpg)
Summary Inspiration: n n What does FOIL stand for? Show the steps on the binomial product below. (a + b) (c + d) Which terms do you usually combine after FOIL-ing? What is special about Differences of Squares? What is special about Perfect Squares? By Marianne Ilmanen
Lesson 6-2 practice a rational exponents answers
Exponents in fractions
Adding and subtracting polynomial fractions worksheet
Exponents and polynomials
Rules for dividing polynomials
Real world applications of polynomials
Radical functions and rational exponents unit test
Unit 6 radical functions homework 6 radical equations
Unit 6 radical functions homework 4 rational exponents
Unit 6 review questions
Classifying polynomials degree
Classify each polynomial according to its degree and type.
Algebra 2 unit 5 polynomial functions
5.2 polynomials linear factors and zeros
Unit 6 polynomials and polynomial functions
Zero negative exponents
Write the exponent of 3 in the prime factorization of 162
Exponents jeopardy
Exponential forn
Solve exponential equations with different bases
Fermat's little theorem example
Rewrite in radical form calculator
Rational exponents
Solving equations with rational exponents
Rational exponents
Exponents algebra 2
Multiplication property of exponents
Laws of exponents division
Evaluate 32 3/5
Properties of integer exponents
Numerical expression example
Greatest common factor with exponents
53 to the power of 3
Vocabulary for exponents
Derivative of logarithmic function
Zero and negative exponents worksheet
Integer exponents
How to undo exponents
How to simplify fraction exponents
I love exponents
4 divided by 3
7 properties of exponents
How to solve exponential equations
Multiplying exponents
7-3 reteaching more multiplication properties of exponents
7-1 zero and negative exponents answer key
7-1 integer exponents
Integer exponents
A number to the negative power
Equating exponents
Do you multiply exponents
6-1 integer exponents
6-1 integer exponents
Evaluating expressions with exponents quiz
Rational exponents
5-6 radical expressions and rational exponents
Property of rational exponents
Quotient of powers property
Exponents warm up
Rational exponent theorem
Exponents warm up
Simplifying expressions using properties of exponents
Properties of exponents algebra 2
Properties of exponents
7-3 multiplication properties of exponents
7-3 practice more multiplication properties of exponents
More multiplication properties of exponents practice
Evaluate 32 3/5
Exponents jeopardy
6³ meaning
6-2 rational exponents
Complex numbers and rational exponents
Adding and subtracting numbers in scientific notation
7-1 practice multiplication properties of exponents page 8
5-6 radical expressions and rational exponents
How to simplify integer exponents
Negative exponenets
Dividing negative exponents
7-3 more multiplication properties of exponents
Language
Use the distributive property to rewrite the expression
Negative exponents chart
Evaluating nth roots
Properties of integral exponents
Section p.2 exponents and scientific notation
Multiplying decimals with exponents