Polynomials and Factoring Adding and Subtracting Polynomials Objective
Polynomials and Factoring
Adding and Subtracting Polynomials Objective: To classify, add, and subtract polynomials.
Objectives • I can find the degree of a monomial. • I can adding and subtract monomials. • I can classify polynomials. • I can add polynomials. • I can subtract polynomials.
Vocabulary • A monomial is a real number, a variable, or a product of a real number and one or more variable with whole-number exponents. • The degree of a monomial is the sum of the exponents of its variables. • The degree of a nonzero constant is 0. Zero has no degree.
Finding the Degree of a Monomial What is the degree of each monomial? 1. 5 x 2. 6 x 3 y 2 3. 4 4. 8 xy 5. – 7 y 4 z 6. 11 7. 2 b 8 c 2 8. – 3
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Adding and Subtracting Monomials What is the sum or difference? 1. 3 x 2 + 5 x 2 2. 4 x 3 y – x 3 y 3. – 6 x 4 + 11 x 4 4. 2 x 2 y 4 – 7 x 2 y 4 5. 3 t 4 + 11 t 4 6. 7 x 2 – 2 x 2 7. 2 m 3 n 3 + 9 m 3 n 3 8. 5 bc 4 – 13 bc 4
Practice •
Vocabulary • A polynomial is a monomial or a sum of monomials. • Standard form of a polynomial means that the degrees of its monomial terms decrease from left to right. • The degree of a polynomial in one variable is the same as the degree of the monomial with the greatest exponent. • A binomial has two terms that are either added or subtracted. • A trinomial has three terms that are either added or subtracted.
Vocabulary
Classifying Polynomials Write each polynomial in standard form. What is the name of the polynomial based on its degree and number of terms? 1. 2. 3. 4. 5. 6. 7. 8. 3 x + 4 x 2 4 x – 1 + 5 x 3 + 7 x 2 x – 3 + 8 x 2 6 x 2 – 13 x 2 – 4 x + 4 5 y – 2 y 2 3 z 4 – 5 z – 2 z 2 c + 8 c 3 – 3 c 7 x 2 + 4 – 3 x
Practice •
Adding Polynomials A researcher studied the number of overnight stays in U. S. National Park Service campgrounds and in the backcountry of the national park system over a 5 year period. Campgrounds: – 7. 1 x 2 – 180 x + 5800 Backcountry: 21 x 2 – 140 x + 1900 In each polynomial, x = 0 corresponds to the first year in the 5 year period. What polynomial models the total number of overnight stays in both campgrounds and backcountry?
Adding Polynomials A nutritionist studied the U. S. consumption of carrots and celery and of broccoli over a 6 year period. The nutritionist modeled the results, in millions of pounds, with the following polynomials. Carrots & Celery: – 12 x 3 + 106 x 2 – 241 x + 4477 Broccoli: 14 x 2 – 14 x + 1545 In each polynomial, x = 0 corresponds to the first year of the 6 year period. What polynomial models the total number of pounds, in millions, of carrots and celery and broccoli consumed in the U. S. over the 6 year period?
Adding Polynomials Simplify each. 1. (5 r 3 + 8) + (6 r 3 + 3) 2. (6 x 2 + 7) + (3 x 2 + 1) 3. (3 z 3 – 4 z + 7 z 2) + (8 z 2 – 6 z – 5) 4. (2 k 2 – k + 3) + (5 k 2 + 3 k – 7)
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Subtracting Polynomials Simplify each. 1. (x 3 – 3 x 2 + 5 x) – (7 x 3 + 5 x 2 – 12) 2. (– 4 m 3 – m + 9) – (– 4 m 2 + m – 12) 3. (x 2 – 2) – (3 x + 5) 4. (14 h 4 + 3 h 3) – (9 h 4 +2 h 3) 5. (– 9 r 3 + 2 r – 1) – (– 5 r 2 + r + 8) 6. (y 3 – 4 y 2 – 2) – (6 y 3 + 4 – 6 y 2)
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Questions • Is it possible to write a trinomial with a degree of 0? • Is the sum of two trinomials always a trinomial?
Exit Ticket •
Multiplying and Factoring Objective: To multiply a monomial by a polynomial. To factor a monomial from a polynomial.
Objective • I can multiply a monomial and a trinomial. • I can find the greatest common factor. • I can factor out a monomial. • I can factor a polynomial model.
Vocabulary • You can use distributive property to multiply a monomial by a polynomial.
Multiplying a Monomial and a Trinomial Simplify: 1. –x 3(9 x 4 – 2 x 3 + 7) 2. 5 n(3 n 3 – n 2 + 8) 3. 4 x(2 x 3 – 7 x 2 + x) 4. – 8 y 3(7 y 2 – 4 y – 1) 5. 3 m 2(10 + m – 4 m 2) 6. –w 2(5 w 4 + 7 w 2 – 15)
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Vocabulary Factoring a polynomial reverses the multiplication process. When factoring a monomial from a polynomial, the first step is to fine the greatest common factor (GCF) of the polynomial’s terms. Once you find the GCF of a polynomial’s terms, you can factor it out of the polynomial.
Finding the Greatest Common Factor Find the GCF of each. 1. 12 x + 20 2. 8 w 2 – 16 w 3. 45 b + 27 4. 5 x 3 + 25 x 2 + 45 x 5. 3 x 4 – 9 x 2 – 12 x 6. 4 x 3 + 12 x – 28 7. a 3 + 6 a 2 – 11 a 8. 14 z 4 – 42 z 3 + 21 z 2
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Factoring Out a Monomial What is the factored form of each? 1. t 2 + 8 t 2. 9 x – 6 3. 4 x 5 – 24 x 3 + 8 x 4. 9 x 6 + 15 x 4 + 12 x 2 5. – 6 x 4 – 18 x 3 – 12 x 2 6. 14 n 3 – 35 n 2 + 28 7. 5 k 3 + 20 k 2 – 15 8. g 4 + 24 g 3 + 12 g 2 + 4 g
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Factoring a Polynomial Model • A helicopter landing pad, or helipad, is sometimes marked with a circle inside a square so that it is visible from the air. What is the area of the shaded region of the helipad? Write your answer in factored 2 x form. • Suppose the side length of the square is 6 x the radius of the circle is 3 x. What is the factored form of the area of the shaded region? x and
Practice • A circular mirror is surrounded by a square metal frame. The radius of the mirror is 5 x. The side length of the metal frame is 15 x. What is the area of the metal frame? Write your answer in factored form. • A circular table is painted yellow with a red square in the middle. The radius of the table is 6 x. The side length of the red square is 3 x. What is the area of the yellow part of the tabletop? Write your answer in factored form.
Multiplying Binomials Objective: To multiply two binomials or a binomial by a trinomial.
Side Notes • There are three different methods that I will be showing you on multiplying binomials. • The three methods are: 1. FOIL, 2. Distributive, 3. Box. • I want you to figure out which is easier for you and you work the problems that way. I do not care which method you use.
Objectives • I can use the distributive property. • I can use a table. • I can use the FOIL method. • I can apply multiplication of binomials. • I can multiply a trinomial and a binomial.
Using the Distributive Property What is the simplified form? 1. 2. 3. 4. 5. 6. (2 x + 4)(3 x – 7) (x – 6)(4 x + 3) (x – 3)(4 x – 5) (3 x + 1)(x + 4) (2 x + 2)(x + 3)(x + 6)
Practice Simplify each product using the Distributive Property. 1. 2. 3. 4. 5. 6. (x + 7)(x + 4) (y – 3)(y + 8) (m + 6)(m – 7) (c – 10)(c – 5) (2 r – 3)(r + 1) (2 x + 7)(3 x – 4)
Vocabulary When you use the Distributive Property to multiply binomials, notice that you multiply each term of the first binomial by each term of the second binomial. A table can help you organize your work.
Using a Table What is the simplified form? 1. (5 x – 3)(2 x + 1) 2. (3 x – 4)(x + 2) 3. (n – 6)(4 n – 7) 4. (2 p 2 + 3)(2 p – 5) 5. (x + 1)(x + 4) 6. (x + 2)(x + 4)
Practice Simplify each product using a table. 1. 2. 3. 4. 5. 6. (x + 5)(x – 4) (a – 1)(a – 11) (w – 2)(w + 6) (2 h – 7)(h + 9) (x – 8)(3 x + 1) (3 p + 4)(2 p + 5)
Vocabulary FOIL stands for First – multiply the first term in both binomials Outer – multiply the outer most terms in both binomials Inner – multiply the inner most terms in both binomials Last – multiply the last term in both binomials
Using FOIL What is the simplified form? 1. (x + 7)(x + 4) 2. (y – 3)(y + 8) 3. (m + 6)(m – 7) 4. (c – 10)(c – 5) 5. (2 r – 3)(r + 1) 6. (2 x + 7)(3 x – 4)
Practice Simplify each product using the FOIL method. 1. 2. 3. 4. 5. 6. 7. 8. 9. (a + 8)(a – 2) (x + 4)(4 x – 5) (k – 6)(k + 8) (b – 3)(b – 9) (5 m – 2)(m + 3) (9 z + 4)(5 z – 3) (3 h + 2)(6 h – 5) (4 w+13)(w+2) (8 c – 1)(6 c – 7)
Applying Multiplication of Binomials A cylinder has the dimensions shown in the diagram. What is the polynomial in standard form that describes the total surface area of the cylinder? X+1 X+4
Applying Multiplication of Binomials What is the total surface area of a cylinder with radius x + 2 and height x + 4? Write your answer as a polynomial in standard form.
Practice 1. What is the total surface area of the cylinder? Write your answer as a polynomial in standard form. X+2 X+5 2. The radius of a cylindrical gift box is (2 x + 3) inches. The height of the gift box is twice the radius. What is the surface area of the cylinder? Write your answer in standard form.
Multiplying a Trinomial and a Binomial What is the simplified form? 1. (3 x 2 + x – 5)(2 x – 7) 2. (2 x 2 – 3 x + 1)(x – 3) 3. (x + 5)(x 2 – 3 x + 1) 4. (k 2 – 4 k + 3)(k – 2) 5. (2 a 2 + 4 a + 5)(5 a – 4) 6. (2 g + 7)(3 g 2 – 5 g + 2)
Practice What is the simplified form? 1. (5 x 3 + 4 x 2 – 7 x)(x – 2) 2. (8 b – 3)(7 b 2 + b – 9) 3. (b 2 + 3)(b 2 – 4 b + 5) 4. (2 x 2 + 4 x – 3)(3 x + 1) 5. (3 c 5 – 4 c + 8)(4 c 2 – 3)
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Multiplying Special Cases Objective: To find the square of a binomial and to find the product of a sum and difference.
Objectives • I can square a binomial. • I can apply squares of binomials. • I can find the product of a sum and difference.
Vocabulary • There are special rules you can use to simplify the square of a binomials or the product of a sum and difference. • Squares of binomials have the form (a + b)2 or (a – b)2. • The square of a binomial: • Words: The square of a binomial is the square of the first term plus twice the product of the two terms plus the square of the last term. • Algebra: • (a + b)2 = a 2 + 2 ab + b 2 (a – b)2 = a 2 – 2 ab + b 2 • Examples: • (x + 4)2 = x 2 + 8 x + 16 (x – 3)2 = x 2 – 6 x + 9
Squaring a Binomial What is the simpler form of each product? 1. (x + 8)2 2. (2 m – 3)2 3. (n – 7)2 4. (2 x + 9)2 5. (g – 2)2 6. (3 x + 1)2 7. (8 + r)2
Practice What is the simpler form of each product? 1. (3 s + 9)2 2. (4 x – 6)2 3. (a – 8)2 4. (w + 5)2 5. (5 m – 2)2 6. (2 n + 7)2 7. (h + 2)2 8. (k – 11)2
Applying Squares of Binomials A square outdoor patio is surrounded by a brick walkway as shown. What is the area of the walkway? X feet 3 feet What if the brick walkway is 4 feet wide. What is the area?
Practice • x+4 x– 1 x
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Finding the Product of a Sum and Difference What is the simplified form? 1. 2. 3. 4. 5. 6. 7. 8. (x 3 + 8)(x 3 – 8) (x + 9)(x – 9) (6 + m 2)(6 – m 2) (3 c – 4)(3 c + 4) (v + 6)(v – 6) (5 r + 9)(5 r – 9) (2 g + 9 h)(2 g – 9 h) (2 p 2 + 7 q)(2 p 2 – 7 q)
Practice Simplify each product. 1. 2. 3. 4. 5. 6. (v + 6)(v – 6) (b + 1)(b – 1) (z – 5)(z + 5) (x – 3)(x + 3) (10 + y)(10 – y) (t – 13)(t + 13)
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Factoring 2 x + bx + c
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Factoring x 2 + bx + c where b > 0, c > 0 •
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Vocabulary Some factorable trinomials have a negative coefficient of x and a positive constant term. In this case, you need to inspect the negative factors of c to find the factors of the trinomial.
Factoring x 2 + bx + c where b < 0, c > 0 •
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Vocabulary • When you factor trinomials with a negative constant term, you need to inspect pairs of positive and negative factors of c.
Factoring x 2 + bx + c where c < 0 •
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Applying Factoring Trinomials 1. The area of a rectangle is given by the trinomial x 2 – 2 x – 35. What are the possible dimensions of the rectangle? Use factoring. 2. A rectangle’s area is x 2 – x – 72. What are the possible dimensions of the rectangle? Use factoring.
Practice 1. The area of a rectangular desk is given by the trinomial d 2 – 7 d – 18. What are the possible dimensions of the desk? Use factoring. 2. The area of a rectangular rug is given by the trinomial r 2 – 3 r – 4. What are the possible dimensions of the rug? Use factoring.
Vocabulary • You can factor some trinomials that have more than one variable. • A trinomial with two variables may be factorable if the first term includes the square of one variable, the middle term includes both variables, and the last term includes the square of the other variable.
Factoring a Trinomial with Two Variables •
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Factoring 2 ax + bx + c Objective: To factor trinomials of the form ax 2 + bx + c.
Objectives • I can factor when ac is positive. • I can factor when ac is negative. • I can apply trinomial factoring. • I can factor out a monomial first.
Vocabulary • You can write some trinomials of the form ax 2 + bx + c as the products of two binomials. • To factor a trinomial of the form ax 2 + bx + c, you should look for factors of the product ac that have a sum of b.
Factoring When ac Is Positive •
Practice What is the factored form? 1. 2. 3. 4. 5. 6. 3 d 2 + 23 d + 14 2 x 2 + 13 x + 6 4 p 2 + 7 p + 3 6 r 2 – 23 r + 20 4 n 2 – 8 n + 3 8 g 2 – 14 g + 3
Factoring When ac is Negative •
Practice What is the factored form? 1. 2. 3. 4. 5. 6. 7. 8. 4 w 2 + w – 3 5 x 2 + 19 x – 4 3 z 2 + 23 z – 36 2 k 2 – 13 k – 24 4 c 2 – 5 c – 6 6 t 2 + 7 t – 5 4 d 2 – 4 d – 35 2 x 2 – x – 3
Applying Trinomial Factoring 1. The area of a rectangle is 2 x 2 – 13 x – 7. What are the possible dimensions of the rectangle? Use factoring. 2. The area of a rectangle is 8 x 2 + 22 x + 15. What are the possible dimensions of the rectangle? Use factoring. 3. The area of a rectangular kitchen tile is 8 x 2 + 30 x + 7. What are the possible dimensions of the tile? Use factoring.
Practice • The area of a rectangle is 6 x 2 – 11 x – 72. What are the possible dimensions of the rectangle? Use factoring. • The area of a rectangular knitted blanket is 15 x 2 – 14 x – 8. What are the possible dimensions of the blanket? Use factoring.
Vocabulary • To factor a polynomial completely, first factor out the GCF (greatest common factor) of the polynomial’s terms. • Then factor the remaining polynomial until it is written as the product of polynomials that cannot be factored further.
Factoring Out a Monomial First •
Practice What is the factored form? 1. 2. 3. 4. 5. 6. 7. 8. 12 p 2 + 20 p – 8 20 w 2 – 45 w + 10 8 v 2 + 34 v – 30 12 x 2 – 46 x – 8 6 s 2 + 57 s + 72 9 r 2 + 3 r – 30 30 x 2 + 14 x – 8 66 k 2 + 57 k + 12
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Factoring Special Cases Objective: To factor perfect – square trinomials and the differences of two squares.
Objectives • I can factor a perfect-square trinomial. • I can factor to find a length. • I can factor a difference of two squares. • I can factor out a common factor.
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Vocabulary • Here is how to recognize a perfect – square trinomial: • The first and the last terms are perfect squares. • The middle term is twice the product of one factor from the first term and one factor from the last term.
Factoring a Perfect – Square Trinomial •
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Factoring to Find a Length •
Practice The given expression represents the area. Find the side length of the square.
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Factoring a Difference of Two Squares •
Factoring a Difference of Two Squares •
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Vocabulary When you factor out the GCF of a polynomial, sometimes the expression that remains is a perfect – square trinomial or the difference of two squares. You can then factor this expression further using the rules that we have already learned.
Factoring out a Common Factor •
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Factoring by Grouping Objective: To factor higher – degree polynomials by grouping.
Objectives • I can factor a cubic polynomial. • I can factor a polynomial completely. • I can find the dimensions of a rectangular prism.
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Factoring a Cubic Polynomial •
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Vocabulary Before factoring by grouping, you may need to factor out the GCF of all the terms.
Factoring a Polynomial Completely •
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Vocabulary • You can sometimes factor to find possible expressions for length, width, and height of a rectangular prism.
Finding the Dimensions of a Rectangular Prism •
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Factoring Polynomials 1. Factor out the greatest common factor (GCF). 2. If the polynomial has two terms or three terms, look for a difference of two squares, a perfect-square trinomial, or a pair of binomial factors. 3. If the polynomial has four or more terms, group terms and factor to find common binomial factors. 4. As a final check, make sure there are no common factors other than 1.
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