Identifying key concepts Polynomials In the previous work

Identifying key concepts - Polynomials In the previous work we have done we have looked at the terms: Variable, coefficients, like and unlike term etc… Now we will add some more concepts. Example: 4 x 3 + 2 x 2 + 3 x -2 Ø There are 4 terms Ø The variable is x Ø The coefficient of x 3 is 4, of x 2 is 2, of x is 3 Ø The constant is 2 Polynomial 3. mp 4

Identifying key concepts - Polynomials An expression having one or more terms is called a polynomial In a polynomial, the variable must have whole number exponents. There are 3 key types of polynomials. Polynomial 4. mp 4 Monomial - has 1 term Eg. 4 x 3, -2 y, 4 xy 2 z, 10 Binomial - has 2 term Eg. 2 x 3 + 4 y, 3 x - 1 Trinomial - has 3 term Eg. 4 x 3+ 2 x - 7, x 2 + 5 x +6

Identifying key concepts - Polynomials Standard form is the proper way to write a polynomial. It means that the polynomial had been written with each term in order of degree – largest to smallest. Degrees: 1. To find the degree of a single term or a monomial * The degree is the sum of the exponents of the variables. Examples 8 x 2 degree 2 -3 x 5 5 2 xy 3 z 2 6 -4 x 1 7 0 Polynomial 5. mp 4

Identifying key concepts - Polynomials Degrees: 2. To find the degree of a polynomial with more than 1 term, but with 1 variable. * The degree is the highest power of the variable in one term. Examples Polynomial 4 x 3 + 2 x + 1 8 x - 3 x 5 - 7 + 2 x 3 2 + 3 x - 2 x 2 - 4 x 3 Degree Standard Form 3 4 x 3 + 2 x + 1 5 -3 x 5 + 2 x 3 + 8 x - 7 3 -4 x 3 - 2 x 2 + 3 x + 2 Polynomial 6. mp 4

Identifying key concepts - Polynomials Degrees: 3. To determine the degree of a polynomial with 2 or more variables * The degree is the largest sum of the exponents in any one term Examples Polynomial Degree Standard Form 4 xy - 2 x 2 y + 3 xy 3 4 3 xy 3 - 2 x 2 y + 4 xy 8 - x 2 y 2 + 5 y 5 5 5 y 5 - x 2 y 2 + 8 ALWAYS leave your final answers in STANDARD FORM Polynomial 7. mp 4

Using Tiles to Model Polynomials Just as you can use tiles to represent equations, we can use tiles to represent polynomials x 2 x 1 Example: a) -3 x 2 – 2 x + 5 b) 2 m 2 – 5 m - 4 Polynomial 8. mp 4

Assessment Pages 241 - 216 Numbers One chart 4, 5, 6, 9 , 7, 12, 13, 15

Like and Unlike terms Polynomial 9. mp 4 Just as we needed common denominators in fractions to add and subtract, we also need “ like” terms to add and subtract polynomials. Like terms have the exact same variable and exponent Examples: 3 x , 5 x , -3 x , ½ x Unlike terms do not have the exact same variable and exponent Examples: 3 x , 5 y , -3 x 2 , ½ y 3 We can easily “see” like terms using tiles x 2 x 1 Same shapes can be grouped together

Like and Unlike terms Examples 1. Simplify: a. 2 x - 3 x = -1 x b. 4 x - 2 y + 6 x - 3 y = 10 x -5 y c. 3 g – 6 – 2 g + 9 = g + 3 2. What polynomial is represented by the tiles 3 x 2 – 5 x - 4 Polynomial 10. mp 4

Like and Unlike terms 3. Write a polynomial to represent the perimeter x a) Px = x + x + x P = 4 x x b) P =1 x + 1 + x + 1 P = 2 x + 2 x c) x d) P= x+x+x+x+x+x P = 10 x x 1 P = 2 x + 4(1) P = 2 x + 4

Assessment Pages 222 - 223 Numbers 8, 9, 12 , 13, 14, 15, 19 plus worksheet

Adding Polynomials Adding polynomials is the same concept as adding monomials. Examples: 1. Remove Brackets a) (x+4) + (2 x+ 8) 2. Reorder to groups of like terms 3. Add the like terms 4. Write in standard form = x + 4 + 2 x + 8 = x + 2 x + 4 + 8 = 3 x + 12 b) (x 2+2) + (2 x 2+ 3) = x 2 + 2 x 2 + 3 = x 2 + 2 + 3 = 3 x 2 + 5 c) (7 x 3 - 2 x 2 + 5 x) + (-3 x 3+ 6 x 2 – 2 x) + ( 4 x 3 - 7 x 2 - 2 x) = 7 x 3 - 2 x 2 + 5 x -3 x 3+ 6 x 2 – 2 x + 4 x 3 - 7 x 2 - 2 x = 7 x 3 -3 x 3 + 4 x 3 - 2 x 2 + 6 x 2 - 7 x 2 + 5 x- 2 x = 8 x 3 - 3 x 2 + x You. Tube Channel Huphupapps Polynomial 11. mp 4

Adding Polynomials with tiles Adding polynomials with tiles is the same as monomials Examples: a) (3 x 2 - 2 x) + (-2 x 2 -5 x) x 2 – 7 x You. Tube Channel Huphupapps Polynomial 12. mp 4

Adding Polynomials Vertically Adding polynomials vertically is the same as adding regular numbers. Examples: 2 + 2 x - 4 3 x 2 2 a) (3 x + 2 x - 4) + (-2 x +5 x - 3) -2 x 2 + 5 x - 3 + x 2 + 7 x - 7 b) (7 s + 14) + (-6 s 2 +1 s - 6) 7 s+14 2 + 1 s - 6 -6 s + -6 s 2 + 8 s + 8 You. Tube Channel Huphupapps Polynomial 13. mp 4

Assessment Pages 228 Numbers 3, 6, 9, 10, 12, 14, 15 plus worksheet

Subtracting Polynomials Subtracting polynomials is the same concept as adding polynomials, except you must distribute the subtraction sign through the second set of brackets. Examples: a) (6 x 2 -5 x) - (2 x 2 -8 x) = 6 x 2 - 5 x - 2 x 2 + 8 x = 6 x 2 - 2 x 2 - 5 x + 8 x = 4 x 2 + 3 x 1. Remove Brackets – Careful with the subtraction symbol 2. Reorder to groups of like terms 3. Perform the appropriate operation 4. Write in standard form b) (7 x 3 - 2 x 2 + 5 x) - (-3 x 3+ 6 x 2 – 2 x) = 7 x 3 - 2 x 2 + 5 x +3 x 3 - 6 x 2 + 2 x = 7 x 3+3 x 3 - 2 x 2 - 6 x 2 + 5 x+ 2 x = 10 x 3 - 8 x 2 + 7 x You. Tube Channel Huphupapps Polynomial 11. mp 4

Subtracting Polynomials with tiles Must use the additive inverse to change the second set of brackets Examples: a) (3 x 2 - 2 x) - (-2 x 2 -5 x) Additive Inverse 5 x 2 + 3 x You. Tube Channel Huphupapps Polynomial 12. mp 4

Subtracting Polynomials Vertically Subtracting polynomials vertically is the same as adding the additive inverse. Examples: 2 - 7 x + 4 5 x 2 2 a) (5 x - 7 x + 4) - (2 x + 5) -2 x 2 - 2 x - 5 Additive Inverse + 3 x 2 - 9 x - 1

Activity Adding and Subtracting Polynomials Math Lib

Assessment Pages 234 Numbers 4, 8, 9, 12, 13, 15, 16 plus worksheet

Multiplying Polynomials by a constant To multiply polynomials we need to use the distributive property. Steps: Ø Multiply the coefficients Ø Multiply the variables using exponent laws Examples a) 4(3 x) =12 x b) 3(2 x + 4) c) -2(-n 2 + 2 n - 1) =3(2 x) + 3(4) = -2(-n 2) + -2(2 n) - -2(1) =6 x + 12 = 2 n 2 – 4 n + 2

Multiplying Polynomials by a constant using tiles Examples a) 4(3 x) b) 3(2 x + 4) 4 c) -2(-n 2 + 2 n - 1) 3 -2 2 x + 4 -n 2 + 2 n - 1 3 x =12 x =6 x + 12 = 2 n 2 – 4 n + 2 Change to the additive inverse due to the -2

Dividing Polynomials by a constant / monomial When dividing a polynomial by a constant or a monomial Steps: Ø Divide each term of the polynomial by the constant /monomial Examples a) 4 s 2 - 8 4 = 4 s 2 - 8 4 4 = s 2 - 2 b) -3 m 2 n + 15 mn - 21 mn 2 -3 mn = m – 5 + 7 n = m + 7 n - 5 -3 mn

Dividing Polynomials by a constant or monomial using tiles Examples 2 x 2 + 8 x 2 2 x x+4 2 2 x x 2 + 4 x = x 2+ 4 x =x+4

Assessment Pages 246 - 247 Numbers 3, 4, 6, 15, 16 plus worksheet

Multiplying Polynomials by a monomial To multiply polynomials we need to use the distributive property. Steps: Ø Multiply the monomial by monomial Ø Multiply the variables using exponent laws Examples a) 4 x(3 x) =12 x 2 b) 3 x(2 x + 4) c) -2 n(-n 2 + 2 n - 1) =3 x(2 x) + 3 x(4) = -2 n(-n 2) + -2 n(2 n) - -2 n(1) = 6 x 2 + 12 x = 2 n 3 – 4 n 2 + 2 n

Multiplying Polynomials by a monomial using tiles Examples a) 2 x(3 x + 4) 2 x 3 x + 4 = 6 x 2 + 8 x

Assessment Pages 255 - 257 Numbers 4, 6, 7, 11, 12, 16, 19, 22 Challenge: Person Puzzle – Multiplying polynomials by polynomials