5.3 WARM-UP Decide whether the function is a polynomial function. If so, write it in standard form and state its degree, type and leading coefficient. Not a function yes Degree: 5 Leading coefficient: -8 Standard form: Use direct substitution to evaluate the following polynomials -49 76 Use synthetic substitution to evaluate each polynomial -49 149

Add, Subtract, and Multiply Polynomials 5.3 Add, Subtract, and Multiply Polynomials To add or subtract polynomials , combine the coefficients of like terms. a. Add 2x3 – 5x2 + 3x – 9 and x3 + 6x2 + 11 in a vertical format. 2x3 – 5x2 + 3x – 9 + x3 + 6x2 + 11 3x3 + x2 + 3x + 2 b. Add 3y3 – 2y2 – 7y and – 4y2 + 2y – 5 in a horizontal format. (3y3 – 2y2 – 7y) + (– 4y2 + 2y – 5) = 3y3 – 2y2 – 4y2 – 7y + 2y – 5 = 3y3 – 6y2 – 5y – 5

8x3 – x2 – 5x + 1 – (3x3 + 2x2 – x + 7) 8x3 – x2 – 5x + 1 c. Subtract 3x3 + 2x2 – x + 7 from 8x3 – x2 – 5x + 1 in a vertical format. (Align like terms, then change all the signs on the second equation, then add) 8x3 – x2 – 5x + 1 – (3x3 + 2x2 – x + 7) 8x3 – x2 – 5x + 1 + – 3x3 – 2x2 + x – 7 5x3 – 3x2 – 4x – 6 d. Subtract 5z2 – z + 3 from 4z2 + 9z – 12 in a horizontal format. (Write the opposite of the subtracted polynomial, then add like terms.) (4z2 + 9z – 12) – (5z2 – z + 3) = 4z2 + 9z – 12 – 5z2 + z – 3 = – z2 + 10z – 15

Find the sum or difference. 1. (t2 – 6t + 2) + (5t2 – t – 8) t2 – 6t + 2 + 5t2 – t – 8 6t2 – 7t – 6 2. (8d – 3 + 9d3) – (d3 – 13d2 – 4) = 8d – 3 + 9d3 – d3 + 13d2 + 4 = 8d3 + 13d2 + 8d + 1

– 2y2 + 3y – 6 y – 2 4y2 – 6y + 12 – 2y3 + 3y2 – 6y – 2y3 +7y2 –12y Multiply – 2y2 + 3y – 6 and y – 2 in a vertical format. – 2y2 + 3y – 6 y – 2 4y2 – 6y + 12 Multiply – 2y2 + 3y – 6 by – 2 . – 2y3 + 3y2 – 6y Multiply – 2y2 + 3y – 6 by y – 2y3 +7y2 –12y + 12 Combine like terms. Multiply x + 3 and 3x2 – 2x + 4 in a horizontal format. (x + 3)(3x2 – 2x + 4) = 3x3 – 2x2 + 4x + 9x2 – 6x + 12 = 3x3 + 7x2 – 2x + 12

Multiply three binomials Multiply x – 5, x + 1, and x + 3 in a horizontal format. (x – 5)(x + 1)(x + 3) = (x2 +1x – 5x – 5) (x + 3) (x2 – 4x – 5)(x + 3) = x3 – 4x2 – 5x + 3x2 – 12x – 15 (x + 3)(x2 – 4x – 5) = = x3 – x2 – 17x – 15

Try finding the product. (x + 2)(3x2 – x – 5) (a – 5)(a + 2)(a + 6) 3x2 – x – 5 x + 2 = (a2 – 3a – 10)(a + 6) = (a2 – 3a – 10)a + (a2 – 3a – 10)6 6x2 – 2x – 10 3x3 – x2 – 5x = (a3 – 3a2 – 10a + 6a2 – 18a – 60) 3x3 + 5x2 – 7x – 10 = (a3 + 3a2 – 28a – 60) = (xy – 4) (xy – 4) (xy – 4)3 = (x2y2 – 4xy – 4xy + 16) (xy – 4) = (x2y2 – 8xy + 16) (xy – 4) = (x3y3 – 8x2y2 + 16xy – 4x2y2 + 32xy – 64 ) = x3y3 – 12x2y2 + 48xy – 64

Use special product patterns Try these: a. (3t + 4)(3t – 4) = (3t)2 – 42 Sum and difference *You can just square the first number and square the last number = 9t2 – 16 b. (8x – 3)2 = (8x)2 – 2(8x)(3) + 32 Square of a binomial = 64x2 – 48x + 9 *square the first number, square the last number and multiply the first and last numbers and double HOMEWORK 5.3 p. 349 #3-25 all, 28-37 EOP