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MULTIPLICATION OF POLYNOMIALS CHAPTER 4 SECTION 5 MTH 10905 Algebra

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Multiply a Monomial by a Monomial A monomial is a Polynomial with one term, such as 8 because 8x 0, 4x because 4x 1, and -6x 2 Multiple their coefficients and use the product rule of exponents to determine the exponents value. Example: (3y 4 )(5y 2 )(-2a 6 )(8a 8 ) (3)(5)(y 4 )(y 2 )(-2)(8)(a 6 )(a 8 ) 15y 4+2 -16a 6+8 15y 6 -16a 14

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Multiply a Monomial by a Monomial Example: (4xy 5 )(2x 7 y 2 ) (4)(2)(x)(x 7 )(y 5 )(y 2 ) 8x 1+7 y 5+2 8x 8 y 7 Example: 7a 2 bc 4 (-2a 5 b 7 c) (7)(-2)(a 2 )(a 5 )(b)(b 7 )(c 4 )(c) -14a 2+5 b 1+7 c 4+1 -14a 7 b 8 c 5

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Multiply a Monomial by a Monomial Example: (-3x 3 z 8 )(-5xy 4 z 2 ) (-3)(-5)(x)(x 3 )(y 4 )(z 2 )(z 8 ) 15x 1+3 y 4 z 2+8 15x 4 y 4 z 10

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Multiply a Polynomial by a Monomial Use the distributive property: a(b + c) = ab + ac Example: 6a(a 2 + 10) (6a)(a 2 ) + (6a)(10) 6a 1+2 + 60a 6a 3 + 60a

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Multiply a Polynomial by a Monomial Example: -2x(2x 2 – 3x – 5) (-2x)(2x 2 ) + (-2x)(-3x) + (-2x)(-5) -4x 1+2 + 6x 1+1 + 10x -4x 3 + 6x 2 + 10x Example: 3x 2 (5x 3 – 3x + 8) (3x 2 )(5x 3 ) + (3x 2 )(-3x) + (3x 2 )(8) 15x 2+3 - 9x 2+1 + 24x 2 15x 5 - 9x 3 + 24x 2

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Multiply a Polynomial by a Monomial Example: 3a(4a 2 b – 7ab + 2) (3a)(4a 2 b) + (3a)(-7ab) + (3a)(2) 12a 1+2 b – 21a 1+1 b + 6a 12a 3 b – 21a 2 b + 6a Example: (5x 2 – 3xy + 5)(3x) Commutative Property (3x)(5x 2 ) + (3x)(-3xy) + (3x)(5) 15x 1+2 - 9x 1+1 y + 15x 15x 3 - 9x 2 y + 15x

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Multiply Binomials using the FOIL Method F = First, multiply the first terms together O = Outer, multiply the two outer terms together I = Inner, multiply the two inner terms together L = Last, multiply the last terms together The product of the two binomials is the sum of these four products. (a + b) (c + d) = ac + ad + bc + bd Each term must multiply every term in the other binomial

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Multiply Binomials using the FOIL Method Example: (5a + 3)(a – 2) (5a)(a) + (5a)(-2) + (3)(a) + (3)(-2) 5a 1+1 + 3a – 10a – 6 5a 2 – 7a – 6 Example: (a + 3)(b – 9) (a)(b) + (a)(-9) + (3)(b) + (3)(-9) ab - 9a + 3b – 27

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Multiply Binomials using the FOIL Method Example: (3x – 4)(x + 2) (3x)(x) + (3x)(2) + (-4)(x) + (-4)(2) 3x 1+1 + 6x – 4x – 8 3x 2 + 2x – 8 Example: (8 – 3b)(7 – 5b) (8)(7) + (8)(-5b) + (-3b)(7) + (-3b)(-5b) 56 – 40b – 21b + 15b 1+1 56 – 61b + 15b 2 15b 2 – 61b + 56

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Multiply Binomials using the FOIL Method Example: (2c + 3)(2c – 3) (2c)(2c) + (2c)(-3) + (3)(2c) + (3)(-3) 4c 1+1 – 6c + 6c – 9 4c 2 – 9

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Multiply Binomials using Formulas for Special Products The product of the Sum and Difference of the Same Two Terms: Difference of Two Squares Formula: (a + b) (a – b) = a 2 - b 2 Example: (y + 10)(y – 10) (y)(y) + (y)(-10) + (10)(y) + (10)(-10) y 2 – 10 2 y 2 – 100

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Multiply Binomials using Formulas for Special Products Example: (7a + 2b)(7a – 2b) (7a)(7a) + (7a)(-2b) + (2b)(7a) + (2b)(-2b) (7a) 2 – (2a) 2 49a 2 – 4b 2

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Multiply Binomials using Formulas for Special Products Example: Using the FOIL method (x + 8) 2 (x + 8)(x + 8) (x)(x) + (x)(8) + (8)(x) + (8)(8) x 1+1 + 8x + 8x + 64 x 2 + 16x + 64

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Multiply Binomials using Formulas for Special Products Square of a Binomial Formula (a + b) 2 = (a + b)(a + b) = a 2 + 2ab + b 2 Example: (3x + 5) 2 = (3x + 5)(3x + 5) 9x 2 + 30x + 25 (3x) 2 + (2)(3x)(5) + 5 2 (3x)(3x) + (3x)(5) + (5)(3x) + (5)(5)

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Square of a Binomial Formula (3 + 5) 2 ≠ 3 2 + 5 2 because 3 2 + 5 2 = 9 + 25 = 34 and (3 + 5) 2 = (8) 2 = 64 (a + b) 2 = (a + b)(a + b) = a 2 + 2ab + b 2 (3 + 5) 2 = 3 2 + (2)(3)(5) + 5 2 = 9 + 30 + 25 = 39 + 25 = 64

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Multiply Binomials using Formulas for Special Products Example: (7r – w) 2 = (7r – w)(7r – w) 49r 2 - 14rw + w 2 (7r) 2 + (2)(7r)(-w) + (-w)(-w) (7r)(7r) + (7r)(-w) + (-w)(7r) + (-w)(-w)

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Multiply any Two Polynomials using a Vertical Procedure Example: (3y + 7)(4y + 5)

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Multiply any Two Polynomials using a Vertical Procedure Example: (3x + 2)(4x 2 + x – 3)

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Multiply any Two Polynomials using a Vertical Procedure Example:

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Multiply Binomials Using Formulas for Special Products Example: (2a 2 + 2a)(4a 3 +2a 2 + a + 4) Multiplication of Polynomials is very important that you understand. In Chapter 5 we will be factoring polynomials, which is the reverse process of multiplication of polynomials.

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HOMEWORK 4.5 Page 275 #21, 25, 29, 36, 43, 45, 47, 53, 59, 75, 76, 77, 79, 80, 93, 95

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