MULTIPLICATION OF

POLYNOMIALS

CHAPTER 4 SECTION 5

MTH 10905Algebra

Multiply a Monomial by a Monomial

A monomial is a Polynomial with one term, such as 8 because 8x0, 4x because 4x1, and -6x2

Multiple their coefficients and use the product rule of exponents to determine the exponents value.

Example: Example:

(3y4)(5y2) (-2a6)(8a8)(3)(5)(y4)(y2) (-2)(8)(a6)(a8)15y4+2 -16a6+8

15y6 -16a14

Multiply a Monomial by a Monomial

Example: (4xy5)(2x7y2)(4)(2)(x)(x7)(y5)(y2)8x1+7y5+2

8x8y7

Example:7a2bc4(-2a5b7c)(7)(-2)(a2)(a5)(b)(b7)(c4)(c)-14a2+5b1+7c4+1

-14a7b8c5

Multiply a Monomial by a Monomial

Example:

(-3x3z8)(-5xy4z2)(-3)(-5)(x)(x3)(y4)(z2)(z8)15x1+3y4z2+8

15x4y4z10

Multiply a Polynomial by a Monomial

Use the distributive property:

a(b + c) = ab + ac

Example:

6a(a2+ 10)(6a)(a2) + (6a)(10)6a1+2 + 60a6a3 + 60a

Multiply a Polynomial by a Monomial

Example:-2x(2x2 – 3x – 5)

(-2x)(2x2) + (-2x)(-3x) + (-2x)(-5) -4x1+2 + 6x1+1 + 10x-4x3 + 6x2 + 10x

Example:3x2(5x3 – 3x + 8)(3x2)(5x3) + (3x2)(-3x) + (3x2)(8) 15x2+3 - 9x2+1 + 24x2

15x5 - 9x3 + 24x2

Multiply a Polynomial by a Monomial

Example:3a(4a2b – 7ab + 2)(3a)(4a2b) + (3a)(-7ab) + (3a)(2) 12a1+2b – 21a1+1b + 6a12a3b – 21a2b + 6a

Example:(5x2 – 3xy + 5)(3x) Commutative Property(3x)(5x2) + (3x)(-3xy) + (3x)(5) 15x1+2 - 9x1+1y + 15x15x3 - 9x2y + 15x

Multiply Binomials using the FOIL Method

F = First, multiply the first terms together O = Outer, multiply the two outer terms togetherI = Inner, multiply the two inner terms togetherL = 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

Multiply Binomials using the FOIL Method

Example:(5a + 3)(a – 2)(5a)(a) + (5a)(-2) + (3)(a) + (3)(-2) 5a1+1 + 3a – 10a – 6 5a2 – 7a – 6

Example:(a + 3)(b – 9) (a)(b) + (a)(-9) + (3)(b) + (3)(-9) ab - 9a + 3b – 27

Multiply Binomials using the FOIL Method

Example:(3x – 4)(x + 2)(3x)(x) + (3x)(2) + (-4)(x) + (-4)(2) 3x1+1 + 6x – 4x – 8 3x2 + 2x – 8

Example:(8 – 3b)(7 – 5b)(8)(7) + (8)(-5b) + (-3b)(7) + (-3b)(-5b)56 – 40b – 21b + 15b1+1 56 – 61b + 15b2

15b2 – 61b + 56

Multiply Binomials using the FOIL Method

Example:(2c + 3)(2c – 3)(2c)(2c) + (2c)(-3) + (3)(2c) + (3)(-3) 4c1+1 – 6c + 6c – 9 4c2 – 9

Multiply Binomials using Formulas for Special Products

The product of the Sum and Difference of theSame Two Terms:

Difference of Two Squares Formula:(a + b) (a – b) = a2 - b2

Example:(y + 10)(y – 10) (y)(y) + (y)(-10) + (10)(y) + (10)(-10)y2 – 102 y2 – 100

Multiply Binomials using Formulas for Special Products

Example:

(7a + 2b)(7a – 2b)(7a)(7a) + (7a)(-2b) + (2b)(7a) + (2b)(-2b)(7a)2 – (2a)2 49a2 – 4b2

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)x1+1 + 8x + 8x + 64 x2 + 16x + 64

Multiply Binomials using Formulas for Special Products

Square of a Binomial Formula

(a + b)2 = (a + b)(a + b) = a2 + 2ab + b2

Example:

(3x + 5)2 = (3x + 5)(3x + 5) 9x2 + 30x + 25

(3x)2 + (2)(3x)(5) + 52

(3x)(3x) + (3x)(5) + (5)(3x) + (5)(5)

Square of a Binomial Formula

(3 + 5)2 ≠ 32 + 52 because 32 + 52 = 9 + 25 = 34

and (3 + 5)2 = (8)2 = 64

(a + b)2 = (a + b)(a + b) = a2 + 2ab + b2

(3 + 5)2 = 32 + (2)(3)(5) + 52 = 9 + 30 + 25 = 39 + 25 = 64

Multiply Binomials using Formulas for Special Products

Example:

(7r – w)2 = (7r – w)(7r – w) 49r2 - 14rw + w2

(7r)2 + (2)(7r)(-w) + (-w)(-w)

(7r)(7r) + (7r)(-w) + (-w)(7r) + (-w)(-w)

Multiply any Two Polynomialsusing a Vertical Procedure

Example: (3y + 7)(4y + 5)

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73

64

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Multiply any Two Polynomialsusing a Vertical Procedure

Example: (3x + 2)(4x2 + x – 3)

671112x

09 312

628

23

34

23

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2

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Multiply any Two Polynomialsusing a Vertical Procedure

Example:

12165123

0 0 9123

12164-

43x

34

234

234

2

2

2

xxxx

xxx

xx

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Multiply Binomials Using Formulas for Special Products

Example: (2a2 + 2a)(4a3 +2a2 + 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

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