§ 4.5

Multiplication of Polynomials

Angel, Elementary Algebra, 7ed 2

Multiplying Polynomials

To multiply a monomial by a monomial, multiply their coefficients and use the product rule of exponents.

a.) (5x5)(2x) = 5 · 2 · x5 ·x = 10x6

b.) (6x2y2)(3xy4) = 6 · 3 · x2 · x · y2 · y4 = 18x3 y6

a.) 3(x - 4) = 3(x) + 3(-4) = 3x - 12

b.) (-3c2 + 5c – 6)(-6c) = (-6c)(-3c2) + (-6c)(5c) + (-6c)(-6) =

18c3 – 30c2 + 36c

To multiply a polynomial by a monomial, use the distributive property.

Angel, Elementary Algebra, 7ed 3

Multiplying PolynomialsTo multiply two binomials, use the distributive property so every term in one polynomial is multiplied by every term in the other polynomial.

Example:

a.) (7x + 3)(2x + 4) = (7x + 3)(2x) + (7x + 3)(4) =

14x2 + 6x + 28x + 12 = 14x2 + 34x + 12

b.) (z + 2y)(4z – 3) = (z + 2y)(4z) + (z + 2y)(-3) =

4z2 + 8yz + (-3z) + (-6y) = 4z2 + 8yz – 3z – 6y

A common method used to multiply two binomials is the FOIL method.

Angel, Elementary Algebra, 7ed 4

The FOIL Method

Consider (Consider (a a + + bb)()(cc + + dd):):

F

O

I

L

Stands for the first – multiply the first terms together. (a + b) (c + d): product ac

F

Stands for the outer – multiply the outer terms together. (a + b) (c + d): product ad

O

Stands for the inner – multiply the inner terms together. (a + b) (c + d): product bc

I

Stands for the last – multiply the last terms together.

L

(a + b) (c + d): product bd

The product of the two binomials is the sum of these four products: (a + b)(c + d) = ac + ad + bc + bd

Angel, Elementary Algebra, 7ed 5

The FOIL Method

Using the FOIL method, multiply (7x + 3)(2x + 4) .

= (7(7xx))(2(2xx))

(7(7xx + 3)(2 + 3)(2xx + + 4)4)

F

F

O

O

+ (7+ (7xx))(4)(4)

+ (3)(2+ (3)(2xx))

I

I

+ (3)+ (3)(4)(4)

L

L

= 1414xx 22 + 28 + 28xx + + 6 6x + x + 1212= 14= 14xx 22 + 34 + 34x + x + 1212

Angel, Elementary Algebra, 7ed 6

Formulas for Special Products

Product of the Sum and Difference of Two Terms

(a + b)(a – b) = a2 – b2

This special product is also called the difference of two squares formula.

Example:

a.) (7x + 3) (7x – 3) = 49x2 - 9

b.) (z3 – 2y4) (z3 + 2y4) = z6 – 4y8

Angel, Elementary Algebra, 7ed 7

Formulas for Special Products

Square of Binomials(a + b)2 = (a + b)(a + b) = a2 + 2ab +

b2

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

To square a binomial, add the square of the first term, twice the product of the terms and the square of the second term.

Example:

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

b.) (z3 – 12y)2 = z6 – 24yz3 + 144y2