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# Multiplying Polynomials

In this section you will learn how to multiply any two polynomials.

## Multiplying Monomials with the Product Rule

To multiply two monomials, such as x3 and x5, recall that

x3 = x Â· x Â· x  and  x5 = x Â· x Â· x Â· x Â· x,

so

The exponent of the product of x3 and x5 is the sum of the exponents 3 and 5. This example illustrates the product rule for multiplying exponential expressions.

Product Rule

If a is any real number and m and n are any positive integers, then

am Â· an = am + n.

Multiplying monomials

Find the indicated products.

a) x2 Â· x4 Â· x

b) (-2ab)(-3ab)

c) -4x2y2 Â· 3xy5

d) (3a)2

Solution

 a) x2 Â· x4 Â· x =  x2 Â· x4 Â· x1 = x7 Product rule
 b) (-2ab)(-3ab) = (-2)(-3) Â· a Â· a Â· b Â· b = 6a2b2 Product rule
 c) (-4x2y2)(3xy5) = (-4)(3)x2 Â· x Â· y2 Â· y5 = -12x3y7 Product rule
 d) (3a)2 = 3a Â· 3a = 9a2

Caution

Be sure to distinguish between adding and multiplying monomials. You can add like terms to get 3x4 + 2x4 = 5x4, but you cannot combine the terms in 3w5 + 6w2. However, you can multiply any two monomials: 3x4 Â· 2x4 = 6x8 and 3w5 Â· 6w2 = 18w7.

## Multiplying Polynomials

To multiply a monomial and a polynomial, we use the distributive property.

Multiply monomials and polynomials

Find each product.

a) 3x2(x3 - 4x)

b) (y2 - 3y + 4)(-2y)

c) -a(b - c)

Solution

Note in part c) that either of the last two binomials is the correct answer. The last one is just a little simpler to read.

Just as we use distributive property to find the product of a monomial and a polynomial, we can use the distributive property to find the product of two binomials as the product of a binomial and a trinomial.

Multiplying polynomials

Use the distributive property to find each product

a) (x + 2)(x + 5)

b) (x + 3)(x2 + 2x - 7)

Solution

a) First multiply each term of x + 5 by x + 2:

b) First multiply each term of the trinomial by x + 3:

Products of polynomials can also be found by arranging the multiplication vertically like multiplication of whole numbers.

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