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Multiplying Polynomials

You probably know how to multiply monomials. For example, -2x3 Â· 4x2 = -8x5.

To multiply a monomial and a polynomial of two or more terms, we apply the distributive property. For example, 3x(x3 - 5) = 3x4 - 15x.

Example 1

Multiplying a monomial and a polynomial

Find the products.

a) 2ab2 Â· 3a2b

b) (-1)(5 - x)

c) (x3 - 5x + 2)(-3x)

Solution

a) 2ab2 Â· 3a2b = 6a3b3

b) (-1)(5 - x) = -5 + x = x - 5

c) Each term of x3 - 5x + 2 is multiplied by -3x:

(x3 - 5x + 2)(-3x) = -3x4 + 15x2 - 6x

Note what happened to the binomial in Example 1(b) when we multiplied it by -1. If we multiply any difference by -1, we get the same type of result:

-1(a - b) = -a + b = b - a.

Because multiplying by -1 is the same as taking the opposite, we can write this equation as

-(a - b) = b - a.

This equation says that a - b and b - a are opposites or additive inverses of each other. Note that the opposite of a b is -a - b, not a - b.

To multiply a binomial and a trinomial, we can use the distributive property or set it up like multiplication of whole numbers.

Example 2

Multiplying a binomial and a trinomial

Find the product (x + 2)(x2 + 3x - 5).

Solution

We can find this product by applying the distributive property twice. First we multiply the binomial and each term of the trinomial:

 (x + 2)(x2 + 3x - 5) = (x + 2)x2 + (x + 2)3x + (x + 2)(-5) Distributive property. = x3 + 2x2 + 3x2 + 6x - 5x - 10 Distributive property. = x3 + 5x2 + x - 10 Combine like terms.

We could have found this product vertically:

 x2 + 3x - 5 x + 2 2x2 + 6x - 10 2(x2 + 3x - 5) = 2x2 + 6x - 10 x3 + 2x2 - 5x x(x2 + 3x - 5) = x3 + 3x2 - 5x x3 + 5x2 + x - 10 Add.