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Multiplying Polynomials

You probably know how to multiply monomials. For example, -2x3 · 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 · 3a2b

b) (-1)(5 - x)

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

Solution

a) 2ab2 · 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.

 

Helpful Hint

Many students find vertical multiplication easier than applying the distributive property twice horizontally. However, you should learn both methods because horizontal multiplication will help you with factoring by grouping.

 

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