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# Multiplying Polynomials

Objective Learn the techniques of multiplying two binomials by using the FOIL method, and multiplying polynomials by using the Distributive Property.

In this lesson, you will use the technique of multiplying a polynomial by a monomial, along with the Distributive Property, to multiply any two polynomials.

## Multiplying Binomials

Foil Method for Multiplying two Binomials

To multiply to binomials, find the sum of the products of

 F the First terms, O the Outer terms, I the Inner terms, and L the Last terms.

Example 1

Find (3x + 2)(2x + 4).

Solution

Apply the FOIL method by adding the product of the First, Outer, Inner, and Last terms of the binomials. Now multiply and add like terms.

= 6x 2 + 12x + 4x + 8

= 6x 2 + 16x + 8

Example 2

Find (2x 2 - 1)(4x 3 - 3x ).

Solution In multiplying more general polynomials, apply the Distributive Property to write the product as a sum of products of polynomials times monomials, and then use basic techniques to multiply polynomials and monomials. Finally, simplify by adding like terms.

While the following examples are relatively long and involve several steps, you are already familiar with each of the individual steps. If you work slowly and carefully, you will be successful in solving these types of problems.

Example 3

Find (3x 2 + 2x +1)( x 3 - x ).

Solution

First use the Distributive Property to write this product as a sum of products of polynomials times monomials.

(3x 2 + 2x +1)( x 3 - x ) = (3x 2 + 2x +1)( x 3 ) + (3x 2 + 2x +1)( - x )

Then multiply each of the terms (again using the Distributive Property).

= [(3x 2 )( x 3 ) + (2 x )( x 3 ) + (1)( x 3 )] + [(3x 2 )(-x) + (2x )(-x) + (1)(-x)]

= 3x 5 + 2x 4 + x 3 - 3x 3 - 2x 2 - x

= 3x 5 + 2x 4 - 2x 3 - 2x 2 - x

Example 4

Find ( a 2 - 2a - 3)( a 2 - 4a - 3).

Solution

First use the Distributive Property to write this product as a sum of products of polynomials times monomials.

( a 2 - 2a - 3)( a 2 - 4a - 3) = ( a 2 - 2a - 3)( a 2 ) + ( a 2 - 2a - 3)(-4a) + ( a 2 - 2a - 3)(-3)

Then use the Distributive Property to multiply each of the terms.

= [( a 2 )( a 2 ) + ( - 2a )( a 2 ) + ( -3)(a 2 )] + [( a 2 )( -4a ) + ( -2a )( - 4a ) + ( -3)( -4a )] +

[( a 2 )( -3) + ( -2a )( -3) + ( - 3)( -3)]

= ( a 4 - 2a 3 - 3a 2 ) + ( - 4a 3 + 8a 2 + 12a ) + ( - 3a 2 + 6a + 9)

Now combine like terms to get the following.

= a 4 + ( -2a 3 - 4a 3 ) + ( - 3a 2 + 8a 2 - 3a 2 ) + (12a + 6a ) + 9 = a 4 - 6a 3 + 2a 2 + 18a + 9