**0.3 Polynomials: Basic Operations 31**

**First Inner**

Product of

**First**

Terms

Product of

**Last**

Terms

**Outer**

**Last**

Product of

**Outer**

Terms

Product of

**Inner**

Terms

Special Products

The method outlined for multiplying polynomials works for *all* products of polynomials. For

the special case when both polynomials are binomials, the **FOIL method** can also be used.

**W _{ORDS}^{M}_{ATH}**

Apply the distributive property. (5 *x*

1)(2 *x*

3) 5 *x* (2 *x*

3) 1(2 *x*

3) Apply the distributive property. 5 *x* (2 *x* ) 5 *x* (3) 1(2 *x* ) 1(3) Multiply each set of monomials. 10 *x*^{2}^{15} *x* 2 *x* 3 Combine like terms. 10 *x*^{2}_{13} *x* 3

The FOIL method finds the

products of the **F** irst terms,

**O**

uter terms, **I** nner terms, and (5 *x*

1)(2 *x*

3) 10 *x*^{2}^{15} *x* 2 *x* 3

**L**

ast terms.

**EXAMPLE 6 Multiplying Binomials Using the FOIL Method**

Multiply (3 *x*

1)(2 *x*

5) using the FOIL method. Solution:

Multiply the **first** terms. (3 *x* )(2 *x* ) 6 *x*^{2}

Multiply the **outer** terms. (3 *x* )(

5) 15 *x* Multiply the **inner** terms. (1)(2 *x* ) 2 *x* Multiply the **last** terms. (1)(

5) 5 Add the first, outer, inner, and last terms,

and identify the like terms. Combine like terms. 6 *x*^{2}^{13} *x* 5

■^{YOUR TURN} Multiply (2 *x*

3)(5 *x*

2).

(3 *x* +

1)(2 *x* -

5) = 6 *x*^{2 }^{-} 15 *x* + 2 *x* - 5

■^{Answer:}

10 *x*^{2}^{19} *x* 6

Study Tip

When the binomials are of the form

( *ax* *b* )( *cx* *d* ), the outer and inner

terms will be like terms and can be

combined.

**EXAMPLE 5 Multiplying Two Polynomials**

Multiply and simplify . Solution:

Multiply each term of the

first trinomial by the

entire second trinomial. Identify like terms.

Combine like terms. 2 *x*^{4}^{13} *x*^{3}^{30} *x*^{2}^{26} *x* 7

■^{YOUR TURN} Multiply and simplify ( - *x*^{3}_{+} 2 *x* -

4)(3 *x*^{2}_{-} *x* +

5).

= 2 *x*^{4}_{-} 10 *x*^{3}_{+} 14 *x*^{2}_{-} 3 *x*^{3}_{+} 15 *x*^{2}_{-} 21 *x* + *x*^{2}_{-} 5 *x* + 7

= 2 *x*^{2}^{(} *x*^{2}^{-} 5 *x* +

7) - 3 *x* ( *x*^{2}^{-} 5 *x* +

7) + 1( *x*^{2}^{-} 5 *x* +

7)

( *x*^{2}_{-} 5 *x* +

7) (2 *x*^{2}_{-} 3 *x* +

1)

■^{Answer:}

3 *x*^{5}_{x}^{4}_{x}^{3}_{14} *x*^{2}_{14} *x* 20

**EXAMPLE 7 Multiplying Binomials Resulting in Special Products**

Find the following:

**a.**

( *x*

5)( *x*

5)

**b.**

( *x*

5)^{2}

**c.**

( *x*

5)^{2}

**Solution:**

**a.**

( *x* -

5)( *x* +

5) = *x*^{2}_{+} 5 *x* - 5 *x* - 5^{2}_{=} *x*^{2}_{-} 5^{2}_{=} *x*^{2}_{-} 25

32_{CHAPTER 0} Prerequisites and Review

Square of a binomial sum: ( *a* *b* )^{2}_{(} *a* *b* )( *a* *b* ) *a*^{2}_{2} *ab* *b*^{2}

Square of a binomial difference: ( *a* *b* )^{2}_{(} *a* *b* )( *a* *b* ) *a*^{2}_{2} *ab* *b*^{2}

**P ERFECT SQUARES**

( *a* + *b* )( *a* - *b* ) = *a*^{2}^{-} *b*^{2}

**D IFFERENCE OF TWO SQUARES**

**EXAMPLE 8 Finding the Square of a Binomial Sum**

Find ( *x*

3)^{2}_{.}

**INCORRECT**

**ERROR:**

Don’t forget the middle term, which is

twice the product of the two terms in

the binomial.

( *x* +

3)^{2 }^{Z} *x*^{2}^{+} 9

**CORRECT**

( *x*

3)^{2}_{(} *x*

3)( *x*

3) x^{2}_{3} *x* 3 *x* 9 x^{2}^{6} *x* 9

Forgetting the middle term, which is *twice* the product of the two terms in

the binomial.

**C _{O M M O N}^{M} I S TA K E**

★

Let *a* and *b* be any real number, variable, or algebraic expression in the following

special products.

Some products of binomials occur frequently in algebra and are given special names.

Example 7 illustrates the *difference of two squares* and *perfect squares* .

r

First

r

Inner

s

Difference of two squares

r Outer r Last

r

First

r

Inner

r Outer

r

First

r

Inner

r Outer r Last

r Last

**b.**

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- Summer '17
- juan alberto
- Algebra, Distributive Property, GCF