## Easy Algebra Step-by-Step: Master High-Frequency Concepts and Skills for Algebra Proficiency—FAST! (2012)

### Chapter 9. Multiplying Polynomials

This chapter presents rules for multiplying polynomials. You use the properties of real numbers and the rules of exponents when you multiply polynomials.

**Multiplying Monomials**

** Multiplying Monomials**

To multiply monomials, (1) multiply the numerical coefficients, (2) multiply the variable factors using rules for exponents, and (3) use the product of the numerical coefficients as the coefficient of the product of the variable factors to obtain the answer.

**Problem** Find the product.

f. **( x)(2)**

**Solution**

*Step 1*. Multiply the numerical coefficients.

*Step 2*. Multiply the variable factors.

To streamline your work when you are multiplying polynomials, arrange the variables in each term alphabetically.

*Step 3*. Use the product in step 1 as the coefficient of *x*^{7} *y*^{9}.

Recall from __Chapter 7__ that when you multiply exponential expressions that have the same base, you *add* the exponents.

*Step 1*. Multiply the numerical coefficients.

*Step 2*. Multiply the variable factors.

If no exponent is written on a variable, the exponent is understood to be 1.

*Step 3*. Use the product in step 1 as the coefficient of *a ^{4} b^{6}*.

*Step 1*. Multiply the numerical coefficients.

*Step 2*. Multiply the variable factors.

*Step 3*. Use the product in step 1 as the coefficient of *x*^{2}.

*Step 1*. Multiply the numerical coefficients.

*Step 2*. Multiply the variable factors.

*Step 3*. Use the product in step 1 as the coefficient of *x*^{5}.

*Step 1*. Multiply the numerical coefficients.

*Step 2*. Multiply the variable factors.

*Step 3*. Use the product in step 1 as the coefficient of *x ^{4} y^{9}*.

*Step 1*. Multiply the numerical coefficients.

(1)(2) = 2

*Step 2*. Multiply the variable factors.

There is only one variable factor, *x*.

*Step 3*. Use the product in step 1 as the coefficient of *x*.

(*x*)(*2*) = 2*x*

**Multiplying Polynomials by Monomials**

** Multiplying a Polynomial by a Monomial**

To multiply a polynomial by a monomial, multiply each term of the polynomial by the monomial.

This rule is a direct application of the distributive property for real numbers.

**Problem** Find the product.

**Solution**

*Step 1*. Multiply each term of the polynomial by the monomial.

*Step 1*. Multiply each term of the polynomial by the monomial.

*Step 1*. Multiply each term of the polynomial by the monomial.

Be careful! Watch your exponents when you are multiplying polynomials by monomials.

*Step 1*. Multiply each term of the polynomial by the monomial.

**Multiplying Binomials**

** Multiplying Two Binomials**

To multiply two binomials, multiply all the terms of the second binomial by each term of the first binomial and then simplify.

**Problem** Find the product.

**Solution**

*Step 1*. Multiply all the terms of the second binomial by each term of the first binomial.

Don’t forget about

*Step 2*. Simplify.

*Step 1*. Multiply all the terms of the second binomial by each term of the first binomial.

*Step 2*. Simplify.

There are no like terms, so is simplified.

**The FOIL Method**

From the last problem, you can see that to find the product of two binomials, you compute four products, called *partial products*, using the terms of the two binomials. The FOIL method is a quick way to get those four partial products. Here is how FOIL works for finding the four partial products for

1. Multiply the two **F**irst terms: *a · c*.

2. Multiply the two **O**uter terms: *a · d*.

3. Multiply the two **I**nner terms: *b · c*.

4. Multiply the two **L**ast terms: *b · d*.

Be aware that the FOIL method works only for the product of two binomials.

Notice that FOIL is an acronym for first, outer, inner, and last. The inner and outer partial products are the *middle terms*.

Forgetting to compute the middle terms is the most common error when finding the product of two binomials.

**Problem** Find the product using the FOIL method.

**Solution**

*Step 1*. Find the partial products using the acronym FOIL.

*Step 2*. Simplify.

*Step 3*. State the main result.

*Step 1*. Find the partial products using the acronym FOIL.

(*x* – 2)(*x* – 5)

*Step 2*. Simplify.

*Step 3*. State the main result.

*Step 1*. Find the partial products using the acronym **FOIL**.

(*x* – 2)(*x* + 5)

*Step 2*. Simplify.

*Step 3*. State the main result.

*Step 1*. Write as a product.

*Step 2*. Find the partial products using the acronym **FOIL**.

Don’t forget the middle terms!

*Step 3*. Simplify.

*Step 4*. State the main result.

**Multiplying Polynomials**

** Multiplying Two Polynomials**

To multiply two polynomials, multipl*y* all the terms of the second polynomial by each term of the first polynomial and then simplify.

**Problem** Find the product.

**Solution**

*Step 1*. Multiply all the terms of the second polynomial by each term of the first pol*y*nomial.

*Step 2*. Simplify.

*Step 1*. Multipl*y* all the terms of the second pol*y*nomial b*y* each term of the first pol*y*nomial.

*Step 2*. Simplify.

*Step 1*. Multipl*y* all the terms of the second pol*y*nomial b*y* each term of the first pol*y*nomial.

*Step 2*. Simplify.

**Special Products**

The answer to the last problem is an example of the “difference of two cubes.” It is a special product. Here is a list of *special products* that you need to know for algebra.

** Special Products**

**Perfect Squares**

Memorizing special products is a winning strategy in algebra.

**Difference of Two Squares**

**Perfect Cubes**

**Sum of Two Cubes**

**Difference of Two Cubes**.

**Exercise 9**

*Find the product*.