### Unit 8: Operations with Polynomials

Polynomials are a special type of algebraic expression that contain two or more terms. For example, a polynomial might look like 5*x* + 2*x*^{3} + *x*^{2} + 2. In this unit, we discuss how to recognize, classify, add, subtract, multiply, and divide polynomials by combining like terms and using the distributive property. These skills help us make calculations that pertain to the motion of two or more objects. For example, we can calculate when and where a runner will overtake a competitor, or how much interest you will earn from two or more savings accounts.

**Completing this unit should take you approximately 4 hours.**

Upon successful completion of this unit, you will be able to:

- identify the degree of a polynomial;
- classify polynomial according to the number of terms;
- add, subtract, and multiply polynomials;
- divide a polynomial by a monomial and a binomial; and
- identify special products of binomials by completing the square and finding the difference of two squares.

### 8.1: Classifying Polynomials

When we discuss polynomials, we are referring to an algebraic expression that includes more than one term. We classify polynomials based on the number of terms in the expression.

Watch this video to review the terminology we will use in this unit.

### 8.2: Adding and Subtracting Polynomials

Like all mathematical expressions, we can perform basic mathematical operations on polynomials. In this section we learn how to add and subtract polynomials using different common methods.

Read this page, which reviews how to classify polynomials. Pay attention to the section on adding and subtracting polynomials. You can rearrange the order of terms in a polynomial to make addition or subtraction possible. Read example 6.7 and its solution to see an example of adding polynomials.

After you review, complete examples 6.8 to 6.12 and check your work.

Watch these videos for more examples of adding and subtracting polynomials.

### 8.3: Multiplying a Polynomial by a Monomial

We can multiply polynomials by monomials or other polynomials. In this section, we will see how to multiply a polynomial by a monomial. We can use the distributive property to do this.

Read this article and watch the videos. It may be helpful to review the section on multiplying monomials. Then, focus on how to use the distributive property and the examples of using it.

After you review, complete review problems 6 to 11 and check your answers.

### 8.4: Multiplying Binomials (FOIL)

When multiplying two binomials (polynomials with two terms), we cannot simply use the distributive property. We need to ensure that each term in one binomial is multiplied by each term in the other binomial. The technique we use to do this is called FOIL. FOIL is a mnemonic that stands for

**F**irst,**O**uter,**I**nner,**L**ast. When we place two binomials next to each other, FOIL reminds us of the order we multiply the terms in.Read this article, which gives many examples of using the FOIL technique to multiply two binomials. Then, try some practice problems.

Watch this video, which gives another example of using the FOIL technique.

### 8.5: Complete the Square and Difference of Two Squares

We sometimes need to take the square of a binomial. In this section, we will look at the specific rules for squaring binomials. These rules are derived from the FOIL method.

Watch this video, which takes square of a binomial in two different ways: using the FOIL method and the quadratic formula. Both methods give the same answer.

Read this article, which gives examples of squaring binomials and the general formulas that always work for them. Example 1 shows how to use the formula. Note the definitions of sum and difference, which are used when multiplying one addition binomial and one subtraction binomial. Example 2 shows how to use this definition.

After you read, complete questions 1 through 5 and 14 through 18 in the practice set and check your answers.

### 8.6: Multiplying Polynomials with Any Number of Terms

Polynomials can have more than two terms, and we will need to know the methods for multiplying these larger polynomials.

Watch this video, which shows a method for ensuring that all terms in the polynomial (in this case a trinomial) are multiplied by both of the terms in the binomial using the distributive property.

Read the section on multiplying a trinomial by a binomial. Review the solution to example 6.45 to see how the distributive property can be used to multiply.

After you read, complete exercises 267, 269, 271, and 273 and check your answers.

### 8.7: Dividing a Polynomial by a Monomial

We also need to be able to divide polynomials. In this section, we will review some techniques for dividing a polynomial by a monomial.

Watch this example of dividing a polynomial by a monomial. This video gives an example of the technique we use to divide.

Read the section on dividing a polynomial by a monomial. Review the solutions for examples 6.77 and 6.78, which show how to divide a polynomial by a number and by a monomial.

After you read, complete examples 6.79 to 6.83 and check your answers.

### 8.8: Dividing a Polynomial by a Binomial

The last mathematical operation we need to discuss is dividing polynomials by binomials.

Read the section on dividing a polynomial by a binomial. Pay attention to the review of long division, since it will help you understand the technique. Review the solution to example 6.84 to see how to divide a trinomial by a binomial. Then, review the solution to example 6.85 to see how we handle dividing by a subtraction binomial. Be careful, and make sure you keep track of the negative sign.

After you study these examples, complete questions 6.167 through 6.170 in the Try It section.Read the section on dividing a polynomial by a binomial. Pay attention to the review of long division, since it will help you understand the technique. Review the solution to example 6.84 to see how to divide a trinomial by a binomial. Then, review the solution to example 6.85 to see how we handle dividing by a subtraction binomial. Be careful, and make sure you keep track of the negative sign.

After you study these examples, complete questions 6.167 through 6.170 in the Try It section.