### Unit 8: Operations with Polynomials

In this unit, you will become familiar with a special type of algebraic expressions, called a polynomial. A polynomial, as opposed to a monomial, is an expression that contains two or more terms. (The word poly means many in Greek.) Usually the polynomials you will work with will look like *x*^{5} + 2*x*^{3} + *x* + 2, or similar. Polynomials have various special properties that you will be analyzing in the future math courses. In this course, you will learn how to recognize, classify, add, subtract, multiply, and divide polynomials. You will apply the skills of combining like terms and using distributive property in order to perform these operations. These skills are helpful when you are dealing with the motion of two or more objects: for example, when you need to calculate when and where one runner will overtake another runner. For another example, these skills are useful if you need to know how much interest you are earning from two or more savings accounts.

**Completing this unit should take you approximately 14 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. (Complete Square and Difference of Two Squares).

### 8.1: Classification of Polynomials

In this subunit, you will be introduced to a certain kind of algebraic expression called a polynomial expression. You will learn how to identify polynomials and classify them according to their degree and number of terms.

Read the page until the section titled "Classification of Polynomial Equations." Note that while it is not necessary to memorize all the new vocabulary words you will encounter, you should know their meaning, as they will be used often in all the materials in this course from now on. After reading and working through the examples, scroll down to the "Exercises" section and complete exercises 10-34. The solutions to the exercises are shown directly below each problem.

Watch this video and take notes. This short video shows that the total value of a variable number of $20, $10, and $5 bills is a polynomial expression. One can substitute the number of each type of bills into this polynomial to calculate the total amount of money.

### 8.2: Addition and Subtraction of Polynomials

### 8.2.1: Horizontal Format

Watch this video and take notes. In this video, you will see examples of polynomial expressions of various degrees and number of terms. Then, you will work through two examples of adding and subtracting two polynomials. You will notice that all you have to do to add or subtract polynomials horizontally is open the parentheses (in case of subtraction) and combine like terms. This video also contains an example of applications of polynomials in geometry.

Watch this video and take notes. In this video, adding and subtracting polynomials is performed in both horizontal and vertical formats. Note that in the vertical format, the terms of the same degree (or like terms) are aligned one under another, much like the digits of the same place value are aligned in addition or subtraction of decimals and large numbers.

### 8.2.2: Vertical Format

Watch this video and take notes. In this video, adding and subtracting polynomials is performed in both horizontal and vertical formats. Note that in the vertical format, the terms of the same degree (or like terms) are aligned one under another, much like the digits of the same place value are aligned in addition or subtraction of decimals and large numbers.

Watch this video and take notes. In this video, two polynomials, each containing two variables, are added vertically. Note how like terms are aligned under each other and the rest of the terms are simply rewritten as they cannot be combined.

Scroll down to the section titled "Practice Problems," and complete problems 1-10. Use either horizontal or vertical format of adding/subtracting. Make sure to write the result as a polynomial in standard form. Once you have completed the practice problems, check your answers against the answer key.

Scroll down to Sample Set D, and work through examples 18-21. Then, complete exercises 25-28 from Practice Set D. To check your answers, click on "Show Solution" link next to each problem.

### 8.3: Multiplying Polynomials

### 8.3.1: Multiplying Polynomial by a Monomial

Scroll down to "Example B," and work through Example B, Example C, and Guided Practice. Watch the videos embedded in the text, if you need help with these examples. Then, complete practice problems 6-11. Once you have completed the practice problems, check your answers against the answer key.

### 8.3.2: Multiplying a Polynomial by a Polynomial

When you multiply polynomials, you have to distribute each term of the first polynomial over the second polynomial, which means you have to multiply each term of the first polynomial by each term of the second polynomial. The resources in this sub-subunit will show you various techniques for keeping track of all the resultant terms. The multiplication result will usually contain like terms, which can be combined.

### 8.3.2.1: Multiplying Binomials (FOIL)

When you multiply a binomial by a binomial (two terms by two terms), your result will have to have four terms. Sometimes two of these terms will be alike, and you will have to combine them to simplify the result. Depending on the binomials you are multiplying, the final result might be a binomial, a trinomial, or a four-term polynomial.

Read this page. This reading explains a mnemonic for multiplying binomials, known as FOIL, which stands for First, Outer, Inner, Last, the order in which the terms are multiplied. Practice the basic binomial multiplication problems on this page in order to gain mastery and to be able to do these types of problems quickly.

Click on the "new problem" button at the end of the reading to try a practice problem, and check your answer. Continue this process by clicking on "new problem" five times. You can also create a worksheet of five problems by clicking on the button at the bottom of the page titled "Click Here for a Randomly Generated Worksheet and Answers." The answers will be provided at the end of the worksheet.

Watch this video and take notes. In this video, Sal Khan uses the distributive property to multiply two binomials and then shows that if the order of the binomials in the multiplication problem is switched, then the result remains the same.

Read this page. Here, you will find slightly more complicated binomial multiplication examples. Click on the "new problem" button at the end of the reading to try a practice problem, and check your answer. Continue this process by clicking on "new problem" five times. You can also create a worksheet of five problems by clicking on the button at the bottom of the page titled "Click Here for a Randomly Generated Worksheet and Answers." The answers will be provided at the end of the worksheet.

### 8.3.2.2: Special Products of Binomials

Special products of binomial are the results of the following multiplication:

Multiplying a binomial by itself (in other words, squaring it):

(a ± b)

_{2}= a_{2}± 2ab +b_{2}A trinomial in the form of a trinomial on the right-hand side of the above formula is known as a complete square trinomial.

Multiplying the binomials that are the sum and difference of the same terms:

(a - b)(a + b) = a

_{2}- b_{2}A binomial in the form of a binomial on the right-hand side of the above formula is known as a difference of two squares.

These formulas, sometimes also called classic quadratic formulas, are important to remember. Recognizing when they can be applied is a necessary skill for this and many other higher-level math courses.

### 8.3.2.2.1: Complete Square

Watch this video and take notes. In this video, Sal Khan performs the multiplication of a binomial by itself twice: once using FOIL and once using a classic quadratic formula (which he also derives). This will help you understand why this formula always works.

Read this page. This reading provides practice with squaring simple binomials. Click on the "new problem" button at the end of the reading to try a practice problem, and check your answer. Continue this process by clicking on "new problem" five times. You can also create a worksheet of five problems by clicking on the button at the bottom of the page titled "Click Here for a Randomly Generated Worksheet and Answers." The answers will be provided at the end of the worksheet.

Read this page, which contains examples of squaring all kind of binomials. Scroll down to the Practice Set, and complete exercises 1-13. Use the "Special Products of Binomials" video embedded in the text for guidance, if you need help. Once you have completed the practice problems, check your answers against the answer key.

### 8.3.2.2.2: Difference of Two Squares

Watch this video and take notes. In this video, Sal Khan performs the multiplication of a binomial by the binomial with same terms but the opposite sign between them twice: once using FOIL and once using a classic quadratic formula (which he also derives). This video will help you understand why this formula always works.

Briefly review this page, which contains examples of applying the difference of two squares formula to all kind of binomials. Scroll down to the Practice Set, and complete exercises 14-23. Use the "Special Products of Binomials" video embedded in the text for guidance, if you need help. Once you have completed the practice problems, check your answers against the answer key.

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

Complete exercises 13-28. This exercise set will allow you to assess your mastery of classification and operations with polynomials (except for division). Once you have completed the practice problems, check your answers against the answer key.

Watch this video and take notes. The video contains an example of multiplication of a binomial of a trinomial. Note the method used to make sure each term of one polynomial is multiplied by each term of another.

Scroll down to the Practice Set, and solve problems 7-21. This set of problems contains exercises involving multiplying polynomials with any number of terms. Some of the problems involve multiplying three polynomials. In this case, multiply any two of the three polynomials first and then multiply the third polynomial by the simplified result. Once you have completed the practice problems, check your answers against the answer key.

### 8.4: Dividing Polynomials

Because polynomials can be multiplied, it follows that they can also be divided. The quotient, or the result of the division, of two polynomials, can be defined the same way as a quotient of two real numbers: it is a polynomial such that if it is multiplied by the divisor, then the result is the original polynomial. To find such a polynomial by guessing and checking, however, is not practical. In this subunit, you will learn the technique for dividing two polynomials. You will find that it is somewhat similar to long division of real numbers.

### 8.4.1: Dividing a Polynomial by a Monomial

Read the section titled "Dividing a Polynomial by a Monomial," work through the examples in Sample Set A, and complete the exercises in Practice Set A. The solutions to the exercises are shown directly below each problem.

Watch this video and take notes. This video shows an example of dividing a polynomial by a monomial.

### 8.4.2: Dividing a Polynomial by a Binomial

Watch these videos and take notes. The first video shows an example of dividing a quadratic trinomial by a binomial. Note that the result can be checked by multiplication, just like as the result of the division of real numbers. The second video introduces the method of dividing polynomials known as long division. Sal Khan first tries out this method by dividing a simple binomial by a monomial, which you already know how to do. Then, he generalizes the procedure in order to divide a trinomial by a binomial. Note that he checks his result later by factoring, a procedure you will learn about in Unit 9. You will see more examples of long division in the next two Khan Academy videos as well. The third video shows an example of dividing a cubic four-term polynomial by a binomial. This time, the long division method produces a remainder.

Scroll down to Sample Set B, work through examples 4 and 5, and then work through example 6 in Sample Set C. (Note that example 6 focuses on dividing a polynomial where one of the coefficients is zero.) Then, complete the exercises 6-9 in Practice Set B and exercises 10-13 in Practice Set C. The solutions to the exercises are shown when you click on the "Show Solution" link.

Complete exercises 1-8. Pay careful attention to the directions in each exercise, as they are not all the same. Once you have completed the exercises, check your answers against the answer key.