### Unit 9: Factoring Polynomials

Factoring, the procedure you will learn to perform in this unit, is multiplication in reverse: instead of multiplying two polynomials, you will need to write a given polynomial as a product of two or more different expressions. Factoring is an important tool that you will use in solving quadratic, and, later, higher degree polynomial equations. Quadratic equations occur a lot in problems that involve rectangular objects and their areas: planning gardens, framing photographs, carpeting the floors, and so on.

**Completing this unit should take you approximately 21 hours.**

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

- rewrite polynomial as a product of two or more polynomials;
- identify polynomials that cannot be factored;
- choose appropriate factoring strategy for a given polynomial;
- solve quadratic equations by factoring; and
- create quadratic equations and use them to solve problems.

### 9.1: Factoring Monomial from a Polynomial

### 9.1.1: Identifying Greatest Common Factor (GCF) of Two or More Monomials and of Other Expressions

Read this page. Reading and understanding algebraic expressions, much like translating sentences from a foreign language, is a skill that takes time to develop. Prior to learning how to factor an algebraic expression (that is, to write the expression as a product), it is essential to have fluency in distinguishing which parts of the expressions are multiplied and which are added/subtracted. This reading provides a useful review of how to identify which algebraic expression is a product of several factors and which one is a sum of several terms.

Click on the "new problem" button at the end of the page to try a practice problem. After answering the problem, click on the "check your answer" button. Continue this process by clicking on "new problem," and solve 10 problems. You can also create a worksheet of 10 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. This set of examples provides practice with identifying common factors, both monomial and binomial, in algebraic expressions. Click on the "new problem" button at the end of the page to try a practice problem. After answering the problem, click on the "check your answer" button. Continue this process by clicking on "new problem," and solve 10 problems. You can also create a worksheet of 10 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.

### 9.1.2: Rewriting Polynomial as a Product of a Monomial and a Different Polynomial

Now that you have practice identifying common factors, you can begin to factor polynomials by rewriting them as a product of a common factor and another polynomial. This process, as Dr. Burns points out below in "Factoring Simple Expressions," is the result of the distributive property of real numbers applied in the opposite direction: instead of multiplying a number by a parenthesis, you will be writing the given result as a product of a number and a parenthesis.

Read this page. Note that some examples involve factoring out a common binomial factor.

Try a few simple factoring problems yourself: click on the "new problem" button at the end of the page to try a practice problem. After answering the problem, click on the "check your answer" button. Continue this process by clicking on "new problem," and solve 10 problems. You can also create a worksheet of 10 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 the section titled "Finding the Greatest Common Binomial Factor." Work through the examples and Guided Practice, and complete practice problems 1-10. Watch the "Polynomial Equations in Factored Form" video embedded in the text, if you need help. Once you have completed the practice problems, check your work against the answer key.

Complete this exercise. Determine if the terms of the given expression have a common factor. If they do, write the factored expression in the tab on the right side of the page. If not, your answer will be the same as the original expression. Select "Check Answer" to see if your answer is correct. If it is incorrect, you will be prompted to try again. If it is correct, you can click on "Correct! Next Question" to move to the next problem.

### 9.2: Factoring Polynomials by Grouping

Watch the first 6 minutes of the video and take notes. Note the steps taken to factor a four-term polynomial.

- Separate the polynomial into two groups of two terms.
- Identify a common factor in each of the group and factor it out.
- Check that the resultant expressions contain a common binomial factor. If they do not, then the polynomial cannot be factored, at least not when the terms are grouped this way.
- Factor out a common binomial factor and rewrite the polynomial as a product of two binomials.

- Separate the polynomial into two groups of two terms.
Scroll down to the section titled "Factoring by Grouping," and work through Example 3. Then, scroll down to the Practice Set and solve problems 11-15, 17, 18, and 23-27. All of these problems are four-term polynomials that can be factored by grouping. Once you have completed the practice problems, check your answers against the answer key.

### 9.3: Factoring Quadratic Trinomials

### 9.3.1: Factoring Trinomials of the Form x^2 + bx + c

### 9.3.1.1: Revisiting FOIL: Working Backwards

You have seen that a polynomial can be written as a product of a monomial and a different polynomial due to distributive property. In this sub-subunit, you will review the FOIL procedure for multiplying two binomials in order to analyze how it can be reversed in order to write the given result (a trinomial) as a product of two binomials.

Read this page.

### 9.3.1.2: Factoring Trinomials of the Form x^2 + bx + c When c Is Positive

Read this page. Note that in one of the examples the trinomial cannot be factored. Then, try factoring trinomials yourself: click on the "new problem" button at the end of the page to try a practice problem. After answering the problem, click on the "check your answer" button. Continue this process by clicking on "new problem," and solve 10 problems. You can also create a worksheet of 10 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. Sal Khan uses both inverse FOIL reasoning and factoring by grouping to factor a quadratic trinomial of the form x

^{2}+ bx + c with positive c.Complete this exercise. Factor each trinomial and enter the result in the tab on the right side of the page. Select "Check Answer" to see if your answer is correct. If it is incorrect, you will be prompted to try again. If it is correct, you can click on "Correct! Next Question" to move to the next problem.

### 9.3.1.3: Factoring Trinomials of the Form x^2 + bx + c When c Is Negative

Read this page. Note that in one of the examples the trinomial cannot be factored. Then, try factoring trinomials yourself: click on the "new problem" button at the end of the page to try a practice problem. After answering the problem, click on the "check your answer" button. Continue this process by clicking on "new problem," and solve 10 problems. You can also create a worksheet of 10 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 the first 7 minutes of the first video (introduction and four examples), and take notes. This video reviews the procedure for factoring quadratic trinomials of the form x

^{2}+ bx + c. The second video shows four examples of factoring polynomials of the form ax^{2}+ bx + c.

### 9.3.2: Factoring Trinomials of the Form ax^2 + bx + c

There are two methods traditionally used to factor quadratic trinomials of the form

*ax*using trial factors (also known as^{2}+ bx + c:*guess and check*) and by grouping (also known as*ac*, or bridge method). You will try out both methods and can use a preferred method in the future. You will also develop a sense of when a trinomial can be factored instantly by guessing and when this would take a long time and factoring by grouping is a better choice.

### 9.3.2.1: Using Trial Factors ("Guess and Check" Method)

Read the page until the section titled "Example: 'Factor by Grouping' Method." Try factoring trinomials using this method: click on the "new problem" button at the end of the page to try a practice problem. After answering the problem, click on the "check your answer" button. Continue this process by clicking on "new problem," and solve five problems. 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.

While Dr. Burns points out in this reading that listing all trial factors and checking their products can be tedious, some trinomials can be factored fairly quickly using this method. For example, if either a or c or both are prime, their only factors are 1 and itself, and this limits the number of trial factors. Also, if c is positive, both binomial factors will have to contain the same sign, and this limits the number of trial factors as well.

Watch the first five minutes of the first video (introduction and two examples), and take notes. This video explains how to factor the trinomials of the form ax

^{2}+ bx + c by guessing and checking. The second video shows two more examples of factoring the trinomials of the form ax^{2}+ bx + c by guessing and checking.

### 9.3.2.2: Factoring by Grouping (Bridge or ac Method)

Revisit this resource. Read the section titled "Example: 'Factor by Grouping' Method." After working through the example, try factoring trinomials using this method yourself: click on the "new problem" button at the end of the page to try a practice problem. After answering the problem, click on the "check your answer" button. Continue this process by clicking on "new problem" and solve five problems. 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.

This page will guide you through factoring trinomials by grouping one step at a time. Try to factor at least five trinomials or more. You can also click on "Information on the AC method" at the bottom of the page and scroll down to the section titled "The AC Method" to review the procedure for the method, examples, and more exercises.

### 9.3.2.3: When a Is Negative

Work through Example E. In this example, the issue of factoring a trinomial with a = -1 is avoided by factoring (-1) out. Then, the resultant trinomial has the form of x

^{2}+ bx + c and can be factored as such. Then, work on practice problems 13-16 and 18. Once you have completed the practice problems, check your answers against the answer key.Complete this exercise to practice factoring various trinomials by grouping.

### 9.3.3: Factoring Trinomials of the Form x^2 + bxy + cy^2 and ax^2 + bxy + cy^2

Watch these videos and take notes. You will see examples of a trinomial containing two variables being factored. Note that the inverse FOIL reasoning can still be applied to determine how the binomial factors are going to look like. Inverse FOIL reasoning, together with the method of trial factors, is again applied to determine how the binomial factors are going to look like.

Complete this exercise. Solve the problem and type your answer in the answer tab.

### 9.3.4: Prime Trinomials

Read this brief example of prime trinomials, and then attempt the practice problems on the third page. When you have finished, you may check your answers against the answer key.

### 9.4: Special Factoring

Recall that the products of some binomial result in special kinds of polynomials, called complete square trinomial and difference of two squares. Because you are working on factoring, which is a procedure inverse to multiplying, and you are given a polynomial that looks like a complete square or difference of two squares, the polynomial can be factored immediately with the help of classic quadraticformulas (used in the reverse direction):

a

^{2}± 2ab +b^{2}= (a ± b)^{2}a^{2}- b^{2}= (a - b)(a + b)There also will be two new special factoring formulas introduced in this subunit, known as sum and difference of two cubes:

a

^{3}+ b^{3}= (a + b)(a^{2}- ab + b^{2}) a^{3}- b^{3}= (a - b)(a^{2}+ ab + b^{2})

### 9.4.1: Identifying and Factoring Complete Square Trinomials

Read the article until the section titled "Solving Quadratic Polynomial Equations by Factoring." Watch the videos embedded in the text, and work through Guided Practice Examples. Then, complete practice problems 1-8. Once you have completed the practice problems, check your answers against the answer key.

Watch this video and take notes. This video shows how to check whether the given trinomial is a complete square and factor it if it is. (If it is not, factoring by grouping should be attempted.)

### 9.4.2: Identifying and Factoring Difference of Two Squares

Read the article, watch the videos embedded in the text and work through Guided Practice Examples. Then, complete practice problems 1-10. Once you have completed the practice problems, check your answers against the answer key.

Watch this video and take notes. This video shows how to check whether the given binomial is a difference of two squares and factor it if it is.

Complete this exercise to practice factoring simple Difference of Two Squares binomials.

Complete this exercise. It contains slightly more complicated Difference of Two Squares binomials.

### 9.4.3: Identifying and Factoring Sum and Difference of Two Cubes

Watch these videos and take notes. In these videos, the sum of two cubes formula is proven by using polynomial division and there is an example of factoring a binomial that is a difference of two cubes.

Watch this video and take notes. In this video, Sal Khan shows why the sum of two cubes formula is true (in a different way than in Dr. Sousa's video) and uses it to factor a binomial.

### 9.5: Factoring General Polynomials

Read the section titled "Guidance," and work through Example A. Scroll down to the Guided Practice, and work through the examples. Then, complete practice problems 1-10. Once you have completed the practice problems, check your answers against the answer key.

Watch these videos and take notes. The first video shows an example of a trinomial that requires two factoring steps in order to be factored completely: factoring out a common monomial factor and factoring by grouping. The second shows an example of a binomial that requires two factoring steps in order to be factored completely: factoring out a common monomial factor and using the difference of two cubes formula.

Watch these videos and take notes. In the first example, the difference of two squares formula is applied twice in order to factor the expression completely: once to the original polynomial, and then to the new binomial factor. The second is an example of an interesting polynomial that can be attempted to be factored either by using the difference of two squares or difference of two cubes formulas. Whichever one you choose, the resultant factors can still be factored further using another special binomial factoring formula. In this video, the difference of two squares formula is used first. As an exercise, try an alternative method (e.g. applying difference of two cubes first) and try to show that the results will in fact be the same.

Complete exercises 3-13. This exercise set will allow you to assess your mastery of factoring polynomials. Click on the "Show Solution" link next to each problem to check your answer.

Complete this exercise. These trinomials require two steps to be factored completely - factoring out the greatest common factor and then factoring the remaining trinomial.

Complete this exercise. It contains binomials that require two steps to be factored - factoring out the greatest common factor and using the Difference of Two Squares formula.

Read these examples and practice problems on factoring a polynomial. The answers to the practice problems are provided at the end of the slides.

### 9.6: Solving Quadratic Equations by Factoring

In Unit 2, you learned how to solve all kinds of linear equations, or equations where the highest degree of the unknown variable is 1. In this subunit, you will be introduced to one of the methods of solving quadratic equations, or equations where the highest degree of the unknown variable is 2 - factoring. Factoring breaks up a quadratic equation into two linear equations. Thus, a quadratic equation that can be solved by factoring will generally have two solutions.

### 9.6.1: Principle of Zero Products and Identifying Solutions

Watch the first five minutes of the video, and take notes. In this video, the sum of two cubes formula is proven by using polynomial division. Also, there is an example of factoring a binomial that is a difference of two cubes.

### 9.6.2: Factoring before Solving

Watch this video and take notes. In this video, you will see a few examples solved by various methods of factoring.

Work through the examples. Then, click on the "new problem" button at the end of the page to try a practice problem. After answering the problem, click on the "check your answer" button. Continue this process by clicking on "new problem," and solve 10 problems. You can also create a worksheet of 10 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.

Work through the examples. Then, click on the "new problem" button at the end of the page to try a practice problem. After answering the problem, click on the "check your answer" button. Continue this process by clicking on "new problem," and solve 10 problems. You can also create a worksheet of 10 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.

### 9.6.3: Solving Application Problems

Read the article, and watch the videos embedded in the text. Then, complete practice problems 1-9. Once you have completed the practice problems, check your answers against the answer key.