Unit 9: Factoring Polynomials
Factoring is multiplication in reverse: rather than multiplying two polynomials, you write a given polynomial as a product of two or more different expressions. Factoring is an important tool for solving advanced equations, such as quadratic equations. Quadratic equations occur in problems that involve rectangular objects and their areas, such as planning gardens, framing photographs, or carpeting a floor.
Completing this unit should take you approximately 7 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; and
- solve quadratic equations by factoring.
9.1: Identifying Greatest Common Factor
Reading and understanding algebraic expressions, much like translating sentences from a foreign language, is a skill that takes time to develop. Before learning how to factor an algebraic expression (or write the expression as a product), you need to be able to break down the parts.
Read this short article which provides an overview of the language and definitions you need to understand for factoring.
Read this page up to the section on factoring by grouping. Factoring takes advantage of the distributive property in the opposite way from how we did it in Unit 8.
After you read, complete examples 7.1 through 7.7 and check your work.
9.2: Factoring Polynomials by Grouping
There are four steps you need to take to factor a four-term polynomial:
- Separate the polynomial into two groups of two terms.
- Identify a common factor in each member of the group and factor it out.
- Check that the resulting 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.
Work through example 3 in the section on factoring by grouping, and solve problems 11 to 15, 17, 18, and 23 through 27 in the practice set. All of these problems are four-term polynomials that you can factor by grouping. After you complete the practice problems, check your answers.
9.3: Revisiting FOIL by Working Backwards
When we factor binomials, we are essentially "undoing" the FOIL technique we learned in the last unit.
Read this article. The section Key Tools for Factoring Trinomials gives a brief overview of how factoring works, and Key Ideas for Finding the Numbers that Work also gives some basic rules to keep in mind when factoring.
9.4: Factoring Trinomials of the Form x² + bx + c when c is Positive
We will study how to factor trinomials in two different forms: when c is positive and when c is negative. In this section, we will study how to factor trinomials with a positive c value.
After you watch, complete this assessment and check your answers.
9.5: Factoring Trinomials of the Form x² + bx + c when c is Negative
Now we can study trinomials where c is a negative number. These can be more complicated.
Read this article. Pay attention to the overview of the steps for these problems. Read the examples and note where factoring is not possible, like in the last example. After you have reviewed the materials, complete a few practice problems until you feel comfortable. Be sure to check your answers when you are done.
9.6: Factoring by Grouping
One useful method for factoring polynomials is factoring by grouping, using the bridge or the AC method.
Watch these two videos which go through examples of using the factoring by grouping method.
After you have watched the videos, complete this assignment and check your answers.
9.7: Factoring Trinomials of the Form x² + bxy + cy² and ax² + bxy + cy²
The previous examples of factoring trinomials only included one variable. Often, algebraic expressions include more than one variable, and we need ways to factor polynomials with two variables. The methods used to factor trinomials with two variables are based on the same principles as those with one variable.
Watch these videos, which demonstrate different methods for factoring trinomials with two variables.
After you have watched the three videos, complete this assignment and check your answers.
9.8: Identifying and Factoring Complete Square Trinomials
We can identify some trinomials as perfect squares. A perfect square trinomial has the form a2 + 2ab + b2 or a2 − 2ab + b2. Because each term is a square, they can be factored into (a + b)2 or (a − b)2, respectively. It is important to be able to identify trinomials in this form to easily factor them after determining the values of a and b.
Read this article up to the section on solving quadratic polynomial equations by factoring. Watch the videos and work through the guided practice examples.
After you review, complete practice problems 1 through 8 and check your answers.
This video shows how to check whether the given trinomial is a complete square. Factor it if it is. You should try to factor by grouping if it is not.
9.9: Identifying and Factoring the Difference of Two Squares
A special case of polynomials is the difference of two squares. For example, the equation x2 − 4 is a difference of two squares because it can be written as x2 − 22.
Read this article, which reviews the sum, difference, and square of the binomial equations we discussed earlier. Make sure you are comfortable with this before moving on. Pay attention to the examples of factoring the difference of two squares, since they show how to use the method.
After you read, complete review questions 1 through 5 and check your answers.
9.10: Identifying and Factoring the Sum and Difference of Two Cubes
Much like factoring the difference of two squares, we can use special techniques to factor the sum and difference of two cubes. An example of the sum of two cubes is x3 + 27, as this can be written as x3 + 33.
Watch this video to see an example of how we treat this special case of factoring.
9.11: Factoring General Polynomials
Sometimes factoring a polynomial requires more than one step. When this occurs, we combine the different techniques we have learned to complete factor the complex polynomial.
Watch these videos. Factoring a trinomial requires two steps. First, a common monomial factor is factored out, and then the resulting polynomial is factored by grouping. The second video gives an example of a multi-step factoring process by factoring out a common monomial and then using the difference of cubes formula.
This article gives more examples and introduces the idea of "prime polynomials" that cannot be factored.
9.12: The Principle of Zero Product and Identifying Solutions
The principle of zero product tells us that if the product of two quantities is 0, one of the quantities must also be 0.
Read the section on solving quadratic equations using the zero product property. Review the solution to example 7.69 to see the steps for using the principle of zero product.
After you read, complete examples 7.70 and 7.71 and check your answers.
9.13: Using Factoring to Solve Quadratic Equations
We are now ready to use factoring to solve quadratic equations.
Read these two articles, which give examples of how factoring is used to solve quadratic equations. Try to identify the factoring methods being used and why each step is performed. After you have reviewed the materials, complete a few practice problems and check your answers.
Read each example slowly and try to identify the factoring methods being used and why each step is performed. After you have reviewed the materials, complete a few practice problems and check your answers.
9.14: Applications for Solving Problems
Now that we have spent so much time learning about factoring, we are ready to see how it applies to everyday life.
Read this article, which shows how we use factoring in geometry and in problems where we need to find an unknown value.
After you read, complete review questions 1, 2, 5, 7, and 9 and check your answers.