Factoring Polynomials Worksheet Library

Download your Factoring Polynomials Worksheets and Answer Keys

Each free factoring polynomials worksheet can be downloaded as a printable PDF file.

Welcome to the Mashup Math Factoring Polynomials Worksheet Library! On this page, you will find a collection of free PDF worksheets focused on variety of problems involving factoring polynomials.

Each factorization of polynomials worksheet shares ten or more practice problems for algebra students, and each includes a complete answer key. You can download any factoring polynomials worksheet as a printable PDF file by clicking on any of the blue text links in the library below.

This page also shares a short review section that includes several step-by-step examples of how to factoring different types of polynomials in case you or your students are struggling with any of the problems shared on the worksheets below.

Each factoring polynomials worksheet is a sample from the Mashup Math Algebra Worksheet Libraries available on our website.

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Factoring Polynomials Worksheet Library

Difference of Two Squares

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Factoring Trinomials (a=1)

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Factoring by Grouping

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Factoring Polynomials Worksheet Review

If you are having difficulty with solving problems on any factorization of polynomials worksheet, then this review section will go over a few example problems that are similar to the ones on the worksheets available in the factor the polynomial worksheet library above.

When it comes to factoring polynomials, you should be familiar with how to solve the following types of related problems:

  • Factoring the Difference of Two Squares

  • Factoring Trinomials

  • Factoring by Completing the Square

Let’s go ahead and do a quick review of each of these ways of factoring polynomials by working through a few example problems and solving them step-by-step.

Factoring the Difference of Two Squares

In algebra, factoring the difference of two squares is a method of factoring polynomials that can be used to factor/simplify any expression of the form:

  • (a² - b²)

Notice that any expression of the form (a² - b²) is a binomial (i.e. an expression with two terms) that are both perfect squares. Also notice that the operation sign must be (-) since we are dealing with the difference of two squares.

If an algebraic expression meets this criteria, meaning that it is in (a² - b²) form, then we can factor it as follows:

  • (a² - b²) = (a + b)(a - b)

Example: Factor: (x² - 25)

Since the given polynomial is of the form (a² - b²), we can factor it using the difference of two squares formula mentioned above, where a=x and b=5 (since 5² = 25).

  • (a² - b²) = (a + b)(a - b)

  • (x² - 25) = (x + 5)(x - 5)

Final Answer: (x + 5)(x - 5)

The breakdown for factoring the difference of two squares is shown in Figure 01 below.

 

Figure 01: Factoring Polynomials Worksheet Review: Difference of Two Squares Method

 

Factoring Trinomials

In algebra, a trinomial is a polynomial with three terms that is of the form:

  • ax² + bx + c

You can often factor polynomials of this form (i.e. a trinomial) by finding two numbers that meet the following criteria:

  • the sum of the two numbers must be equal to b

  • the product of the two numbers must be equal to c

In other words, you can often factor a trinomial by finding two numbers that add to b and multiply to c.

Example: Factor: x² +2x -15

In this example, we know that:

  • a = 1

  • b = 2

  • c= -15

We can find the factors by finding two numbers that add to b (2) and multiply to c (-15). After some trial-and-error, we can determine that two numbers that meet this criteria are 5 and -3 because:

  • 5 + -3 = 2 (and b=2)

  • 5 x -3 = -15 (and c=-15)

Now that we have our two numbers, we can use them to determine the factors of the trinomial x² +2x -15 as follows:

Final Answer: (x + 5)(x - 3)

The steps for factoring a trinomial this way are highlighted in Figure 02 below. Note that this method will work on many trinomials, but not all trinomials.

 

Figure 02: Factor the Polynomial Worksheet Review: Trinomials

 

Factoring by Completing the Square

While the above method for factoring trinomials of the form ax² + bx + c can be very effective, it can not be used to factor every polynomial of this form.

Why? Because sometimes you will not be able to find two numbers that add to b and multiply to c.

In cases like this, you can use a method called completing the square to find the factors of a trinomial.

Example: Factor: x² +10x + 5

In this example, a=1, b=10, and c=5. Notice that it is not possible to find two numbers with a sum of 10 and a product of 5.

However, this does not mean that the polynomial is not factorable. We can factor by completing the square as follows:

Start by rewriting the polynomial as an equation equal to 0 and the rearrange it so that the constant term is isolated (in this example, the constant term is 5):

  • x² +10x + 5 = 0 → x² +10x = -5

Next, divide the b coefficient value by two, then square it, then add and subtract it to the left side of the equation. The b coefficient is 10, and half of 10 is 5, and 5² = 25, so:

  • x² +10x +25 -25 = -5

  • (x+5)² − 25 = −5

Finally, you can simplify and solve to find the factors of the trinomial as follows:

  • (x+5)² − 25 = −5

  • (x+5)² = 20

  • x+5 = ± √20​

  • x = -5 ± 2√5

Final Answer: x = -5 + 2√5, x = -5 - 2√5

Learning how to factor polynomials by completing the square can be tricky and learning the procedure will take some practice. We highly recommend that you work through the problems on the factoring polynomials worksheets in the library above until you get the hang of it. And, if you want a more in-depth step-by-step tutorial on how to factor by completing the square, the click here to access our free student guide.

 

Free Step-by-Step Guide to Completing the Square

 

While learning how to solve all of the different types of problems you will find in our factor the polynomial worksheet library can be challenging, factoring polynomials in general is a key algebra skill that you must master if you want to be successful in higher levels of math. Learning how to successfully factor different types of polynomials will take time, effort, and lots of practice, so be sure to put in the time, study often, and give yourself plenty of opportunities to develop this important math skill.