A Comprehensive Guide to Factoring Polynomials

If you are a control systems engineer or an algebra student, factoring polynomials is a daily exercise in problem-solving.

Real life applications often require breaking down equations into their parts to visualize how and when variables interact. To do that, you need to know how to factor polynomials.

For those just learning, we have prepared a comprehensive guide to our polynomial factoring calculator. We use simple language and steps to cover the definition of a polynomial, factoring by grouping and how to factor polynomials.

Read on to get the explanations you need.

Definition of a Polynomial

Like so many words in math and science, polynomial has Greek roots. The prefix -poly meaning “many” and -nomial meaning “terms”.

According to Merriam-Webster, a polynomial is … ” a mathematical expression of one or more algebraic terms each of which consists of a constant multiplied by one or more variables raised to a non-negative integral power.”

So a polynomial is an expression with constants, variables, and whole number exponents. It can’t be divided by a variable. It can’t have infinite terms.

The exponents can only be 0, 1, 2, 3 …, etc. Negative or fractional exponents are not allowed. It can be combined using addition, subtraction, multiplication, and division.

-3x^2 + 2x + 4

y^5 – x^4 – x^2

6a^4 + 12a^2 – a

The above are all polynomials.

These three expressions:

(3y^2 – 3y)/z

3a^-3 +a^2 +a

1/7x ^3.5

are not polynomials.

Constants are the numbers to the left (6, 1/7, -3)

Variables are the unknowns (a, x, y)

How to Factor Polynomials

Factoring a polynomial is the process of breaking the -poly (many) into the product of smaller terms like a bi- (two) or tri- (three) -nomial.

Let’s look at a polynomial.

4x^4 + 12x + 24x^2 +16x

Pretty ugly, isn’t it?

The first step is to place the equation in standard form. That is, from largest to smallest exponential order from left to right.

4x^4 + 24x^2 + 16x^2 + 12x

Still pretty ugly. But now you can see your greatest common factor (GCF), right? If you can’t see it, have no fear.

Break the ugly equation above into its prime factors.

2*2*x*x*x*x + 2*2*2*3*x*x*x* + 2*2*2*2*x*x + 2*2*3*x

Look for the factors in every single term separated by the addition signs to get the greatest common factor (GCF). In this case, 2*2*x.

Re-group and use the distributive property.

2*2*x (x*x*x + 2*3*x*x* + 2*2*x + 3)

Now change it back into simplified form.

4x (x^3 + 6x^2 + 4x +3)

To check your work, multiply it back out to see if you get the original equation.

Factoring Polynomials by Grouping

Let’s go back to another ugly equation. This one is slightly different.

4x^4 + 12x^3 + 4x^2 + 12x

Let’s regroup our equation.

(4x^4 + 12x^3) + (4x^2 + 12x)

Break the ugly equation above into its prime factors.

(2*2*x*x*x*x + 2*2*3*x*x*x*) + ( 2*2*x*x + 2*2*3*x)

Now use the distributive property again to get:

4x^2 (x^2 + 3) + 4 (x^2 +3)

Simplify one more time and re-group.

(4x^2 + 4) (x^2 + 3)

Pretty direct, right? What if you don’t want to do the guesswork to find the right pairs to factor out trinomials?

What About Factoring Trinomials?

Sometimes after taking out the GCF, you end up with an equation that can be broken down to even smaller terms. So what happens when you factor out your GCF and you end up with a trinomial? Let’s look at a trinomial like the standard quadratic equation.

Pretend we did the above steps and ended up with this:

3x^2 (x^2 + 5x – 6)

3x^2 is a monomial and cannot be factored further. But what about the other part of the equation?

Step 1 of factoring a trinomial is to identify the parts.

The standard form of a quadratic equation is:

ax^2 + bx +c

In our example, a = 1, b = 5 and c = -6.

Step 2 is to create a table of all the factor pairs of c.

1 * -6

-1* 6

2 *-3

-2* 3

Step 3 is to single out a factor pair that equals b.

1 + -6 = -5

-1 + 6 = 5

2+-3 = -1

-2 + 3 = 1

b = 5

Step 4 is to substitute the matching factor pair into two binomials.

(x – 1)(x + 6)

Multiply it out to check your answer.

In our example, the factored equation is (3x^2)(x -1)(x + 6).

I Still Don’t Understand — What Now?

Factoring polynomials improves with practice. There are several mathematics tutorials available online. On the other hand, if you use our online calculator, the answer will be correct. Explaining your answer will still take some effort.

We recommend taking the long form of writing out your answer to both explain your process and check for errors.

However, if you are still finding it hard to factor polynomials, we have listed several references to help you. Of course, seek your teacher or instructor’s guidance first.

Great places for assistance include:

  • your local community college or university faculty
  • your local librarian can direct you
  • a local tutor offered by your school

Tutoring may be offered for free or at low cost. If in-person tutors are unavailable, try online.

You may find online video references more helpful. Almost all of the lessons are free. Use your favorite search engine. Try these websites:

For more practical examples and worksheets, many textbooks offer online resources, articles, and charts. If you have a graphing calculator, problem sets and practice can be downloaded directly.

We hope this guide has been helpful to you as you navigate polynomials. Please bookmark this calculator for easy access. Better yet, save this information and calculator to the home screen of your smartphone, for quick checks.

For other algebra helps and articles, check out our Algebra section today.

Leave a Reply