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Factor by grouping - Factoring Polynomials

Factor By Grouping

Factoring

Before we get into the details of factoring polynomials by grouping, let’s do a quick review of the general process of factoring itself.

First, we need to know what exactly a “factor” is. The understanding of what factors are is crucial to all of mathematics, and it is a term you will hear again and again as you progress with your studies.

With regards to division, a factor is just a term or expression that, when another term or expression is divided by this factor, the remainder is equal to zero. In its simplest terms, consider the following: 4 is a factor of 20 because, when 20 is divided by 4, we get the whole number 5 and no remainder. 7 is not a factor of 20 because, when 20 is divided by 7, we get 2.86, which is not a whole number (this is the same as saying 2 and a remainder of 0.86).

When we’re looking at factoring polynomials, in particular, the meaning of a factor isn’t all that different. The factor of a polynomial is just a value of the independent value (usually x) that makes an entire polynomial equation to zero. Not too complicated after all!

Check out our videos covering how to find the greatest common factor of polynomials, factoring polynomials with common factor, as well as factoring trinomials with leading coefficient not 1.

Factoring Polynomials by Grouping

Now that we have a good understanding of what it means to factor in its most general terms, let’s look at factoring by grouping. Factoring polynomials by grouping is just another technique we can use, similar to others you’ve likely seen in the past. What makes factoring by grouping so powerful, however, is its ability to help us to factor higher degree polynomials like cubics with relative ease.

It is important to understand what we mean when we say “grouping”. When we are factoring by grouping, all we are really doing is breaking up our polynomial into easier-to-factor groups or “families” so that we can better approach the problem. Once we break it up into groups, we can factor using the methods we’ve learned from factoring quadratic and simpler polynomials.

The process is the same for any degree of polynomial, whether we be factoring quadratics by grouping, cubics by grouping, and beyond. Remember, the degree of a polynomial just related to the highest power on the independent variable x.

Lastly, for a video explanation of all of this, see our video on how to factor by grouping.

How to Factor by Grouping

The best way to learn this technique is to do some factoring by grouping examples!

Example:

Factor the following polynomial by grouping:

x37x2+2x14x^3-7x^2+2x-14

Step 1: Divide Polynomial Into Groups

This is the trickiest part of solving these kinds of problems. Choosing what groups to make varies from problem to problem, but, in most cases, we are usually going to group the 2 highest powers together and then the lowest 2 or 1 powers together. You will see later that this doesn’t necessarily matter, but it is the easiest way to do it.

In this case, the groups we will make are:

(x37x2)(x^3-7x^2) and (+2x14)(+2x-14)

Step 2: Factor Individual Groups

Now that we have done our grouping step, next we need to factor each of these groups using skills we’ve developed in the past.

In the group that is (x37x2)(x^3-7x^2), we can take out the common factor x2x^2 of this family. After taking x2x^2 out, we end up with:

x2(x7)x^2(x-7)

Going on to the next group which is (+2x14)(+2x-14), the common factor here is 2. So we take 2 out of the family and we end up with:

2(x7)2(x-7)

Step 3: Factor the Entire Polynomial

After factoring each group individually, we now need to put the groups together. This gives us a more complicated looking polynomial that is:

x2(x7)+2(x7)x^2(x-7)+2(x-7)

What is important now is to consider each of our “groups” from before as their own terms in the polynomial, with x2(x7)x^2(x-7) being the first term, and +2(x7)+2(x-7) being the second term. With that in mind, we can factor this entire polynomial by recognizing there is a common factor between the two terms: (x7)(x-7). So we factor it out and that leaves us with our final answer:

(x7)(x2+2)(x-7)(x^2+2)

Now, you may ask is it possible to group different terms into a family and still get the same final answer. In fact, it is possible!

Alternative Method:

Step 1: Divide Polynomial Into Groups

In this case, the groups we will make are:

(x3+2x)(x^3+2x) and (7x214)(-7x^2-14)

Step 2: Factor Individual Groups

Now that we have done our grouping step, next we need to factor each of these groups using skills we’ve developed in the past, just like in the first method to solving this problem.

For (x3+2x)(x^3+2x), xx is the factor. Giving us a factored group of: x(x2+2)x(x^2+2)

For (7x214)(-7x^2-14), -7 is the factor. Giving us a factored group of: 7(x2+2)-7(x^2+2)

Step 3: Factor the Entire Polynomial

After factoring each group individually, again, we now need to put the groups together. This gives us a more complicated looking polynomial that is:

x(x2+2)7(x2+2)x(x^2+2)-7(x^2+2)

Once again, what is important now is to consider each of our “groups” from before as their own terms in the polynomial with x(x2+2)x(x^2+2) being the first term, and 7(x2+2)-7(x^2+2) being the second term.

Now, look for the common factor between the two terms, and hopefully you’ll notice (x2+2)(x^2+2) as the common factor. So, we factor it out, and arrive at the same final answer of:

(x2+2)(x7)(x^2+2)(x-7)

And that’s all there is to it! For more practice, take a look at this factoring by grouping worksheet, which has many more examples to try. You can find it here. Lastly, for some related work, see our videos on completing the square and the conversion of standard form to vertex form quadratic equations.

Factor by grouping

This is a more advanced lesson on how to factor polynomials by grouping. Tips: 1) Look for the common factors and then 2) group them together.

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Factor by grouping

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