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Factoring By Grouping

In this video, we are going to look at how to factor by grouping.

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For example:
To factor 12x^3+18x^2+10x+15, first think about it in two pieces, cutting it in half. Then we will look at the first two terms, and take out the greatest common factor, which is 6x^2. This will leave us with
6x^2(2x+3)
Then, we do the same thing for the next two terms. The greatest common factor here is 5. When we take out the greatest common factor, we are left with
5(2x+3)
Notice that the same exact factors are written in both sets of parentheses. From here we can factor out the (2x+3) from each term. This will result in a final answer of
(2x+3)(6x^2+5)

Factoring by Grouping

Video-Lesson Transcript

Let’s go over factoring by grouping.

We have 12x^3 + 18x^2 + 10x + 15

So we’re going to factor this four-term polynomial by grouping.

We’re going to split this in half.

Let’s look at the first two terms of the polynomial.

12x^3 + 18x^2

And we’ll find out what is the greatest common factor of these two terms.

It’s 6x^2.

Let’s factor.

12x^3 + 18x^2
6x^2 (2x + 3)

Now let’s do the same with the other two terms.

10x + 15

The greatest common factor for these two is 5.

Let’s factor.

10x + 15
+5 (2x + 3)

Let’s put the plus sign.

At this point, we have the same exact factors:

2x + 3

We can actually factor this out of both of these terms.

We have

6x^2 (2x + 3) +5 (2x + 3)

So let’s take the same factors out.

Our final answer is

(2x + 3) (6x^2 + 5)

Let’s take a look at another example.

6x^3 + 7x^2 - 42x - 49

Again, let’s split the polynomial into two.

Then factor the first two terms first.

6x^3 + 7x^2

There’s no greatest common factor for these two coefficients. The only factor we have is the variable.

6x^3 + 7x^2
x^2 (6x + 7)

Then do the same with the remaining two terms.

- 42x - 49

7 goes into both of these but since they are negative, we would want to take the negative out.

-7 (6x + 7)

So now we have

x^2 (6x + 7) -7 (6x + 7)

Let’s take the common factor out. And we’re left with our final answer.

(6x + 7) (x^2 - 7)

This is polynomial factoring.