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 $(2x+3)(6x^2+5)$ ## 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. $(2x + 3) (6x^2 + 5)$

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.