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### Course: Integrated math 2>Unit 2

Lesson 6: Factoring quadratics by grouping

# Factoring by grouping

Learn about a factorization method called "grouping." For example, we can use grouping to write 2x²+8x+3x+12 as (2x+3)(x+4).

#### What you need to know for this lesson

Factoring a polynomial involves writing it as a product of two or more polynomials. It reverses the process of polynomial multiplication.
We have seen several examples of factoring already. However, for this article, you should be especially familiar with taking common factors using the distributive property. For example, $6{x}^{2}+4x=2x\left(3x+2\right)$ .

#### What you will learn in this lesson

In this article, we will learn how to use a factoring method called grouping.

## Example 1: Factoring $2{x}^{2}+8x+3x+12$‍

First, notice that there is no factor common to all terms in $2{x}^{2}+8x+3x+12$. However, if we group the first two terms together and the last two terms together, each group has its own GCF, or greatest common factor:
$\underset{\text{first grouping}}{\underset{⏟}{\left(2{x}^{2}+8x\right)}}+\underset{\text{second grouping}}{\underset{⏟}{\left(3x+12\right)}}$
In particular, there is a GCF of $2x$ in the first grouping and a GCF of $3$ in the second grouping. We can factor these out to obtain the following expression:
$2x\left(x+4\right)+3\left(x+4\right)$
Notice that this reveals yet another common factor between the two terms: $x+4$. We can use the distributive property to factor out this common factor.
$2x\left(x+4\right)+3\left(x+4\right)=\left(x+4\right)\left(2x+3\right)$
Since the polynomial is now expressed as a product of two binomials, it is in factored form. We can check our work by multiplying and comparing it to the original polynomial.

## Example 2: Factoring $3{x}^{2}+6x+4x+8$‍

Let's summarize what was done above by factoring another polynomial.
The factored form is $\left(x+2\right)\left(3x+4\right)$.

1) Factor $9{x}^{2}+6x+12x+8$.

2) Factor $5{x}^{2}+10x+2x+4$.

3) Factor $8{x}^{2}+6x+4x+3$.

## Example 3: Factoring $3{x}^{2}-6x-4x+8$‍

Extra care should be taken when using the grouping method to factor a polynomial with negative coefficients.
For example, the steps below can be used to factor $3{x}^{2}-6x-4x+8$.
The factored form of the polynomial is $\left(x-2\right)\left(3x-4\right)$. We can multiply the binomials to check our work.
A few of the steps above may seem different than what you saw in the first example, so you may have a few questions.
Where did the "+" sign between the groupings come from?
In step $\left(1\right)$, a "+" sign was added between the groupings $\left(3{x}^{2}-6x\right)$ and $\left(-4x+8\right)$. This is because the third term $\left(-4x\right)$ is negative, and the sign of the term must be included within the grouping.
Keeping the minus sign outside the second grouping is tricky. For example, a common error is to group $3{x}^{2}-6x-4x+8$ as $\left(3{x}^{2}-6x\right)-\left(4x+8\right)$. This grouping, however, simplifies to $3{x}^{2}-6x-4x-8$, which is not the same as the original expression.
Why factor out $-4$ instead of $4$?
In step $\left(2\right)$, we factored out a $-4$ to reveal a common factor of $\left(x-2\right)$ between the terms. If we instead factored out a positive $4$, we would not obtain that common binomial factor seen above:
$\begin{array}{rl}\left(3{x}^{2}-6x\right)+\left(-4x+8\right)& =3x\left(x-2\right)+4\left(-x+2\right)\end{array}$
When the leading term in a group is negative, we will often need to factor out a negative common factor.

4) Factor $2{x}^{2}-3x-4x+6$.

5) Factor $3{x}^{2}+3x-10x-10$.

6) Factor $3{x}^{2}+6x-x-2$.

## Challenge problem

7*) Factor $2{x}^{3}+10{x}^{2}+3x+15$.

### When can we use the grouping method?

The grouping method can be used to factor polynomials whenever a common factor exists between the groupings.
For example, we can use the grouping method to factor $3{x}^{2}+9x+2x+6$ since it can be written as follows:
$\begin{array}{rl}\left(3{x}^{2}+9x\right)+\left(2x+6\right)& =3x\left(x+3\right)+2\left(x+3\right)\end{array}$
We cannot, however, use the grouping method to factor $2{x}^{2}+3x+4x+12$ because factoring out the GCF from both groupings does not yield a common factor!
$\begin{array}{rl}\left(2{x}^{2}+3x\right)+\left(4x+12\right)& =x\left(2x+3\right)+4\left(x+3\right)\end{array}$

#### Using grouping to factor trinomials

You can also use grouping to factor certain three termed quadratics (i.e. trinomials) like $2{x}^{2}+7x+3$. This is because we can rewrite the expression as follows:
$2{x}^{2}+7x+3=2{x}^{2}+1x+6x+3$
Then we can use grouping to factor $2{x}^{2}+1x+6x+3$ as $\left(x+3\right)\left(2x+1\right)$.
For more on factoring quadratic trinomials like these using the grouping method, check out our next article.

## Want to join the conversation?

• This may sound a bit dumb, but is there any significant difference between factoring, grouping trinomials, difference of squares, and GCF?
• Nothing significant, but there are important (however small) differences and they are used for different things.
• How would you work out the problem if there were only 3 terms?
• Find two numbers that add to the middle coefficient and multiply to give the product of the first and last coefficients (or constants). This is called the ac method.

Example: Factor 6x^2 + 19x + 10.
6*10 = 60, so we need to find two numbers that add to 19 and multiply to give 60. These numbers (after some trial and error) are 15 and 4. So split up 19x into 15x + 4x (or 4x + 15x), then factor by grouping:

6x^2 + 19x + 10 = 6x^2 + 15x + 4x + 10
= 3x(2x + 5) + 2(2x + 5)
= (3x + 2)(2x + 5).

Have a blessed, wonderful day!
• what the meaning of GCF?
• GCF is the abbreviation for Greatest Common Factor.
It is the value that you can evenly divide all terms by. It can be a number, a variable, or a mix of numbers and variables.
• Why are they called quadratics? They are typically trinomials with a leading term raised to the second degree.
• Quadratics actually are derived from the Latin word, 'quadratum', which literally means 'square'.
A quadratic is basically a type of problem that deals with a variable multiplied by itself — an operation known as squaring.
So, you could relate the word quadratic to Latin, not mathematics
• Wow I got this better than how I learned it in school thanks Sal!
• Lets say there is no GCF in anything. Then What?
• There is always a GCF, it just depends on how many there are.
Take 8 and 6, the factors of those two numbers are 1 and 2.
2 is bigger, so the GCF of 8 and 6 is 2.
That works the same with every other set of numbers.

Let's take 4 and 3.
The factors of 3 are 1,3
The factors of 4 are 1,2,4

The only common factor here is 1, so 1 is the GCF of 4 and 3!

So to answer your question, if you can't find the greatest common factor of two numbers, it's 1.
• In problem 3, I solved the expression and got the answer (4x+2)(2x+1.5). It said my answer was correct, but when I checked my answer, I got the same expression. What did I do wrong?
• u used 1.5 which is not a natural number. Therefore answer is incorrect. Correct answer must include only natural numbers: (3x+2)(3x+4)
• Why does it say we cannot use grouping on this one 2x^2+3x+4x+12. It is usable though isn't it? Answer would be (x+2)(2x+3) right?
(1 vote)
• Your factors actually don't create the polynomial. If you multiply your factors, you will get: 2x^2+3x+4x+6, which does not match the original polynomial.

Now, how do you know if isn't working before checking the factors by multiplying them.

Pull the GCF from each pair of terms to get:
2x^2+3x+4x+12 = x(2x+3) + 4(x+3)

Notice, the 2 binomials in parentheses do not match. To go to factors from this point, we need to remove a common binomial factor from each term. There is no common binomial factor. This tells use that the polynomial is not factorable.

Hope this helps.
• Hi Professor,
Why dont you use the formula Ax^2+Bx-C where A*C=a*b AND B=a+b ? It helps me to learn it this way, since it opens up for future equations that take the square of A and C.