# 9.13: Factoring by Grouping

**At Grade**Created by: CK-12

**Practice**Factoring by Grouping

What if you had a polynomial expression like \begin{align*}3x^2 - 6x + 2x - 4\end{align*}

### Watch This

CK-12 Foundation: 0913S Factoring By Grouping

### Guidance

Sometimes, we can factor a polynomial containing four or more terms by factoring common monomials from groups of terms. This method is called **factor by grouping.**

The next example illustrates how this process works.

#### Example A

*Factor* \begin{align*}2x+2y+ax+ay\end{align*}

**Solution**

There is no factor common to all the terms. However, the first two terms have a common factor of 2 and the last two terms have a common factor of \begin{align*}a\end{align*}

\begin{align*}2x + 2y + ax + ay = 2(x + y) + a(x + y)\end{align*}

Now we notice that the binomial \begin{align*}(x + y)\end{align*}

\begin{align*}(x + y)(2 + a)\end{align*}

#### Example B

*Factor* \begin{align*}3x^2+6x+4x+8\end{align*}

**Solution**

We factor 3x from the first two terms and factor 4 from the last two terms:

\begin{align*}3x(x+2)+4(x+2)\end{align*}

Now factor \begin{align*}(x+2)\end{align*}

Now the polynomial is factored completely.

**Factor Quadratic Trinomials Where a ≠ 1**

Factoring by grouping is a very useful method for factoring quadratic trinomials of the form \begin{align*}ax^2+bx+c\end{align*}

A quadratic like this doesn’t factor as \begin{align*}(x \pm m)(x \pm n)\end{align*}

To factor a quadratic polynomial where \begin{align*}a \neq 1\end{align*}

- We find the product \begin{align*}ac\end{align*}
ac . - We look for two numbers that multiply to \begin{align*}ac\end{align*}
ac and add up to \begin{align*}b\end{align*}b . - We rewrite the middle term using the two numbers we just found.
- We factor the expression by grouping.

Let’s apply this method to the following examples.

#### Example C

*Factor the following quadratic trinomials by grouping.*

a) \begin{align*}3x^2+8x+4\end{align*}

b) \begin{align*}6x^2-11x+4\end{align*}

**Solution:**

Let’s follow the steps outlined above:

a) \begin{align*}3x^2+8x+4\end{align*}

*Step 1:* \begin{align*}ac = 3 \cdot 4 = 12\end{align*}

*Step 2:* The number 12 can be written as a product of two numbers in any of these ways:

\begin{align*}12 &= 1 \cdot 12 && \text{and} && 1 + 12 = 13\\
12 &= 2 \cdot 6 && \text{and} && 2 + 6 = 8 \qquad This \ is \ the \ correct \ choice.\\
12 &= 3 \cdot 4 && \text{and} && 3 + 4 = 7\end{align*}

*Step 3:* Re-write the middle term: \begin{align*}8x = 2x + 6x\end{align*}

\begin{align*}3x^2+8x+4=3x^2+2x+6x+4\end{align*}

*Step 4:* Factor an \begin{align*}x\end{align*}

\begin{align*}x(3x+2)+2(3x+2)\end{align*}

\begin{align*}(3x+2)(x+2) \qquad This \ is \ the \ answer.\end{align*}

To check if this is correct we multiply \begin{align*}(3x+2)(x+2)\end{align*}

\begin{align*}& \qquad \ \ 3x+2\\
& \underline{\;\;\;\;\;\;\;\;\;\;\;x+2\;}\\
& \quad \quad \ \ 6x+4\\
& \underline{3x^2+2x \;\;\;\;\;}\\
& 3x^2+8x+4\end{align*}

The solution checks out.

b) \begin{align*}6x^2-11x+4\end{align*}

*Step 1:* \begin{align*}ac = 6 \cdot 4 = 24\end{align*}

*Step 2:* The number 24 can be written as a product of two numbers in any of these ways:

\begin{align*}24 &= 1 \cdot 24 && \text{and} && 1 + 24 = 25\\
24 &= -1 \cdot (-24) && \text{and} && -1 + (-24) = -25\\
24 &= 2 \cdot 12 && \text{and} && 2 + 12 = 14\\
24 &= -2 \cdot (-12) && \text{and} && -2 + (-12) = -14\\
24 &= 3 \cdot 8 && \text{and} && 3 + 8 = 11\\
24 &= -3 \cdot (-8) && \text{and} && -3 + (-8) = -11 \qquad (Correct \ choice) \\
24 &= 4 \cdot 6 && \text{and} && 4 + 6 = 10\\
24 &= -4 \cdot (-6) && \text{and} && -4 + (-6) = -10\end{align*}

*Step 3:* Re-write the middle term: \begin{align*}-11x = -3x - 8x\end{align*}

\begin{align*}6x^2-11x+4=6x^2-3x-8x+4\end{align*}

*Step 4:* Factor by grouping: factor a \begin{align*}3x\end{align*}

\begin{align*}3x(2x-1)-4(2x-1)\end{align*}

Now factor the common binomial \begin{align*}(2x - 1)\end{align*}

\begin{align*}(2x-1)(3x-4) \qquad This \ is \ the \ answer.\end{align*}

Watch this video for help with the Examples above.

CK-12 Foundation: Factoring By Grouping

### Vocabulary

- It is possible to factor a polynomial containing four or more terms by factoring common monomials from groups of terms. This method is called
**factoring by grouping**.

### Guided Practice

*Factor \begin{align*}5x^2-6x+1\end{align*} 5x2−6x+1 by grouping.*

**Solution:**

Let’s follow the steps outlined above:

\begin{align*}5x^2-6x+1\end{align*}

*Step 1:* \begin{align*}ac = 5 \cdot 1 = 5\end{align*}

*Step 2:* The number 5 can be written as a product of two numbers in any of these ways:

\begin{align*}5 &= 1 \cdot 5 && \text{and} && 1 + 5 = 6\\
5 &= -1 \cdot (-5) && \text{and} && -1 + (-5) = -6 \qquad (Correct \ choice)\end{align*}

*Step 3:* Re-write the middle term: \begin{align*}-6x = -x - 5x\end{align*}, so the problem becomes:

\begin{align*}5x^2-6x+1=5x^2-x-5x+1\end{align*}

*Step 4:* Factor by grouping: factor an \begin{align*}x\end{align*} from the first two terms and \begin{align*}a - 1\end{align*} from the last two terms:

\begin{align*}x(5x-1)-1(5x-1)\end{align*}

Now factor the common binomial \begin{align*}(5x - 1)\end{align*}:

\begin{align*}(5x-1)(x-1) \qquad This \ is \ the \ answer.\end{align*}

### Practice

Factor by grouping.

- \begin{align*}6x^2-9x+10x-15\end{align*}
- \begin{align*}5x^2-35x+x-7\end{align*}
- \begin{align*}9x^2-9x-x+1\end{align*}
- \begin{align*}4x^2+32x-5x-40\end{align*}
- \begin{align*}2a^2-6ab+3ab-9b^2\end{align*}
- \begin{align*}5x^2+15x-2xy-6y\end{align*}

Factor the following quadratic trinomials by grouping.

- \begin{align*}4x^2+25x-21\end{align*}
- \begin{align*}6x^2+7x+1\end{align*}
- \begin{align*}4x^2+8x-5\end{align*}
- \begin{align*}3x^2+16x+21\end{align*}
- \begin{align*}6x^2-2x-4\end{align*}
- \begin{align*}8x^2-14x-15\end{align*}

Factor by Grouping

Factoring by grouping is a method of factoring a polynomial by factoring common monomials from groups of terms.Grouping Symbols

Grouping symbols are parentheses or brackets used to group numbers and operations.Volume

Volume is the amount of space inside the bounds of a three-dimensional object.### Image Attributions

Here you'll learn how to group a polynomial's terms to help you factor it.

## Concept Nodes:

Factor by Grouping

Factoring by grouping is a method of factoring a polynomial by factoring common monomials from groups of terms.Grouping Symbols

Grouping symbols are parentheses or brackets used to group numbers and operations.Volume

Volume is the amount of space inside the bounds of a three-dimensional object.