# 10.6 Introduction to factoring polynomials  (Page 2/6)

 Page 2 / 6

Find the greatest common factor: $16{x}^{2},\phantom{\rule{0.2em}{0ex}}24{x}^{3}.$

8 x 2

Find the greatest common factor: $27{y}^{3},\phantom{\rule{0.2em}{0ex}}18{y}^{4}.$

9 y 3

Find the greatest common factor of $14{x}^{3},\phantom{\rule{0.2em}{0ex}}8{x}^{2},\phantom{\rule{0.2em}{0ex}}10x.$

## Solution

 Factor each coefficient into primes and write the variables with exponents in expanded form. Circle the common factors in each column. Bring down the common factors. Multiply the factors. $\text{The GCF of}\phantom{\rule{0.2em}{0ex}}14{x}^{3}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}8{x}^{2}\text{, and}10x\phantom{\rule{0.2em}{0ex}}\text{is}\phantom{\rule{0.2em}{0ex}}2x$

Find the greatest common factor: $21{x}^{3},\phantom{\rule{0.2em}{0ex}}9{x}^{2},\phantom{\rule{0.2em}{0ex}}15x.$

3 x

Find the greatest common factor: $25{m}^{4},\phantom{\rule{0.2em}{0ex}}35{m}^{3},\phantom{\rule{0.2em}{0ex}}20{m}^{2}.$

5 m 2

## Factor the greatest common factor from a polynomial

Just like in arithmetic, where it is sometimes useful to represent a number in factored form (for example, $12$ as $2·6\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}3·4\text{),}$ in algebra it can be useful to represent a polynomial in factored form. One way to do this is by finding the greatest common factor of all the terms. Remember that you can multiply a polynomial by a monomial as follows:

$\begin{array}{ccc}\hfill 2\left(x& +& 7\right)\phantom{\rule{0.2em}{0ex}}\text{factors}\hfill \\ \hfill 2·x& +& 2·7\hfill \\ \hfill 2x& +& 14\phantom{\rule{0.2em}{0ex}}\text{product}\hfill \end{array}$

Here, we will start with a product, like $2x+14,$ and end with its factors, $2\left(x+7\right).$ To do this we apply the Distributive Property “in reverse”.

## Distributive property

If $a,\phantom{\rule{0.2em}{0ex}}b,\phantom{\rule{0.2em}{0ex}}c$ are real numbers, then

$a\left(b+c\right)=ab+ac\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}ab+ac=a\left(b+c\right)$

The form on the left is used to multiply. The form on the right is used to factor.

So how do we use the Distributive Property to factor a polynomial? We find the GCF of all the terms and write the polynomial as a product!

Factor: $2x+14.$

## Solution

 Step 1: Find the GCF of all the terms of the polynomial. Find the GCF of 2x and 14. Step 2: Rewrite each term as a product using the GCF. Rewrite 2x and 14 as products of their GCF, 2. $2x=2\cdot x$ $14=2\cdot 7$ Step 3: Use the Distributive Property 'in reverse' to factor the expression. $2\left(x+7\right)$ Step 4: Check by multiplying the factors. Check:

Factor: $4x+12.$

4( x + 3)

Factor: $6a+24.$

6( a + 4)

Notice that in [link] , we used the word factor as both a noun and a verb:

$\begin{array}{cccc}\text{Noun}\hfill & & & 7\phantom{\rule{0.2em}{0ex}}\text{is a factor of}\phantom{\rule{0.2em}{0ex}}14\hfill \\ \text{Verb}\hfill & & & \text{factor}\phantom{\rule{0.2em}{0ex}}2\phantom{\rule{0.2em}{0ex}}\text{from}\phantom{\rule{0.2em}{0ex}}2x+14\hfill \end{array}$

## Factor the greatest common factor from a polynomial.

1. Find the GCF of all the terms of the polynomial.
2. Rewrite each term as a product using the GCF.
3. Use the Distributive Property ‘in reverse’ to factor the expression.
4. Check by multiplying the factors.

Factor: $3a+3.$

## Solution

 Rewrite each term as a product using the GCF. Use the Distributive Property 'in reverse' to factor the GCF. Check by multiplying the factors to get the original polynomial.

Factor: $9a+9.$

9( a + 1)

Factor: $11x+11.$

11( x + 1)

The expressions in the next example have several factors in common. Remember to write the GCF as the product of all the common factors.

Factor: $12x-60.$

## Solution

 Rewrite each term as a product using the GCF. Factor the GCF. Check by multiplying the factors.

Factor: $11x-44.$

11( x − 4)

Factor: $13y-52.$

13( y − 4)

Now we’ll factor the greatest common factor    from a trinomial    . We start by finding the GCF of all three terms.

Factor: $3{y}^{2}+6y+9.$

## Solution

 Rewrite each term as a product using the GCF. Factor the GCF. Check by multiplying.

Factor: $4{y}^{2}+8y+12.$

4( y 2 + 2 y + 3)

Factor: $6{x}^{2}+42x-12.$

6( x 2 + 7 x − 2)

In the next example, we factor a variable from a binomial    .

Factor: $6{x}^{2}+5x.$

## Solution

 $6{x}^{2}+5x$ Rewrite each term as a product. Factor the GCF. $x\left(6x+5\right)$ Check by multiplying. $x\left(6x+5\right)$ $x\cdot 6x+x\cdot 5$ $6{x}^{2}+5x✓$

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