7.1 Greatest common factor and factor by grouping (Page 2/5)

Elementary algebra Page 2 / 5

\begin{matrix} 2 (x + 7) & factors \\ 2 \cdot x + 2 \cdot 7 \\ 2 x + 14 & product \end{matrix}

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

We state the Distributive Property here just as you saw it in earlier chapters and “in reverse.”

Distributive property

If $a, b, c$ are real numbers, then

\begin{matrix} a (b + c) = a b + a c & and & a b + a c = a (b + c) \end{matrix}

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

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

How to factor the greatest common factor from a polynomial

Factor: $4 x + 12$ .

Solution

This table has three columns. In the first column are the steps for factoring. The first row has the first step, “Find the G C F of all the terms of the polynomial”. The second column in the first row has “find the G C F of 4 x and 12”. The third column in the first row has 4 x factored as 2 times 2 times x and below it 18 factored as 2 times 2 times 3. Then, below the factors are the statements, “G C F = 2 times 2” and “G C F = 4”.

The second row has the second step “rewrite each term as a product using the G C F”. The second column in the second row has the statement “Rewrite 4 x and 12 as products of their G C F, 4” Then the two equations 4 x = 4 times x and 12 = 4 times 3. The third column in the second row has the expressions 4x + 12 and below this 4 times x + 4 times 3.

The third row has the step “Use the reverse distributive property to factor the expression”. The second column in the third row is blank. The third column in the third row has “4(x + 3)”.

The fourth row has the fourth step “check by multiplying the factors”. The second column in the fourth row is blank. The third column in the fourth row has three expressions. The first is 4(x + 3), the second is 4 times x + 4 times 3. The third is 4 x + 12.

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Factor: $6 a + 24$ .

$6 (a + 4)$

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Factor: $2 b + 14$ .

$2 (b + 7)$

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Factor the greatest common factor from a polynomial.

Find the GCF of all the terms of the polynomial.
Rewrite each term as a product using the GCF.
Use the “reverse” Distributive Property to factor the expression.
Check by multiplying the factors.

Factor as a noun and a verb

We use “factor” as both a noun and a verb.

This figure has two statements. The first statement has “noun”. Beside it the statement “7 is a factor of 14” labeling the word factor as the noun. The second statement has “verb”. Beside this statement is “factor 3 from 3a + 3 labeling factor as the verb.

Factor: $5 a + 5$ .

Solution

Find the GCF of 5 a and 5.

Rewrite each term as a product using the GCF.
Use the Distributive Property "in reverse" to factor the GCF.
Check by mulitplying the factors to get the orginal polynomial.
$5 (a + 1)$
$5 \cdot a + 5 \cdot 1$
$5 a + 5 ✓$

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Factor: $14 x + 14$ .

$14 (x + 1)$

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Factor: $12 p + 12$ .

$12 (p + 1)$

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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: $12 x - 60$ .

Solution

Find the GCF of 12 x and 60.

Rewrite each term as a product using the GCF.
Factor the GCF.
Check by mulitplying the factors.
$12 (x - 5)$
$12 \cdot x - 12 \cdot 5$
$12 x - 60 ✓$

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Factor: $18 u - 36$ .

$8 (u - 2)$

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Factor: $30 y - 60$ .

$30 (y - 2)$

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

Factor: $4 y^{2} + 24 y + 28$ .

Solution

We start by finding the GCF of all three terms.

Find the GCF of $4 y^{2}$ , $24 y$ and 28.

Rewrite each term as a product using the GCF.
Factor the GCF.
Check by mulitplying.
$4 (y^{2} + 6 y + 7)$
$4 \cdot y^{2} + 4 \cdot 6 y + 4 \cdot 7$
$4 y^{2} + 24 y + 28 ✓$

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Factor: $5 x^{2} - 25 x + 15$ .

$5 (x^{2} - 5 x + 3)$

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Factor: $3 y^{2} - 12 y + 27$ .

$3 (y^{2} - 4 y + 9)$

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Factor: $5 x^{3} - 25 x^{2}$ .

Solution

Find the GCF of $5 x^{3}$ and $25 x^{2} .$

Rewrite each term.
Factor the GCF.
Check.
$5 x^{2} (x - 5)$
$5 x^{2} \cdot x - 5 x^{2} \cdot 5$
$5 x^{3} - 25 x^{2} ✓$

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Factor: $2 x^{3} + 12 x^{2}$ .

$2 x^{2} (x + 6)$

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Factor: $6 y^{3} - 15 y^{2}$ .

$3 y^{2} (2 y - 5)$

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Factor: $21 x^{3} - 9 x^{2} + 15 x$ .

Solution

In a previous example we found the GCF of $21 x^{3}, 9 x^{2}, 15 x$ to be $3 x$ .


Rewrite each term using the GCF, 3 x .
Factor the GCF.
Check.
$3 x (7 x^{2} - 3 x + 5)$
$3 x \cdot 7 x^{2} - 3 x \cdot 3 x + 3 x \cdot 5$
$21 x^{3} - 9 x^{2} + 15 x ✓$

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Factor: $20 x^{3} - 10 x^{2} + 14 x$ .

$2 x (10 x^{2} - 5 x + 7)$

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Factor: $24 y^{3} - 12 y^{2} - 20 y$ .

$4 y (6 y^{2} - 3 y - 5)$

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Factor: $8 m^{3} - 12 m^{2} n + 20 m n^{2}$ .

Solution

Find the GCF of $8 m^{3}$ , $12 m^{2} n$ , $20 m n^{2}$ .

Rewrite each term.
Factor the GCF.
Check.
$4 m (2 m^{2} - 3 m n + 5 n^{2})$
$4 m \cdot 2 m^{2} - 4 m \cdot 3 m n + 4 m \cdot 5 n^{2}$
$8 m^{3} - 12 m^{2} n + 20 m n^{2} ✓$

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Factor: $9 x y^{2} + 6 x^{2} y^{2} + 21 y^{3}$ .

$3 y^{2} (3 x + 2 x^{2} + 7 y)$

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Factor: $3 p^{3} - 6 p^{2} q + 9 p q^{3}$ .

$3 p (p^{2} - 2 p q + 3 q^{2})$

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When the leading coefficient is negative, we factor the negative out as part of the GCF.

Factor: $−8 y - 24$ .

Solution

When the leading coefficient is negative, the GCF will be negative.

Ignoring the signs of the terms, we first find the GCF of 8 y and 24 is 8. Since the expression −8 y − 24 has a negative leading coefficient, we use −8 as the GCF.
Rewrite each term using the GCF.
Factor the GCF.
Check.
$−8 (y + 3)$
$−8 \cdot y + (−8) \cdot 3$
$−8 y - 24 ✓$

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Source: OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2

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