Factoring the Greatest Common Factor From a Polynomial

Factoring 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 \(2·6\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}3·4),\) in algebra, it can be useful to represent a polynomial in factored form. One way to do this is by finding the GCF of all the terms. Remember, we multiply a polynomial by a monomial as follows:

Now we will start with a product, like \(2x+14\), 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 tutorials and “in reverse.”

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!

Example: How to Factor the Greatest Common Factor from a Polynomial

Factor: \(4x+12\).

Solution

Factor as a Noun and a Verb

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

Example

Factor: \(5a+5\).

Solution

Find the GCF of 5a 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.
\(5(a+1)\)
\(5\cdot a+5\cdot 1\)
\(5a+5✓\)

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

Example

Factor: \(12x-60\).

Solution

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

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

Example

Factor: \(4{y}^{2}+24y+28\).

Solution

We start by finding the GCF of all three terms.

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

Example

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.
\(5{x}^{2}(x-5)\)
\(5{x}^{2}\cdot x-5{x}^{2}\cdot 5\)
\(5{x}^{3}-25{x}^{2}✓\)

Example

Factor: \(21{x}^{3}-9{x}^{2}+15x\).

Solution

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

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

Example

Factor: \(8{m}^{3}-12{m}^{2}n+20m{n}^{2}\).

Solution

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

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

Example

Factor: \(-8y-24\).

Solution

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

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

Example

Factor: \(-6{a}^{2}+36a\).

Solution

The leading coefficient is negative, so the GCF will be negative.?

Since the leading coefficient is negative, the GCF is negative, −6a.
Rewrite each term using the GCF.
Factor the GCF.
Check.
\(-6a(a-6)\)
\(-6a\cdot a+(-6a)(-6)\)
\(-6{a}^{2}+36a✓\)

Example

Factor: \(5q(q+7)-6(q+7)\).

Solution

The GCF is the binomial \(q+7\).

Factor the GCF, (q + 7).
Check on your own by multiplying.