Factor the Greatest Common Factor from a Polynomial
It is sometimes useful to represent a number as a product of factors, for example, 12 as \(2·6\) or \(3·4\). In algebra, it can also be useful to represent a polynomial in factored form. We will start with a product, such as \(3x^2+15x\), and end with its factors, \(3x(x+5)\). 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, and c are real numbers, then
\[a(b+c)=ab+ac \quad \text{and} \quad ab+ac=a(b+c)\nonumber\]
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!
Example \(\PageIndex{5}\)
Factor: \(9xy^2+6x^2y^2+21y^3\).
 Answer

\(3y^2(3x+2x^2+7y)\)
Example \(\PageIndex{6}\)
Factor: \(3p^3−6p^2q+9pq^3\).
 Answer

\(3p(p^2−2pq+3q^3)\)
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:
\[\begin{array} {ll} \text{Noun:} &\hspace{50mm} 7 \text{ is a factor of }14 \\ \text{Verb:} &\hspace{50mm} \text{factor }3 \text{ from }3a+3\end{array}\nonumber\]
Example \(\PageIndex{7}\)
Factor: \(5x^3−25x^2\).
 Answer

Example \(\PageIndex{8}\)
Factor: \(2x^3+12x^2\).
 Answer

\(2x^2(x+6)\)
Example \(\PageIndex{9}\)
Factor: \(6y^3−15y^2\).
 Answer

\(3y^2(2y−5)\)
Example \(\PageIndex{10}\)
Factor: \(8x^3y−10x^2y^2+12xy^3\).
 Answer

Example \(\PageIndex{11}\)
Factor: \(15x^3y−3x^2y^2+6xy^3\).
 Answer

\(3xy(5x^2−xy+2y^2)\)
Example \(\PageIndex{12}\)
Factor: \(8a^3b+2a^2b^2−6ab^3\).
 Answer

\(2ab(4a^2+ab−3b^2)\)
When the leading coefficient is negative, we factor the negative out as part of the GCF.
Example \(\PageIndex{13}\)
Factor: \(−4a^3+36a^2−8a\).
 Answer

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


Rewrite each term using the GCF, \(−4a\). 

Factor the GCF. 

Check:
\[−4a(a^2−9a+2)\nonumber\]
\[−4a·a^2−(−4a)·9a+(−4a)·2\nonumber\]
\[−4a^3+36a^2−8a\checkmark\nonumber\]


Example \(\PageIndex{14}\)
Factor: \(−4b^3+16b^2−8b\).
 Answer

\(−4b(b^2−4b+2)\)
Example \(\PageIndex{15}\)
Factor: \(−7a^3+21a^2−14a\).
 Answer

\(−7a(a^2−3a+2)\)
So far our greatest common factors have been monomials. In the next example, the greatest common factor is a binomial.
Example \(\PageIndex{16}\)
Factor: \(3y(y+7)−4(y+7)\).
 Answer

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



Factor the GCF, \((y+7)\). 

\((y+7)(3 y4)\) 
Check on your own by multiplying. 


Example \(\PageIndex{17}\)
Factor: \(4m(m+3)−7(m+3)\).
 Answer

\((m+3)(4m−7)\)
Example \(\PageIndex{18}\)
Factor: \(8n(n−4)+5(n−4)\).
 Answer

\((n−4)(8n+5)\)