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Now 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.”
We state the Distributive Property here just as you saw it in earlier chapters and “in reverse.”
If $a,b,c$ are real numbers, then
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!
We use “factor” as both a noun and a verb.
Factor: $5a+5$ .
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.  
$5(a+1)$  
$5\cdot a+5\cdot 1$  
$5a+5\u2713$ 
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: $12x60$ .
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.  
$12(x5)$  
$12\cdot x12\cdot 5$  
$12x60\u2713$ 
Now we’ll factor the greatest common factor from a trinomial. We start by finding the GCF of all three terms.
Factor: $4{y}^{2}+24y+28$ .
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.  
$4({y}^{2}+6y+7)$  
$4\cdot {y}^{2}+4\cdot 6y+4\cdot 7$  
$4{y}^{2}+24y+28\u2713$ 
Factor: $5{x}^{3}25{x}^{2}$ .
Find the GCF of $5{x}^{3}$ and $25{x}^{2}.$  


Rewrite each term.  
Factor the GCF.  
Check.  
$5{x}^{2}(x5)$  
$5{x}^{2}\cdot x5{x}^{2}\cdot 5$  
$5{x}^{3}25{x}^{2}\u2713$ 
Factor: $21{x}^{3}9{x}^{2}+15x$ .
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, 3 x .  
Factor the GCF.  
Check.  
$3x(7{x}^{2}3x+5)$  
$3x\cdot 7{x}^{2}3x\cdot 3x+3x\cdot 5$  
$21{x}^{3}9{x}^{2}+15x\u2713$ 
Factor: $20{x}^{3}10{x}^{2}+14x$ .
$2x\left(10{x}^{2}5x+7\right)$
Factor: $24{y}^{3}12{y}^{2}20y$ .
$4y\left(6{y}^{2}3y5\right)$
Factor: $8{m}^{3}12{m}^{2}n+20m{n}^{2}$ .
Find the GCF of $8{m}^{3}$ , $12{m}^{2}n$ , $20m{n}^{2}$ .  


Rewrite each term.  
Factor the GCF.  
Check.  
$4m(2{m}^{2}3mn+5{n}^{2})$  
$4m\cdot 2{m}^{2}4m\cdot 3mn+4m\cdot 5{n}^{2}$  
$8{m}^{3}12{m}^{2}n+20m{n}^{2}\u2713$ 
Factor: $9x{y}^{2}+6{x}^{2}{y}^{2}+21{y}^{3}$ .
$3{y}^{2}\left(3x+2{x}^{2}+7y\right)$
Factor: $3{p}^{3}6{p}^{2}q+9p{q}^{3}$ .
$3p\left({p}^{2}2pq+3{q}^{2}\right)$
When the leading coefficient is negative, we factor the negative out as part of the GCF.
Factor: $\mathrm{8}y24$ .
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.  
$\mathrm{8}(y+3)$  
$\mathrm{8}\cdot y+\left(\mathrm{8}\right)\cdot 3$  
$\mathrm{8}y24\u2713$ 
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