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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 $14{x}^{3},\phantom{\rule{0.2em}{0ex}}8{x}^{2},\phantom{\rule{0.2em}{0ex}}10x.$
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}
Just like in arithmetic, where it is sometimes useful to represent a number in factored form (for example, $12$ as $2\xb76\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}3\xb74\text{),}$ 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:
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”.
If $a,\phantom{\rule{0.2em}{0ex}}b,\phantom{\rule{0.2em}{0ex}}c$ 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 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: $2x+14.$
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.  $2(x+7)$  
Step 4: Check by multiplying the factors.  Check:

Notice that in [link] , we used the word factor as both a noun and a verb:
Factor: $3a+3.$
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.  
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: $12x60.$
Rewrite each term as a product using the GCF.  
Factor the GCF.  
Check by multiplying the factors.  
Now we’ll factor the greatest common factor from a trinomial . We start by finding the GCF of all three terms.
Factor: $3{y}^{2}+6y+9.$
Rewrite each term as a product using the GCF.  
Factor the GCF.  
Check by multiplying.  
In the next example, we factor a variable from a binomial .
Factor: $6{x}^{2}+5x.$
$6{x}^{2}+5x$  
Rewrite each term as a product.  
Factor the GCF.  $x(6x+5)$ 
Check by multiplying.  
$x(6x+5)$
$x\cdot 6x+x\cdot 5$ $6{x}^{2}+5x\u2713$ 
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