Question 203222
<ol><li>Factor each term (mentally or on paper)<ul><li>Factor the coefficient into prime factors</li><li>Rewrite any variables with exponents without exponents. For example, {{{x^6}}} becomes {{{x*x*x*x*x*x}}}</li></ul></li><li>Now you have a list of factors for each term. Compare these lists looking for the longest string of factors which is <b>in every list</b> of factors. Remember, order of factors is <b>not</b> significant! This longest list of common factors is the GCF.</li><li>Rewrite the polynomial in factored form:<ol><li>Write the GCF</li><li>Write a left parenthesis</li><li>Write each term from its list of factors. <b>Omit the factors of the GCF</b>and separate the terms with "+" and "-" as they were in the original polynomial.</li><li>Write a right parenthesis</li></ol></li>Recombine the factors. (Do not use the distributive property!)<ol><li>Multiply together the individual factors of the GCF. </li><li>Multiply together the individual factors of the terms in the parentheses.</li></ol></li></ol>
Here's an example:
{{{12x^7 + 30x^5 - 42x^3 + 60x^2}}}
1. Factor each term
{{{12x^7 = 2*2*3*x*x*x*x*x*x*x}}}
{{{30x^5 = 2*3*5*x*x*x*x*x}}}
{{{42x^3 = 2*3*7*x*x*x}}}
{{{60x^2 = 2*2*3*5*x*x}}}<br>
2. Compare these lists looking for the longest string of factors found in every term. Until you develop a better eye for this you might want to line up the factors like below. Note how the spacing of factors is adjusted as needed to line up the common factors:
<pre>
2*2*3*    x*x*x*x*x*x*x
2*  3*5*  x*x*x*x*x
2*  3*  7*x*x*x
2*2*3*5*  x*x
</pre>
From this point of view we should be able to see the following factors are <b>in all four terms</b>: 2, 3, x and a second x. So the GCF is 2*3*x*x<br>
3. Write the GCF, a left parenthesis, the "remainders" of the other terms and a right parenthsis. Please make sure you understand where all of the following comes from:
{{{2*3*x*x(2*x*x*x*x*x + 5*x*x*x - 7*x + 2*5)}}}<br>
4. Recombine the factors
{{{6x^2(2x^5 + 5x^3 - 7x + 10)}}}