document.write( "Question 109539: Factor completely. If it is prime, say so.\r
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Algebra.Com's Answer #79849 by jim_thompson5910(35256) $\"\"$ $\"About$

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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

$\"9%2Ax%5E2-9%2Ax-54\"$ Start with the given expression.

$\"9%28x%5E2-x-6%29\"$ Factor out the GCF $\"9\"$ .

Now let's try to factor the inner expression $\"x%5E2-x-6\"$

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Looking at the expression $\"x%5E2-x-6\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-1\"$ , and the last term is $\"-6\"$ .

Now multiply the first coefficient $\"1\"$ by the last term $\"-6\"$ to get $\"%281%29%28-6%29=-6\"$ .

Now the question is: what two whole numbers multiply to $\"-6\"$ (the previous product) and add to the second coefficient $\"-1\"$ ?

To find these two numbers, we need to list all of the factors of $\"-6\"$ (the previous product).

Factors of $\"-6\"$ :

1,2,3,6

-1,-2,-3,-6

Note: list the negative of each factor. This will allow us to find all possible combinations.

These factors pair up and multiply to $\"-6\"$ .

1*(-6) = -6
2*(-3) = -6
(-1)*(6) = -6
(-2)*(3) = -6

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"-1\"$ :

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First Number	Second Number	Sum
1	-6	1+(-6)=-5
2	-3	2+(-3)=-1
-1	6	-1+6=5
-2	3	-2+3=1

From the table, we can see that the two numbers $\"2\"$ and $\"-3\"$ add to $\"-1\"$ (the middle coefficient).

So the two numbers $\"2\"$ and $\"-3\"$ both multiply to $\"-6\"$ and add to $\"-1\"$

Now replace the middle term $\"-1x\"$ with $\"2x-3x\"$ . Remember, $\"2\"$ and $\"-3\"$ add to $\"-1\"$ . So this shows us that $\"2x-3x=-1x\"$ .

$\"x%5E2%2Bhighlight%282x-3x%29-6\"$ Replace the second term $\"-1x\"$ with $\"2x-3x\"$ .

$\"%28x%5E2%2B2x%29%2B%28-3x-6%29\"$ Group the terms into two pairs.

$\"x%28x%2B2%29%2B%28-3x-6%29\"$ Factor out the GCF $\"x\"$ from the first group.

$\"x%28x%2B2%29-3%28x%2B2%29\"$ Factor out $\"3\"$ from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.

$\"%28x-3%29%28x%2B2%29\"$ Combine like terms. Or factor out the common term $\"x%2B2\"$

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So $\"9%28x%5E2-x-6%29\"$ then factors further to $\"9%28x-3%29%28x%2B2%29\"$

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Answer:

So $\"9%2Ax%5E2-9%2Ax-54\"$ completely factors to $\"9%28x-3%29%28x%2B2%29\"$ .

In other words, $\"9%2Ax%5E2-9%2Ax-54=9%28x-3%29%28x%2B2%29\"$ .

Note: you can check the answer by expanding $\"9%28x-3%29%28x%2B2%29\"$ to get $\"9%2Ax%5E2-9%2Ax-54\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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