document.write( "Question 896620: Factor the expression.
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Algebra.Com's Answer #543656 by richwmiller(17219) $\"\"$ $\"About$

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

Looking at the expression $\"x%5E2-18x%2B81\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-18\"$ , and the last term is $\"81\"$ .

Now multiply the first coefficient $\"1\"$ by the last term $\"81\"$ to get $\"%281%29%2881%29=81\"$ .

Now the question is: what two whole numbers multiply to $\"81\"$ (the previous product) and add to the second coefficient $\"-18\"$ ?

To find these two numbers, we need to list all of the factors of $\"81\"$ (the previous product).

Factors of $\"81\"$ :

1,3,9,27,81

-1,-3,-9,-27,-81

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

These factors pair up and multiply to $\"81\"$ .

1*81 = 81
3*27 = 81
9*9 = 81
(-1)*(-81) = 81
(-3)*(-27) = 81
(-9)*(-9) = 81

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"-18\"$ :

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First Number	Second Number	Sum
1	81	1+81=82
3	27	3+27=30
9	9	9+9=18
-1	-81	-1+(-81)=-82
-3	-27	-3+(-27)=-30
-9	-9	-9+(-9)=-18

From the table, we can see that the two numbers $\"-9\"$ and $\"-9\"$ add to $\"-18\"$ (the middle coefficient).

So the two numbers $\"-9\"$ and $\"-9\"$ both multiply to $\"81\"$ and add to $\"-18\"$

Now replace the middle term $\"-18x\"$ with $\"-9x-9x\"$ . Remember, $\"-9\"$ and $\"-9\"$ add to $\"-18\"$ . So this shows us that $\"-9x-9x=-18x\"$ .

$\"x%5E2%2Bhighlight%28-9x-9x%29%2B81\"$ Replace the second term $\"-18x\"$ with $\"-9x-9x\"$ .

$\"%28x%5E2-9x%29%2B%28-9x%2B81%29\"$ Group the terms into two pairs.

$\"x%28x-9%29%2B%28-9x%2B81%29\"$ Factor out the GCF $\"x\"$ from the first group.

$\"x%28x-9%29-9%28x-9%29\"$ Factor out $\"9\"$ 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.

$\"%28x-9%29%28x-9%29\"$ Combine like terms. Or factor out the common term $\"x-9\"$

$\"%28x-9%29%5E2\"$ Condense the terms.

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Answer:

So $\"x%5E2-18%2Ax%2B81\"$ factors to $\"%28x-9%29%5E2\"$ .

In other words, $\"x%5E2-18%2Ax%2B81=%28x-9%29%5E2\"$ .

Note: you can check the answer by expanding $\"%28x-9%29%5E2\"$ to get $\"x%5E2-18%2Ax%2B81\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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