document.write( "Question 112183: How do I completely factor the polynomial x^2-6x-27 \n" ); document.write( "

Algebra.Com's Answer #81832 by jim_thompson5910(35256) $\"\"$ $\"About$

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

Looking at the expression $\"x%5E2-6x-27\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-6\"$ , and the last term is $\"-27\"$ .

Now multiply the first coefficient $\"1\"$ by the last term $\"-27\"$ to get $\"%281%29%28-27%29=-27\"$ .

Now the question is: what two whole numbers multiply to $\"-27\"$ (the previous product) and add to the second coefficient $\"-6\"$ ?

To find these two numbers, we need to list all of the factors of $\"-27\"$ (the previous product).

Factors of $\"-27\"$ :

1,3,9,27

-1,-3,-9,-27

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

These factors pair up and multiply to $\"-27\"$ .

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

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"-6\"$ :

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First Number	Second Number	Sum
1	-27	1+(-27)=-26
3	-9	3+(-9)=-6
-1	27	-1+27=26
-3	9	-3+9=6

From the table, we can see that the two numbers $\"3\"$ and $\"-9\"$ add to $\"-6\"$ (the middle coefficient).

So the two numbers $\"3\"$ and $\"-9\"$ both multiply to $\"-27\"$ and add to $\"-6\"$

Now replace the middle term $\"-6x\"$ with $\"3x-9x\"$ . Remember, $\"3\"$ and $\"-9\"$ add to $\"-6\"$ . So this shows us that $\"3x-9x=-6x\"$ .

$\"x%5E2%2Bhighlight%283x-9x%29-27\"$ Replace the second term $\"-6x\"$ with $\"3x-9x\"$ .

$\"%28x%5E2%2B3x%29%2B%28-9x-27%29\"$ Group the terms into two pairs.

$\"x%28x%2B3%29%2B%28-9x-27%29\"$ Factor out the GCF $\"x\"$ from the first group.

$\"x%28x%2B3%29-9%28x%2B3%29\"$ 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.

$\"%28x-9%29%28x%2B3%29\"$ Combine like terms. Or factor out the common term $\"x%2B3\"$

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Answer:

So $\"x%5E2-6%2Ax-27\"$ factors to $\"%28x-9%29%28x%2B3%29\"$ .

In other words, $\"x%5E2-6%2Ax-27=%28x-9%29%28x%2B3%29\"$ .

Note: you can check the answer by expanding $\"%28x-9%29%28x%2B3%29\"$ to get $\"x%5E2-6%2Ax-27\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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