document.write( "Question 461734: Hi.
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Algebra.Com's Answer #316577 by MathLover1(20849) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"w%5E2-19w%2B88\"$ \r
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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

Looking at the expression $\"x%5E2-19x%2B88\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-19\"$ , and the last term is $\"88\"$ .

Now multiply the first coefficient $\"1\"$ by the last term $\"88\"$ to get $\"%281%29%2888%29=88\"$ .

Now the question is: what two whole numbers multiply to $\"88\"$ (the previous product) and add to the second coefficient $\"-19\"$ ?

To find these two numbers, we need to list all of the factors of $\"88\"$ (the previous product).

Factors of $\"88\"$ :

1,2,4,8,11,22,44,88

-1,-2,-4,-8,-11,-22,-44,-88

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

These factors pair up and multiply to $\"88\"$ .

1*88 = 88
2*44 = 88
4*22 = 88
8*11 = 88
(-1)*(-88) = 88
(-2)*(-44) = 88
(-4)*(-22) = 88
(-8)*(-11) = 88

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"-19\"$ :

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First Number	Second Number	Sum
1	88	1+88=89
2	44	2+44=46
4	22	4+22=26
8	11	8+11=19
-1	-88	-1+(-88)=-89
-2	-44	-2+(-44)=-46
-4	-22	-4+(-22)=-26
-8	-11	-8+(-11)=-19

From the table, we can see that the two numbers $\"-8\"$ and $\"-11\"$ add to $\"-19\"$ (the middle coefficient).

So the two numbers $\"-8\"$ and $\"-11\"$ both multiply to $\"88\"$ and add to $\"-19\"$

Now replace the middle term $\"-19x\"$ with $\"-8x-11x\"$ . Remember, $\"-8\"$ and $\"-11\"$ add to $\"-19\"$ . So this shows us that $\"-8x-11x=-19x\"$ .

$\"x%5E2%2Bhighlight%28-8x-11x%29%2B88\"$ Replace the second term $\"-19x\"$ with $\"-8x-11x\"$ .

$\"%28x%5E2-8x%29%2B%28-11x%2B88%29\"$ Group the terms into two pairs.

$\"x%28x-8%29%2B%28-11x%2B88%29\"$ Factor out the GCF $\"x\"$ from the first group.

$\"x%28x-8%29-11%28x-8%29\"$ Factor out $\"11\"$ 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.

$\"%28x-11%29%28x-8%29\"$ Combine like terms. Or factor out the common term $\"x-8\"$

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Answer:

So $\"x%5E2-19%2Ax%2B88\"$ factors to $\"%28x-11%29%28x-8%29\"$ .

In other words, $\"x%5E2-19%2Ax%2B88=%28x-11%29%28x-8%29\"$ .

Note: you can check the answer by expanding $\"%28x-11%29%28x-8%29\"$ to get $\"x%5E2-19%2Ax%2B88\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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