document.write( "Question 165723: Factor completely. If the polynomial is prime, state this.\r
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Algebra.Com's Answer #122171 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"-3b-88%2Bb%5E2\"$ Start with the given expression.\r
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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

Looking at the expression $\"b%5E2-3b-88\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-3\"$ , and the last term is $\"-88\"$ .

Now multiply the first coefficient $\"1\"$ by the last term $\"-88\"$ to get $\"%281%29%28-88%29=-88\"$ .

Now the question is: what two whole numbers multiply to $\"-88\"$ (the previous product) and add to the second coefficient $\"-3\"$ ?

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 $\"-3\"$ :

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First Number	Second Number	Sum
1	-88	1+(-88)=-87
2	-44	2+(-44)=-42
4	-22	4+(-22)=-18
8	-11	8+(-11)=-3
-1	88	-1+88=87
-2	44	-2+44=42
-4	22	-4+22=18
-8	11	-8+11=3

From the table, we can see that the two numbers $\"8\"$ and $\"-11\"$ add to $\"-3\"$ (the middle coefficient).

So the two numbers $\"8\"$ and $\"-11\"$ both multiply to $\"-88\"$ and add to $\"-3\"$

Now replace the middle term $\"-3b\"$ with $\"8b-11b\"$ . Remember, $\"8\"$ and $\"-11\"$ add to $\"-3\"$ . So this shows us that $\"8b-11b=-3b\"$ .

$\"b%5E2%2Bhighlight%288b-11b%29-88\"$ Replace the second term $\"-3b\"$ with $\"8b-11b\"$ .

$\"%28b%5E2%2B8b%29%2B%28-11b-88%29\"$ Group the terms into two pairs.

$\"b%28b%2B8%29%2B%28-11b-88%29\"$ Factor out the GCF $\"b\"$ from the first group.

$\"b%28b%2B8%29-11%28b%2B8%29\"$ 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.

$\"%28b-11%29%28b%2B8%29\"$ Combine like terms. Or factor out the common term $\"b%2B8\"$

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Answer:

So $\"b%5E2-3%2Ab-88\"$ factors to $\"%28b-11%29%28b%2B8%29\"$ .

In other words, $\"b%5E2-3%2Ab-88=%28b-11%29%28b%2B8%29\"$ .

Note: you can check the answer by expanding $\"%28b-11%29%28b%2B8%29\"$ to get $\"b%5E2-3%2Ab-88\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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