SOLUTION: Factor completely. If the polynomial is prime, state this. -3b-88+b^2

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Question 165723: Factor completely. If the polynomial is prime, state this.
-3b-88+b^2
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
-3b-88%2Bb%5E2 Start with the given expression.


b%5E2-3b-88 Rearrange the terms.


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:



First NumberSecond NumberSum
1-881+(-88)=-87
2-442+(-44)=-42
4-224+(-22)=-18
8-118+(-11)=-3
-188-1+88=87
-244-2+44=42
-422-4+22=18
-811-8+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).