SOLUTION: Factor polynomial completely 9bn3+15bn2-14bn

Question 127507: Factor polynomial completely
9bn3+15bn2-14bn
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!

Start with the given expression

Factor out the GCF

Now let's focus on the inner expression

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Looking at we can see that the first term is and the last term is where the coefficients are 9 and -14 respectively.

Now multiply the first coefficient 9 and the last coefficient -14 to get -126. Now what two numbers multiply to -126 and add to the middle coefficient 15? Let's list all of the factors of -126:

Factors of -126:
1,2,3,6,7,9,14,18,21,42,63,126

-1,-2,-3,-6,-7,-9,-14,-18,-21,-42,-63,-126 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -126
(1)*(-126)
(2)*(-63)
(3)*(-42)
(6)*(-21)
(7)*(-18)
(9)*(-14)
(-1)*(126)
(-2)*(63)
(-3)*(42)
(-6)*(21)
(-7)*(18)
(-9)*(14)

note: remember, the product of a negative and a positive number is a negative number

Now which of these pairs add to 15? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 15

First Number	Second Number	Sum
1	-126	1+(-126)=-125
2	-63	2+(-63)=-61
3	-42	3+(-42)=-39
6	-21	6+(-21)=-15
7	-18	7+(-18)=-11
9	-14	9+(-14)=-5
-1	126	-1+126=125
-2	63	-2+63=61
-3	42	-3+42=39
-6	21	-6+21=15
-7	18	-7+18=11
-9	14	-9+14=5

From this list we can see that -6 and 21 add up to 15 and multiply to -126

Now looking at the expression , replace with (notice adds up to . So it is equivalent to )

Now let's factor by grouping:

Group like terms

Factor out the GCF of out of the first group. Factor out the GCF of out of the second group

Since we have a common term of , we can combine like terms

So factors to

So this also means that factors to (since is equivalent to )

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So our expression goes from and factors further to

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Answer:

So factors to

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