SOLUTION: Factor the polymomial completely. If a ploynomial is prime say so. 9bn^3+15bn^2-14bn

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Question 120310: Factor the polymomial completely. If a ploynomial is prime say so.
9bn^3+15bn^2-14bn
Found 2 solutions by checkley71, jim_thompson5910:
Answer by checkley71(8403) About Me  (Show Source):
You can put this solution on YOUR website!
9BN^3+15BN^2-14BN
BN(9N^2+15N-14)
BN(3N-2)(3N+7) ANSWER.

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

9bn%5E3%2B15bn%5E2-14bn Start with the given expression


bn%289n%5E2%2B15n-14%29 Factor out the GCF bn


Now let's focus on the inner expression 9n%5E2%2B15n-14




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Looking at 9n%5E2%2B15n-14 we can see that the first term is 9n%5E2 and the last term is -14 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 NumberSecond NumberSum
1-1261+(-126)=-125
2-632+(-63)=-61
3-423+(-42)=-39
6-216+(-21)=-15
7-187+(-18)=-11
9-149+(-14)=-5
-1126-1+126=125
-263-2+63=61
-342-3+42=39
-621-6+21=15
-718-7+18=11
-914-9+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 9n%5E2%2B15n-14, replace 15n with -6n%2B21n (notice -6n%2B21n adds up to 15n. So it is equivalent to 15n)

9n%5E2%2Bhighlight%28-6n%2B21n%29%2B-14


Now let's factor 9n%5E2-6n%2B21n-14 by grouping:


%289n%5E2-6n%29%2B%2821n-14%29 Group like terms


3n%283n-2%29%2B7%283n-2%29 Factor out the GCF of 3n out of the first group. Factor out the GCF of 7 out of the second group


%283n%2B7%29%283n-2%29 Since we have a common term of 3n-2, we can combine like terms

So 9n%5E2-6n%2B21n-14 factors to %283n%2B7%29%283n-2%29


So this also means that 9n%5E2%2B15n-14 factors to %283n%2B7%29%283n-2%29 (since 9n%5E2%2B15n-14 is equivalent to 9n%5E2-6n%2B21n-14)



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So our expression goes from bn%289n%5E2%2B15n-14%29 and factors further to bn%283n%2B7%29%283n-2%29


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Answer:

So 9bn%5E3%2B15bn%5E2-14bn factors to bn%283n%2B7%29%283n-2%29