SOLUTION: Hi, I am trying to figure out this factoring of trinomials. I didn't see where there were any questions on this particular subject, but thought you could help. 14b^3-33b^2+

Algebra ->  Distributive-associative-commutative-properties -> SOLUTION: Hi, I am trying to figure out this factoring of trinomials. I didn't see where there were any questions on this particular subject, but thought you could help. 14b^3-33b^2+      Log On


   

Question 148737: Hi, I am trying to figure out this factoring of trinomials. I didn't see where there were any questions on this particular subject, but thought you could help.
14b^3-33b^2+18b
Thank you for any help you are able to give.
Found 2 solutions by oscargut, jim_thompson5910:
Answer by oscargut(2103) About Me  (Show Source):
You can put this solution on YOUR website!
14b^3-33b^2+18b=b(14b^2-33b+18)

14b^2-33b+18=0 then
b+=+%28-%28-33%29+%2B-+sqrt%28+%28-33%29%5E2-4%2A14%2A18+%29%29%2F%282%2A14%29+ =
b+=+%28-%28-33%29+%2B-+sqrt%2881%29%29%2F%2828%29+=
b+=+%2833+%2B-+9%29%2F%2828%29+
then b=42/28 or x=24/28
then b=3/2 or x=6/7
then
14b^2-33b+18=14(b-3/2)(b-6/7)
Answer:14b^3-33b^2+18b = 14b(b-3/2)(b-6/7)


Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
14b%5E3-33b%5E2%2B18b Start with the given expression


b%2814b%5E2-33b%2B18%29 Factor out the GCF b


Now let's focus on the inner expression 14b%5E2-33b%2B18




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Looking at the expression 14b%5E2-33b%2B18, we can see that the first coefficient is 14, the second coefficient is -33, and the last term is 18.


Now multiply the first coefficient 14 by the last term 18 to get %2814%29%2818%29=252.


Now the question is: what two whole numbers multiply to 252 (the previous product) and add to the second coefficient -33?


To find these two numbers, we need to list all of the factors of 252 (the previous product).


Factors of 252:
1,2,3,4,6,7,9,12,14,18,21,28,36,42,63,84,126,252
-1,-2,-3,-4,-6,-7,-9,-12,-14,-18,-21,-28,-36,-42,-63,-84,-126,-252


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


These factors pair up and multiply to 252. For instance, 1%2A252=252, 2%2A126=252, etc.


Since 252 is positive, this means that either
a) both factors are positive, or...
b) both factors are negative.


Now let's add up each pair of factors to see if one pair adds to the middle coefficient -33:


First NumberSecond NumberSum
12521+252=253
21262+126=128
3843+84=87
4634+63=67
6426+42=48
7367+36=43
9289+28=37
122112+21=33
141814+18=32
-1-252-1+(-252)=-253
-2-126-2+(-126)=-128
-3-84-3+(-84)=-87
-4-63-4+(-63)=-67
-6-42-6+(-42)=-48
-7-36-7+(-36)=-43
-9-28-9+(-28)=-37
-12-21-12+(-21)=-33
-14-18-14+(-18)=-32



From the table, we can see that the two numbers -12 and -21 add to -33 (the middle coefficient).


So the two numbers -12 and -21 both multiply to 252 and add to -33


Now replace the middle term -33b with -12b-21b. Remember, -12 and -21 add to -33. So this shows us that -12b-21b=-33b.


14b%5E2%2Bhighlight%28-12b-21b%29%2B18 Replace the second term with -12b-21b.


%2814b%5E2-12b%29%2B%28-21b%2B18%29 Group the terms into two pairs.


2b%287b-6%29%2B%28-21b%2B18%29 Factor out the GCF 2b from the first group.


2b%287b-6%29-3%287b-6%29 Factor out 3 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.


%282b-3%29%287b-6%29 Combine like terms. Or factor out the common term 7b-6


So 14b%5E2-33b%2B18 factors to %282b-3%29%287b-6%29.



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b%2814b%5E2-33b%2B18%29 Now go back to the second step.



b%282b-3%29%287b-6%29 Replace 14b%5E2-33b%2B18 with %282b-3%29%287b-6%29.


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Answer:



So 14b%5E3-33b%5E2%2B18b factors to b%282b-3%29%287b-6%29.