SOLUTION: factor by grouping: 8x^2-34x+21

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Question 152455: factor by grouping:
8x^2-34x+21
Found 2 solutions by checkley77, jim_thompson5910:
Answer by checkley77(12844) About Me  (Show Source):
You can put this solution on YOUR website!
8x^2-34x+21
(4x-3)(2x-7)

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

Looking at the expression 8x%5E2-34x%2B21, we can see that the first coefficient is 8, the second coefficient is -34, and the last term is 21.


Now multiply the first coefficient 8 by the last term 21 to get %288%29%2821%29=168.


Now the question is: what two whole numbers multiply to 168 (the previous product) and add to the second coefficient -34?


To find these two numbers, we need to list all of the factors of 168 (the previous product).


Factors of 168:
1,2,3,4,6,7,8,12,14,21,24,28,42,56,84,168
-1,-2,-3,-4,-6,-7,-8,-12,-14,-21,-24,-28,-42,-56,-84,-168


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


These factors pair up and multiply to 168.
1*168
2*84
3*56
4*42
6*28
7*24
8*21
12*14
(-1)*(-168)
(-2)*(-84)
(-3)*(-56)
(-4)*(-42)
(-6)*(-28)
(-7)*(-24)
(-8)*(-21)
(-12)*(-14)

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -34:


First NumberSecond NumberSum
11681+168=169
2842+84=86
3563+56=59
4424+42=46
6286+28=34
7247+24=31
8218+21=29
121412+14=26
-1-168-1+(-168)=-169
-2-84-2+(-84)=-86
-3-56-3+(-56)=-59
-4-42-4+(-42)=-46
-6-28-6+(-28)=-34
-7-24-7+(-24)=-31
-8-21-8+(-21)=-29
-12-14-12+(-14)=-26



From the table, we can see that the two numbers -6 and -28 add to -34 (the middle coefficient).


So the two numbers -6 and -28 both multiply to 168 and add to -34


Now replace the middle term -34x with -6x-28x. Remember, -6 and -28 add to -34. So this shows us that -6x-28x=-34x.


8x%5E2%2Bhighlight%28-6x-28x%29%2B21 Replace the second term -34x with -6x-28x.


%288x%5E2-6x%29%2B%28-28x%2B21%29 Group the terms into two pairs.


2x%284x-3%29%2B%28-28x%2B21%29 Factor out the GCF 2x from the first group.


2x%284x-3%29-7%284x-3%29 Factor out 7 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.


%282x-7%29%284x-3%29 Combine like terms. Or factor out the common term 4x-3

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Answer:


So 8x%5E2-34x%2B21 factors to %282x-7%29%284x-3%29.


Note: you can check the answer by FOILing %282x-7%29%284x-3%29 to get 8x%5E2-34x%2B21 or by graphing the original expression and the answer (the two graphs should be identical).