SOLUTION: Does anyone have a easy way to factor? Her is a problem I just don't know how to do. 21x^2-25x-4 Thanks!!

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Question 582096: Does anyone have a easy way to factor? Her is a problem I just don't know how to do.
21x^2-25x-4
Thanks!!
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression 21x%5E2-25x-4, we can see that the first coefficient is 21, the second coefficient is -25, and the last term is -4.


Now multiply the first coefficient 21 by the last term -4 to get %2821%29%28-4%29=-84.


Now the question is: what two whole numbers multiply to -84 (the previous product) and add to the second coefficient -25?


To find these two numbers, we need to list all of the factors of -84 (the previous product).


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


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


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

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -25:


First NumberSecond NumberSum
1-841+(-84)=-83
2-422+(-42)=-40
3-283+(-28)=-25
4-214+(-21)=-17
6-146+(-14)=-8
7-127+(-12)=-5
-184-1+84=83
-242-2+42=40
-328-3+28=25
-421-4+21=17
-614-6+14=8
-712-7+12=5



From the table, we can see that the two numbers 3 and -28 add to -25 (the middle coefficient).


So the two numbers 3 and -28 both multiply to -84 and add to -25


Now replace the middle term -25x with 3x-28x. Remember, 3 and -28 add to -25. So this shows us that 3x-28x=-25x.


21x%5E2%2Bhighlight%283x-28x%29-4 Replace the second term -25x with 3x-28x.


%2821x%5E2%2B3x%29%2B%28-28x-4%29 Group the terms into two pairs.


3x%287x%2B1%29%2B%28-28x-4%29 Factor out the GCF 3x from the first group.


3x%287x%2B1%29-4%287x%2B1%29 Factor out 4 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.


%283x-4%29%287x%2B1%29 Combine like terms. Or factor out the common term 7x%2B1


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Answer:


So 21x%5E2-25x-4 factors to %283x-4%29%287x%2B1%29.


In other words, 21x%5E2-25x-4=%283x-4%29%287x%2B1%29.


Note: you can check the answer by expanding %283x-4%29%287x%2B1%29 to get 21x%5E2-25x-4 or by graphing the original expression and the answer (the two graphs should be identical).