SOLUTION: 6. How do I complete this using factoring? I appreciate the help. 8x^2 - 22x - 21

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: 6. How do I complete this using factoring? I appreciate the help. 8x^2 - 22x - 21      Log On


   

Question 156340: 6. How do I complete this using factoring? I appreciate the help.

8x^2 - 22x - 21
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression 8x%5E2-22x-21, we can see that the first coefficient is 8, the second coefficient is -22, and the last term is -21.


Now multiply the first coefficient 8 by the last term -21 to get %288%29%28-21%29=-168.


Now the question is: what two whole numbers multiply to -168 (the previous product) and add to the second coefficient -22?


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 -22:


First NumberSecond NumberSum
1-1681+(-168)=-167
2-842+(-84)=-82
3-563+(-56)=-53
4-424+(-42)=-38
6-286+(-28)=-22
7-247+(-24)=-17
8-218+(-21)=-13
12-1412+(-14)=-2
-1168-1+168=167
-284-2+84=82
-356-3+56=53
-442-4+42=38
-628-6+28=22
-724-7+24=17
-821-8+21=13
-1214-12+14=2



From the table, we can see that the two numbers 6 and -28 add to -22 (the middle coefficient).


So the two numbers 6 and -28 both multiply to -168 and add to -22


Now replace the middle term -22x with 6x-28x. Remember, 6 and -28 add to -22. So this shows us that 6x-28x=-22x.


8x%5E2%2Bhighlight%286x-28x%29-21 Replace the second term -22x with 6x-28x.


%288x%5E2%2B6x%29%2B%28-28x-21%29 Group the terms into two pairs.


2x%284x%2B3%29%2B%28-28x-21%29 Factor out the GCF 2x from the first group.


2x%284x%2B3%29-7%284x%2B3%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%2B3%29 Combine like terms. Or factor out the common term 4x%2B3

---------------------------------------------


Answer:


So 8x%5E2-22x-21 factors to %282x-7%29%284x%2B3%29.


Note: you can check the answer by FOILing %282x-7%29%284x%2B3%29 to get 8x%5E2-22x-21 or by graphing the original expression and the answer (the two graphs should be identical).