SOLUTION: Can you help me factor this polynomial completely. -3s^2 - 10s + 8 Thanks for the help.

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Question 101293: Can you help me factor this polynomial completely.
-3s^2 - 10s + 8
Thanks for the help.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


-3%2As%5E2-10%2As%2B8 Start with the given expression.



-%283s%5E2%2B10s-8%29 Factor out the GCF -1.



Now let's try to factor the inner expression 3s%5E2%2B10s-8



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Looking at the expression 3s%5E2%2B10s-8, we can see that the first coefficient is 3, the second coefficient is 10, and the last term is -8.



Now multiply the first coefficient 3 by the last term -8 to get %283%29%28-8%29=-24.



Now the question is: what two whole numbers multiply to -24 (the previous product) and add to the second coefficient 10?



To find these two numbers, we need to list all of the factors of -24 (the previous product).



Factors of -24:

1,2,3,4,6,8,12,24

-1,-2,-3,-4,-6,-8,-12,-24



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



These factors pair up and multiply to -24.

1*(-24) = -24
2*(-12) = -24
3*(-8) = -24
4*(-6) = -24
(-1)*(24) = -24
(-2)*(12) = -24
(-3)*(8) = -24
(-4)*(6) = -24


Now let's add up each pair of factors to see if one pair adds to the middle coefficient 10:



First NumberSecond NumberSum
1-241+(-24)=-23
2-122+(-12)=-10
3-83+(-8)=-5
4-64+(-6)=-2
-124-1+24=23
-212-2+12=10
-38-3+8=5
-46-4+6=2




From the table, we can see that the two numbers -2 and 12 add to 10 (the middle coefficient).



So the two numbers -2 and 12 both multiply to -24 and add to 10



Now replace the middle term 10s with -2s%2B12s. Remember, -2 and 12 add to 10. So this shows us that -2s%2B12s=10s.



3s%5E2%2Bhighlight%28-2s%2B12s%29-8 Replace the second term 10s with -2s%2B12s.



%283s%5E2-2s%29%2B%2812s-8%29 Group the terms into two pairs.



s%283s-2%29%2B%2812s-8%29 Factor out the GCF s from the first group.



s%283s-2%29%2B4%283s-2%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.



%28s%2B4%29%283s-2%29 Combine like terms. Or factor out the common term 3s-2



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So -1%283s%5E2%2B10s-8%29 then factors further to -%28s%2B4%29%283s-2%29



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Answer:



So -3%2As%5E2-10%2As%2B8 completely factors to -%28s%2B4%29%283s-2%29.



In other words, -3%2As%5E2-10%2As%2B8=-%28s%2B4%29%283s-2%29.



Note: you can check the answer by expanding -%28s%2B4%29%283s-2%29 to get -3%2As%5E2-10%2As%2B8 or by graphing the original expression and the answer (the two graphs should be identical).