SOLUTION: Would someone show me how to do this step by step? Factor completely 5s^2 - 22s + 8

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Question 590911: Would someone show me how to do this step by step?
Factor completely
5s^2 - 22s + 8
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression 5s%5E2-22s%2B8, we can see that the first coefficient is 5, the second coefficient is -22, and the last term is 8.


Now multiply the first coefficient 5 by the last term 8 to get %285%29%288%29=40.


Now the question is: what two whole numbers multiply to 40 (the previous product) and add to the second coefficient -22?


To find these two numbers, we need to list all of the factors of 40 (the previous product).


Factors of 40:
1,2,4,5,8,10,20,40
-1,-2,-4,-5,-8,-10,-20,-40


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


These factors pair up and multiply to 40.
1*40 = 40
2*20 = 40
4*10 = 40
5*8 = 40
(-1)*(-40) = 40
(-2)*(-20) = 40
(-4)*(-10) = 40
(-5)*(-8) = 40

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -22:


First NumberSecond NumberSum
1401+40=41
2202+20=22
4104+10=14
585+8=13
-1-40-1+(-40)=-41
-2-20-2+(-20)=-22
-4-10-4+(-10)=-14
-5-8-5+(-8)=-13



From the table, we can see that the two numbers -2 and -20 add to -22 (the middle coefficient).


So the two numbers -2 and -20 both multiply to 40 and add to -22


Now replace the middle term -22s with -2s-20s. Remember, -2 and -20 add to -22. So this shows us that -2s-20s=-22s.


5s%5E2%2Bhighlight%28-2s-20s%29%2B8 Replace the second term -22s with -2s-20s.


%285s%5E2-2s%29%2B%28-20s%2B8%29 Group the terms into two pairs.


s%285s-2%29%2B%28-20s%2B8%29 Factor out the GCF s from the first group.


s%285s-2%29-4%285s-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-4%29%285s-2%29 Combine like terms. Or factor out the common term 5s-2


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Answer:


So 5s%5E2-22s%2B8 factors to %28s-4%29%285s-2%29.


In other words, 5s%5E2-22s%2B8=%28s-4%29%285s-2%29.


Note: you can check the answer by expanding %28s-4%29%285s-2%29 to get 5s%5E2-22s%2B8 or by graphing the original expression and the answer (the two graphs should be identical).

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