SOLUTION: - 8 w2 + 6 w + 5

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Question 103801: - 8 w2 + 6 w + 5
Answer by jim_thompson5910(35256) About Me  (Show Source):
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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


-8%2Aw%5E2%2B6%2Aw%2B5 Start with the given expression.



-%288w%5E2-6w-5%29 Factor out the GCF -1.



Now let's try to factor the inner expression 8w%5E2-6w-5



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Looking at the expression 8w%5E2-6w-5, we can see that the first coefficient is 8, the second coefficient is -6, and the last term is -5.



Now multiply the first coefficient 8 by the last term -5 to get %288%29%28-5%29=-40.



Now the question is: what two whole numbers multiply to -40 (the previous product) and add to the second coefficient -6?



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 -6:



First NumberSecond NumberSum
1-401+(-40)=-39
2-202+(-20)=-18
4-104+(-10)=-6
5-85+(-8)=-3
-140-1+40=39
-220-2+20=18
-410-4+10=6
-58-5+8=3




From the table, we can see that the two numbers 4 and -10 add to -6 (the middle coefficient).



So the two numbers 4 and -10 both multiply to -40 and add to -6



Now replace the middle term -6w with 4w-10w. Remember, 4 and -10 add to -6. So this shows us that 4w-10w=-6w.



8w%5E2%2Bhighlight%284w-10w%29-5 Replace the second term -6w with 4w-10w.



%288w%5E2%2B4w%29%2B%28-10w-5%29 Group the terms into two pairs.



4w%282w%2B1%29%2B%28-10w-5%29 Factor out the GCF 4w from the first group.



4w%282w%2B1%29-5%282w%2B1%29 Factor out 5 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.



%284w-5%29%282w%2B1%29 Combine like terms. Or factor out the common term 2w%2B1



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So -1%288w%5E2-6w-5%29 then factors further to -%284w-5%29%282w%2B1%29



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Answer:



So -8%2Aw%5E2%2B6%2Aw%2B5 completely factors to -%284w-5%29%282w%2B1%29.



In other words, -8%2Aw%5E2%2B6%2Aw%2B5=-%284w-5%29%282w%2B1%29.



Note: you can check the answer by expanding -%284w-5%29%282w%2B1%29 to get -8%2Aw%5E2%2B6%2Aw%2B5 or by graphing the original expression and the answer (the two graphs should be identical).