SOLUTION: Factor Completely 4x^4 + 8x^2 - 5

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Question 474273: Factor Completely
4x^4 + 8x^2 - 5
Answer by MathLover1(20850) About Me  (Show Source):
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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression 4x%5E2%2B8x-5, we can see that the first coefficient is 4, the second coefficient is 8, and the last term is -5.



Now multiply the first coefficient 4 by the last term -5 to get %284%29%28-5%29=-20.



Now the question is: what two whole numbers multiply to -20 (the previous product) and add to the second coefficient 8?



To find these two numbers, we need to list all of the factors of -20 (the previous product).



Factors of -20:

1,2,4,5,10,20

-1,-2,-4,-5,-10,-20



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



These factors pair up and multiply to -20.

1*(-20) = -20
2*(-10) = -20
4*(-5) = -20
(-1)*(20) = -20
(-2)*(10) = -20
(-4)*(5) = -20


Now let's add up each pair of factors to see if one pair adds to the middle coefficient 8:



First NumberSecond NumberSum
1-201+(-20)=-19
2-102+(-10)=-8
4-54+(-5)=-1
-120-1+20=19
-210-2+10=8
-45-4+5=1




From the table, we can see that the two numbers -2 and 10 add to 8 (the middle coefficient).



So the two numbers -2 and 10 both multiply to -20 and add to 8



Now replace the middle term 8x with -2x%2B10x. Remember, -2 and 10 add to 8. So this shows us that -2x%2B10x=8x.



4x%5E2%2Bhighlight%28-2x%2B10x%29-5 Replace the second term 8x with -2x%2B10x.



%284x%5E2-2x%29%2B%2810x-5%29 Group the terms into two pairs.



2x%282x-1%29%2B%2810x-5%29 Factor out the GCF 2x from the first group.



2x%282x-1%29%2B5%282x-1%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.



%282x%2B5%29%282x-1%29 Combine like terms. Or factor out the common term 2x-1



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Answer:



So 4%2Ax%5E2%2B8%2Ax-5 factors to %282x%2B5%29%282x-1%29.



In other words, 4%2Ax%5E2%2B8%2Ax-5=%282x%2B5%29%282x-1%29.



Note: you can check the answer by expanding %282x%2B5%29%282x-1%29 to get 4%2Ax%5E2%2B8%2Ax-5 or by graphing the original expression and the answer (the two graphs should be identical).