SOLUTION: Can someone please, please help me with this one? 16x^4 - 40x^2 + 9

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Question 102564: Can someone please, please help me with this one?
16x^4 - 40x^2 + 9
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let t=x%5E2

So we now get

16t%5E2+-+40t+%2B+9

Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression 16t%5E2-40t%2B9, we can see that the first coefficient is 16, the second coefficient is -40, and the last term is 9.



Now multiply the first coefficient 16 by the last term 9 to get %2816%29%289%29=144.



Now the question is: what two whole numbers multiply to 144 (the previous product) and add to the second coefficient -40?



To find these two numbers, we need to list all of the factors of 144 (the previous product).



Factors of 144:

1,2,3,4,6,8,9,12,16,18,24,36,48,72,144

-1,-2,-3,-4,-6,-8,-9,-12,-16,-18,-24,-36,-48,-72,-144



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



These factors pair up and multiply to 144.

1*144 = 144
2*72 = 144
3*48 = 144
4*36 = 144
6*24 = 144
8*18 = 144
9*16 = 144
12*12 = 144
(-1)*(-144) = 144
(-2)*(-72) = 144
(-3)*(-48) = 144
(-4)*(-36) = 144
(-6)*(-24) = 144
(-8)*(-18) = 144
(-9)*(-16) = 144
(-12)*(-12) = 144


Now let's add up each pair of factors to see if one pair adds to the middle coefficient -40:



First NumberSecond NumberSum
11441+144=145
2722+72=74
3483+48=51
4364+36=40
6246+24=30
8188+18=26
9169+16=25
121212+12=24
-1-144-1+(-144)=-145
-2-72-2+(-72)=-74
-3-48-3+(-48)=-51
-4-36-4+(-36)=-40
-6-24-6+(-24)=-30
-8-18-8+(-18)=-26
-9-16-9+(-16)=-25
-12-12-12+(-12)=-24




From the table, we can see that the two numbers -4 and -36 add to -40 (the middle coefficient).



So the two numbers -4 and -36 both multiply to 144 and add to -40



Now replace the middle term -40t with -4t-36t. Remember, -4 and -36 add to -40. So this shows us that -4t-36t=-40t.



16t%5E2%2Bhighlight%28-4t-36t%29%2B9 Replace the second term -40t with -4t-36t.



%2816t%5E2-4t%29%2B%28-36t%2B9%29 Group the terms into two pairs.



4t%284t-1%29%2B%28-36t%2B9%29 Factor out the GCF 4t from the first group.



4t%284t-1%29-9%284t-1%29 Factor out 9 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.



%284t-9%29%284t-1%29 Combine like terms. Or factor out the common term 4t-1



===============================================================



Answer:



So 16%2At%5E2-40%2At%2B9 factors to %284t-9%29%284t-1%29.



In other words, 16%2At%5E2-40%2At%2B9=%284t-9%29%284t-1%29.



Note: you can check the answer by expanding %284t-9%29%284t-1%29 to get 16%2At%5E2-40%2At%2B9 or by graphing the original expression and the answer (the two graphs should be identical).





Since t=x%5E2, we can replace each t with x%5E2 to get

%284x%5E2-9%29%284x%5E2-1%29