SOLUTION: Factor Completely 16x^4-40x^2+9

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Question 158061: Factor Completely
16x^4-40x^2+9
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let z=x%5E2. So this means that z%5E2=x%5E4


So the expression goes from 16x%5E4-40x%5E2%2B9 to 16z%5E2-40z%2B9




Looking at the expression 16z%5E2-40z%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
2*72
3*48
4*36
6*24
8*18
9*16
12*12
(-1)*(-144)
(-2)*(-72)
(-3)*(-48)
(-4)*(-36)
(-6)*(-24)
(-8)*(-18)
(-9)*(-16)
(-12)*(-12)

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 -40z with -4z-36z. Remember, -4 and -36 add to -40. So this shows us that -4z-36z=-40z.


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


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


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


4z%284z-1%29-9%284z-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.


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


%284x%5E2-9%29%284x%5E2-1%29 Plug in z=x%5E2


%282x%2B3%29%282x-3%29%282x%2B1%29%282x-1%29 Factor each group using the difference of squares.

Note: 4x%5E2-9 factors to %282x%2B3%29%282x-3%29 and 4x%5E2-1 factors to %282x%2B1%29%282x-1%29


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Answer:


So 16x%5E4-40x%5E2%2B9 completely factors to %282x%2B3%29%282x-3%29%282x%2B1%29%282x-1%29.