SOLUTION: x^4-20x^3+96x^2 could you help me solve this problem on factoring trinomials.

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Question 171902This question is from textbook
: x^4-20x^3+96x^2 could you help me solve this problem on factoring trinomials. This question is from textbook

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

x%5E4-20x%5E3%2B96x%5E2 Start with the given expression


x%5E2%28x%5E2-20x%2B96%29 Factor out the GCF x%5E2


Now let's focus on the inner expression x%5E2-20x%2B96




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Looking at 1x%5E2-20x%2B96 we can see that the first term is 1x%5E2 and the last term is 96 where the coefficients are 1 and 96 respectively.

Now multiply the first coefficient 1 and the last coefficient 96 to get 96. Now what two numbers multiply to 96 and add to the middle coefficient -20? Let's list all of the factors of 96:



Factors of 96:
1,2,3,4,6,8,12,16,24,32,48,96

-1,-2,-3,-4,-6,-8,-12,-16,-24,-32,-48,-96 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 96
1*96
2*48
3*32
4*24
6*16
8*12
(-1)*(-96)
(-2)*(-48)
(-3)*(-32)
(-4)*(-24)
(-6)*(-16)
(-8)*(-12)

note: remember two negative numbers multiplied together make a positive number


Now which of these pairs add to -20? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -20

First NumberSecond NumberSum
1961+96=97
2482+48=50
3323+32=35
4244+24=28
6166+16=22
8128+12=20
-1-96-1+(-96)=-97
-2-48-2+(-48)=-50
-3-32-3+(-32)=-35
-4-24-4+(-24)=-28
-6-16-6+(-16)=-22
-8-12-8+(-12)=-20



From this list we can see that -8 and -12 add up to -20 and multiply to 96


Now looking at the expression 1x%5E2-20x%2B96, replace -20x with -8x%2B-12x (notice -8x%2B-12x adds up to -20x. So it is equivalent to -20x)

1x%5E2%2Bhighlight%28-8x%2B-12x%29%2B96


Now let's factor 1x%5E2-8x-12x%2B96 by grouping:


%281x%5E2-8x%29%2B%28-12x%2B96%29 Group like terms


x%28x-8%29-12%28x-8%29 Factor out the GCF of x out of the first group. Factor out the GCF of -12 out of the second group


%28x-12%29%28x-8%29 Since we have a common term of x-8, we can combine like terms

So 1x%5E2-8x-12x%2B96 factors to %28x-12%29%28x-8%29


So this also means that 1x%5E2-20x%2B96 factors to %28x-12%29%28x-8%29 (since 1x%5E2-20x%2B96 is equivalent to 1x%5E2-8x-12x%2B96)



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So our expression goes from x%5E2%28x%5E2-20x%2B96%29 and factors further to x%5E2%28x-12%29%28x-8%29


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Answer:

So x%5E4-20x%5E3%2B96x%5E2 factors to x%5E2%28x-12%29%28x-8%29