Question 171902


{{{x^4-20x^3+96x^2}}} Start with the given expression



{{{x^2(x^2-20x+96)}}} Factor out the GCF {{{x^2}}}



Now let's focus on the inner expression {{{x^2-20x+96}}}





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Looking at {{{1x^2-20x+96}}} we can see that the first term is {{{1x^2}}} 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


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">96</td><td>1+96=97</td></tr><tr><td align="center">2</td><td align="center">48</td><td>2+48=50</td></tr><tr><td align="center">3</td><td align="center">32</td><td>3+32=35</td></tr><tr><td align="center">4</td><td align="center">24</td><td>4+24=28</td></tr><tr><td align="center">6</td><td align="center">16</td><td>6+16=22</td></tr><tr><td align="center">8</td><td align="center">12</td><td>8+12=20</td></tr><tr><td align="center">-1</td><td align="center">-96</td><td>-1+(-96)=-97</td></tr><tr><td align="center">-2</td><td align="center">-48</td><td>-2+(-48)=-50</td></tr><tr><td align="center">-3</td><td align="center">-32</td><td>-3+(-32)=-35</td></tr><tr><td align="center">-4</td><td align="center">-24</td><td>-4+(-24)=-28</td></tr><tr><td align="center">-6</td><td align="center">-16</td><td>-6+(-16)=-22</td></tr><tr><td align="center">-8</td><td align="center">-12</td><td>-8+(-12)=-20</td></tr></table>



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



Now looking at the expression {{{1x^2-20x+96}}}, replace {{{-20x}}} with {{{-8x+-12x}}} (notice {{{-8x+-12x}}} adds up to {{{-20x}}}. So it is equivalent to {{{-20x}}})


{{{1x^2+highlight(-8x+-12x)+96}}}



Now let's factor {{{1x^2-8x-12x+96}}} by grouping:



{{{(1x^2-8x)+(-12x+96)}}} Group like terms



{{{x(x-8)-12(x-8)}}} Factor out the GCF of {{{x}}} out of the first group. Factor out the GCF of {{{-12}}} out of the second group



{{{(x-12)(x-8)}}} Since we have a common term of {{{x-8}}}, we can combine like terms


So {{{1x^2-8x-12x+96}}} factors to {{{(x-12)(x-8)}}}



So this also means that {{{1x^2-20x+96}}} factors to {{{(x-12)(x-8)}}} (since {{{1x^2-20x+96}}} is equivalent to {{{1x^2-8x-12x+96}}})




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So our expression goes from {{{x^2(x^2-20x+96)}}} and factors further to {{{x^2(x-12)(x-8)}}}



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Answer:


So {{{x^4-20x^3+96x^2}}} factors to {{{x^2(x-12)(x-8)}}}