Question 176141


{{{2x^3-26x^2+80x}}} Start with the given expression



{{{2x(x^2-13x+40)}}} Factor out the GCF {{{2x}}}



Now let's focus on the inner expression {{{x^2-13x+40}}}





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Looking at {{{1x^2-13x+40}}} we can see that the first term is {{{1x^2}}} and the last term is {{{40}}} where the coefficients are 1 and 40 respectively.


Now multiply the first coefficient 1 and the last coefficient 40 to get 40. Now what two numbers multiply to 40 and add to the  middle coefficient -13? Let's list all of the factors of 40:




Factors of 40:

1,2,4,5,8,10,20,40


-1,-2,-4,-5,-8,-10,-20,-40 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to 40

1*40

2*20

4*10

5*8

(-1)*(-40)

(-2)*(-20)

(-4)*(-10)

(-5)*(-8)


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



Now which of these pairs add to -13? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -13


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">40</td><td>1+40=41</td></tr><tr><td align="center">2</td><td align="center">20</td><td>2+20=22</td></tr><tr><td align="center">4</td><td align="center">10</td><td>4+10=14</td></tr><tr><td align="center">5</td><td align="center">8</td><td>5+8=13</td></tr><tr><td align="center">-1</td><td align="center">-40</td><td>-1+(-40)=-41</td></tr><tr><td align="center">-2</td><td align="center">-20</td><td>-2+(-20)=-22</td></tr><tr><td align="center">-4</td><td align="center">-10</td><td>-4+(-10)=-14</td></tr><tr><td align="center">-5</td><td align="center">-8</td><td>-5+(-8)=-13</td></tr></table>



From this list we can see that -5 and -8 add up to -13 and multiply to 40



Now looking at the expression {{{1x^2-13x+40}}}, replace {{{-13x}}} with {{{-5x+-8x}}} (notice {{{-5x+-8x}}} adds up to {{{-13x}}}. So it is equivalent to {{{-13x}}})


{{{1x^2+highlight(-5x+-8x)+40}}}



Now let's factor {{{1x^2-5x-8x+40}}} by grouping:



{{{(1x^2-5x)+(-8x+40)}}} Group like terms



{{{x(x-5)-8(x-5)}}} Factor out the GCF of {{{x}}} out of the first group. Factor out the GCF of {{{-8}}} out of the second group



{{{(x-8)(x-5)}}} Since we have a common term of {{{x-5}}}, we can combine like terms


So {{{1x^2-5x-8x+40}}} factors to {{{(x-8)(x-5)}}}



So this also means that {{{1x^2-13x+40}}} factors to {{{(x-8)(x-5)}}} (since {{{1x^2-13x+40}}} is equivalent to {{{1x^2-5x-8x+40}}})




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So our expression goes from {{{2x(x^2-13x+40)}}} and factors further to {{{2x(x-8)(x-5)}}}



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Answer:


So {{{2x^3-26x^2+80x}}} completely factors to {{{2x(x-8)(x-5)}}}