Question 169053


{{{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 the expression {{{x^2-13x+40}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{-13}}}, and the last term is {{{40}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{40}}} to get {{{(1)(40)=40}}}.



Now the question is: what two whole numbers multiply to {{{40}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{-13}}}?



To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of {{{40}}} (the previous product).



Factors of {{{40}}}:

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

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



Note: list the negative of each factor. 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)


Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{-13}}}:



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



From the table, we can see that the two numbers {{{-5}}} and {{{-8}}} add to {{{-13}}} (the middle coefficient).



So the two numbers {{{-5}}} and {{{-8}}} both multiply to {{{40}}} <font size=4><b>and</b></font> add to {{{-13}}}



Now replace the middle term {{{-13x}}} with {{{-5x-8x}}}. Remember, {{{-5}}} and {{{-8}}} add to {{{-13}}}. So this shows us that {{{-5x-8x=-13x}}}.



{{{x^2+highlight(-5x-8x)+40}}} Replace the second term {{{-13x}}} with {{{-5x-8x}}}.



{{{(x^2-5x)+(-8x+40)}}} Group the terms into two pairs.



{{{x(x-5)+(-8x+40)}}} Factor out the GCF {{{x}}} from the first group.



{{{x(x-5)-8(x-5)}}} Factor out {{{8}}} 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.



{{{(x-8)(x-5)}}} Combine like terms. Or factor out the common term {{{x-5}}}



So {{{x^2-13x+40}}} factors to {{{(x-8)(x-5)}}}.



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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)}}}