Question 582002


Looking at the expression {{{x^2-xy-56y^2}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{-1}}}, and the last coefficient is {{{-56}}}.



Now multiply the first coefficient {{{1}}} by the last coefficient {{{-56}}} to get {{{(1)(-56)=-56}}}.



Now the question is: what two whole numbers multiply to {{{-56}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{-1}}}?



To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of {{{-56}}} (the previous product).



Factors of {{{-56}}}:

1,2,4,7,8,14,28,56

-1,-2,-4,-7,-8,-14,-28,-56



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{-56}}}.

1*(-56) = -56
2*(-28) = -56
4*(-14) = -56
7*(-8) = -56
(-1)*(56) = -56
(-2)*(28) = -56
(-4)*(14) = -56
(-7)*(8) = -56


Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{-1}}}:



<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>-56</font></td><td  align="center"><font color=black>1+(-56)=-55</font></td></tr><tr><td  align="center"><font color=black>2</font></td><td  align="center"><font color=black>-28</font></td><td  align="center"><font color=black>2+(-28)=-26</font></td></tr><tr><td  align="center"><font color=black>4</font></td><td  align="center"><font color=black>-14</font></td><td  align="center"><font color=black>4+(-14)=-10</font></td></tr><tr><td  align="center"><font color=red>7</font></td><td  align="center"><font color=red>-8</font></td><td  align="center"><font color=red>7+(-8)=-1</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>56</font></td><td  align="center"><font color=black>-1+56=55</font></td></tr><tr><td  align="center"><font color=black>-2</font></td><td  align="center"><font color=black>28</font></td><td  align="center"><font color=black>-2+28=26</font></td></tr><tr><td  align="center"><font color=black>-4</font></td><td  align="center"><font color=black>14</font></td><td  align="center"><font color=black>-4+14=10</font></td></tr><tr><td  align="center"><font color=black>-7</font></td><td  align="center"><font color=black>8</font></td><td  align="center"><font color=black>-7+8=1</font></td></tr></table>



From the table, we can see that the two numbers {{{7}}} and {{{-8}}} add to {{{-1}}} (the middle coefficient).



So the two numbers {{{7}}} and {{{-8}}} both multiply to {{{-56}}} <font size=4><b>and</b></font> add to {{{-1}}}



Now replace the middle term {{{-1xy}}} with {{{7xy-8xy}}}. Remember, {{{7}}} and {{{-8}}} add to {{{-1}}}. So this shows us that {{{7xy-8xy=-1xy}}}.



{{{x^2+highlight(7xy-8xy)-56y^2}}} Replace the second term {{{-1xy}}} with {{{7xy-8xy}}}.



{{{(x^2+7xy)+(-8xy-56y^2)}}} Group the terms into two pairs.



{{{x(x+7y)+(-8xy-56y^2)}}} Factor out the GCF {{{x}}} from the first group.



{{{x(x+7y)-8y(x+7y)}}} Factor out {{{-8y}}} from the second group.



{{{(x-8y)(x+7y)}}} Factor out {{{x+7y}}}



So {{{x^2-xy-56y^2}}} completely factors to {{{(x-8y)(x+7y)}}}