Question 526716


Looking at the expression {{{x^2-11xy+24y^2}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{-11}}}, and the last coefficient is {{{24}}}.



Now multiply the first coefficient {{{1}}} by the last coefficient {{{24}}} to get {{{(1)(24)=24}}}.



Now the question is: what two whole numbers multiply to {{{24}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{-11}}}?



To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of {{{24}}} (the previous product).



Factors of {{{24}}}:

1,2,3,4,6,8,12,24

-1,-2,-3,-4,-6,-8,-12,-24



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



These factors pair up and multiply to {{{24}}}.

1*24 = 24
2*12 = 24
3*8 = 24
4*6 = 24
(-1)*(-24) = 24
(-2)*(-12) = 24
(-3)*(-8) = 24
(-4)*(-6) = 24


Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{-11}}}:



<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>24</font></td><td  align="center"><font color=black>1+24=25</font></td></tr><tr><td  align="center"><font color=black>2</font></td><td  align="center"><font color=black>12</font></td><td  align="center"><font color=black>2+12=14</font></td></tr><tr><td  align="center"><font color=black>3</font></td><td  align="center"><font color=black>8</font></td><td  align="center"><font color=black>3+8=11</font></td></tr><tr><td  align="center"><font color=black>4</font></td><td  align="center"><font color=black>6</font></td><td  align="center"><font color=black>4+6=10</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>-24</font></td><td  align="center"><font color=black>-1+(-24)=-25</font></td></tr><tr><td  align="center"><font color=black>-2</font></td><td  align="center"><font color=black>-12</font></td><td  align="center"><font color=black>-2+(-12)=-14</font></td></tr><tr><td  align="center"><font color=red>-3</font></td><td  align="center"><font color=red>-8</font></td><td  align="center"><font color=red>-3+(-8)=-11</font></td></tr><tr><td  align="center"><font color=black>-4</font></td><td  align="center"><font color=black>-6</font></td><td  align="center"><font color=black>-4+(-6)=-10</font></td></tr></table>



From the table, we can see that the two numbers {{{-3}}} and {{{-8}}} add to {{{-11}}} (the middle coefficient).



So the two numbers {{{-3}}} and {{{-8}}} both multiply to {{{24}}} <font size=4><b>and</b></font> add to {{{-11}}}



Now replace the middle term {{{-11xy}}} with {{{-3xy-8xy}}}. Remember, {{{-3}}} and {{{-8}}} add to {{{-11}}}. So this shows us that {{{-3xy-8xy=-11xy}}}.



{{{x^2+highlight(-3xy-8xy)+24y^2}}} Replace the second term {{{-11xy}}} with {{{-3xy-8xy}}}.



{{{(x^2-3xy)+(-8xy+24y^2)}}} Group the terms into two pairs.



{{{x(x-3y)+(-8xy+24y^2)}}} Factor out the GCF {{{x}}} from the first group.



{{{x(x-3y)-8y(x-3y))}}} Factor {{{-8y}}} from the second group.



{{{(x-8y)(x-3y))}}} Factor {{{x-3y}}} from the entire expression.



So {{{x^2-11xy+24y^2}}} completely factors to {{{(x-8y)(x-3y))}}}



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