Question 208123
I'll do the first three to get you started.


A)



{{{40x^3-64x^2y}}} Start with the given expression.



{{{8x^2(5x-8y)}}} Factor out the GCF {{{8x^2}}}.




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Answer:



So {{{40x^3-64x^2y}}} factors to {{{8x^2(5x-8y)}}}.



In other words, {{{40x^3-64x^2y=8x^2(5x-8y)}}}.




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B)




{{{125r^3-8s^3}}} Start with the given expression.



{{{(5r)^3-(2s)^3}}} Rewrite {{{125r^3}}} as {{{(5r)^3}}}. Rewrite {{{8s^3}}} as {{{(2s)^3}}}.



{{{(5r-2s)((5r)^2+(5r)(2s)+(2s)^2)}}} Now factor by using the difference of cubes formula. Remember the <a href="http://www.purplemath.com/modules/specfact2.htm">difference of cubes formula</a> is {{{A^3-B^3=(A-B)(A^2+AB+B^2)}}}



{{{(5r-2s)(25r^2+10rs+4s^2)}}} Multiply


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Answer:


So {{{125r^3-8s^3}}} factors to {{{(5r-2s)(25r^2+10rs+4s^2)}}}.



In other words, {{{125r^3-8s^3=(5r-2s)(25r^2+10rs+4s^2)}}}



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C)



{{{13-14y+y^2}}} Start with the given expression.



{{{y^2-14y+13}}} Rearrange the terms.





Looking at the expression {{{y^2-14y+13}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{-14}}}, and the last term is {{{13}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{13}}} to get {{{(1)(13)=13}}}.



Now the question is: what two whole numbers multiply to {{{13}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{-14}}}?



To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of {{{13}}} (the previous product).



Factors of {{{13}}}:

1,13

-1,-13



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



These factors pair up and multiply to {{{13}}}.

1*13 = 13
(-1)*(-13) = 13


Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{-14}}}:



<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>13</font></td><td  align="center"><font color=black>1+13=14</font></td></tr><tr><td  align="center"><font color=red>-1</font></td><td  align="center"><font color=red>-13</font></td><td  align="center"><font color=red>-1+(-13)=-14</font></td></tr></table>



From the table, we can see that the two numbers {{{-1}}} and {{{-13}}} add to {{{-14}}} (the middle coefficient).



So the two numbers {{{-1}}} and {{{-13}}} both multiply to {{{13}}} <font size=4><b>and</b></font> add to {{{-14}}}



Now replace the middle term {{{-14y}}} with {{{-y-13y}}}. Remember, {{{-1}}} and {{{-13}}} add to {{{-14}}}. So this shows us that {{{-y-13y=-14y}}}.



{{{y^2+highlight(-y-13y)+13}}} Replace the second term {{{-14y}}} with {{{-y-13y}}}.



{{{(y^2-y)+(-13y+13)}}} Group the terms into two pairs.



{{{y(y-1)+(-13y+13)}}} Factor out the GCF {{{y}}} from the first group.



{{{y(y-1)-13(y-1)}}} Factor out {{{13}}} 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.



{{{(y-13)(y-1)}}} Combine like terms. Or factor out the common term {{{y-1}}}



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Answer:



So {{{13-14y+y^2}}} factors to {{{(y-13)(y-1)}}}.



In other words, {{{13-14y+y^2=(y-13)(y-1)}}}.