Question 645121


Looking at the expression {{{x^2+12xy+32y^2}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{12}}}, and the last coefficient is {{{32}}}.



Now multiply the first coefficient {{{1}}} by the last coefficient {{{32}}} to get {{{(1)(32)=32}}}.



Now the question is: what two whole numbers multiply to {{{32}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{12}}}?



To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of {{{32}}} (the previous product).



Factors of {{{32}}}:

1,2,4,8,16,32

-1,-2,-4,-8,-16,-32



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



These factors pair up and multiply to {{{32}}}.

1*32 = 32
2*16 = 32
4*8 = 32
(-1)*(-32) = 32
(-2)*(-16) = 32
(-4)*(-8) = 32


Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{12}}}:



<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>32</font></td><td  align="center"><font color=black>1+32=33</font></td></tr><tr><td  align="center"><font color=black>2</font></td><td  align="center"><font color=black>16</font></td><td  align="center"><font color=black>2+16=18</font></td></tr><tr><td  align="center"><font color=red>4</font></td><td  align="center"><font color=red>8</font></td><td  align="center"><font color=red>4+8=12</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>-32</font></td><td  align="center"><font color=black>-1+(-32)=-33</font></td></tr><tr><td  align="center"><font color=black>-2</font></td><td  align="center"><font color=black>-16</font></td><td  align="center"><font color=black>-2+(-16)=-18</font></td></tr><tr><td  align="center"><font color=black>-4</font></td><td  align="center"><font color=black>-8</font></td><td  align="center"><font color=black>-4+(-8)=-12</font></td></tr></table>



From the table, we can see that the two numbers {{{4}}} and {{{8}}} add to {{{12}}} (the middle coefficient).



So the two numbers {{{4}}} and {{{8}}} both multiply to {{{32}}} <font size=4><b>and</b></font> add to {{{12}}}



Now replace the middle term {{{12xy}}} with {{{4xy+8xy}}}. Remember, {{{4}}} and {{{8}}} add to {{{12}}}. So this shows us that {{{4xy+8xy=12xy}}}.



{{{x^2+highlight(4xy+8xy)+32y^2}}} Replace the second term {{{12xy}}} with {{{4xy+8xy}}}.



{{{(x^2+4xy)+(8xy+32y^2)}}} Group the terms into two pairs.



{{{x(x+4y)+(8xy+32y^2)}}} Factor out the GCF {{{x}}} from the first group.



{{{x(x+4y)+8y(x+4y)}}} Factor out {{{8y}}} 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+8y)(x+4y)}}} Combine like terms. Or factor out the common term {{{x+4y}}}



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Answer:



So {{{x^2+12xy+32y^2}}} factors to {{{(x+8y)(x+4y)}}}.



In other words, {{{x^2+12xy+32y^2=(x+8y)(x+4y)}}}.



Note: you can check the answer by expanding {{{(x+8y)(x+4y)}}} to get {{{x^2+12xy+32y^2}}} or by graphing the original expression and the answer (the two graphs should be identical).


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