Question 132850
If you factor this expression, then it will be much easier to calculate the result





{{{x^3-2x^2y+xy^2}}} Start with the given expression



{{{x(x^2-2xy+y^2)}}} Factor out the GCF {{{x}}}



Now let's focus on the inner expression {{{x^2-2xy+y^2}}}





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Looking at {{{x^2-2xy+y^2}}} we can see that the first term is {{{1x^2}}} and the last term is {{{y^2}}} where the coefficients are 1 and 1 respectively.


Now multiply the first coefficient 1 and the last coefficient 1 to get 1. Now what two numbers multiply to 1 and add to the  middle coefficient -2? Let's list all of the factors of 1:




Factors of 1:

1


-1 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to 1

1*1

(-1)*(-1)


note: remember two negative numbers multiplied together make a positive number



Now which of these pairs add to -2? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -2


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">1</td><td>1+1=2</td></tr><tr><td align="center">-1</td><td align="center">-1</td><td>-1+(-1)=-2</td></tr></table>



From this list we can see that -1 and -1 add up to -2 and multiply to 1



Now looking at the expression {{{x^2-2xy+y^2}}}, replace {{{-2xy}}} with {{{-xy-xy}}} (notice {{{-xy-xy}}} adds up to {{{-2xy}}}. So it is equivalent to {{{-2xy}}})


{{{x^2+highlight(-xy-xy)+y^2}}}



Now let's factor {{{x^2-xy-xy+y^2}}} by grouping:



{{{(x^2-xy)+(-xy+y^2)}}} Group like terms



{{{x(x-y)-y(x-y)}}} Factor out the GCF of {{{x}}} out of the first group. Factor out the GCF of {{{-y}}} out of the second group



{{{(x-y)(x-y)}}} Since we have a common term of {{{x-y}}}, we can combine like terms


So {{{x^2-xy-xy+y^2}}} factors to {{{(x-y)(x-y)}}}



So this also means that {{{x^2-2xy+y^2}}} factors to {{{(x-y)(x-y)}}} (since {{{x^2-2xy+y^2}}} is equivalent to {{{x^2-xy-xy+y^2}}})



note:  {{{(x-y)(x-y)}}} is equivalent to  {{{(x-y)^2}}} since the term {{{x-y}}} occurs twice. So {{{x^2-2xy+y^2}}} also factors to {{{(x-y)^2}}}




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So our expression goes from {{{x(x^2-2xy+y^2)}}} and factors further to {{{x(x-y)^2}}}







So {{{x^3-2x^2y+xy^2}}} factors to {{{x(x-y)^2}}}





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Now let's evaluate the expression when  x=21 and y=19




{{{21(21-19)^2}}} Now plug in x=21 and y=19



{{{21(2)^2}}} Subtract 



{{{21(4)}}} Square 2 to get 4



{{{84}}} Multiply 

    

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Answer:


So {{{x^3-2x^2y+xy^2=84}}} when x=21 and y=19