SOLUTION: Rewrite the expression x^3 - 2x^2 y + x y^2 so that it is easy to calculate the its value without a calculator when x= 21 and y = 19

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Question 133324: Rewrite the expression x^3 - 2x^2 y + x y^2 so that it is easy to calculate the its value without a calculator when x= 21 and y = 19
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

x%5E3-2x%5E2y%2B1xy%5E2 Start with the given expression


x%28x%5E2-2xy%2By%5E2%29 Factor out the GCF x


Now let's focus on the inner expression x%5E2-2xy%2By%5E2




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Looking at 1x%5E2-2xy%2B1y%5E2 we can see that the first term is 1x%5E2 and the last term is 1y%5E2 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

First NumberSecond NumberSum
111+1=2
-1-1-1+(-1)=-2



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


Now looking at the expression 1x%5E2-2xy%2B1y%5E2, replace -2xy with -1xy%2B-1xy (notice -1xy%2B-1xy adds up to -2xy. So it is equivalent to -2xy)

1x%5E2%2Bhighlight%28-1xy%2B-1xy%29%2B1y%5E2


Now let's factor 1x%5E2-1xy-1xy%2B1y%5E2 by grouping:


%281x%5E2-1xy%29%2B%28-1xy%2B1y%5E2%29 Group like terms


x%28x-y%29-y%28x-y%29 Factor out the GCF of x out of the first group. Factor out the GCF of -y out of the second group


%28x-y%29%28x-y%29 Since we have a common term of x-y, we can combine like terms

So 1x%5E2-1xy-1xy%2B1y%5E2 factors to %28x-y%29%28x-y%29


So this also means that 1x%5E2-2xy%2B1y%5E2 factors to %28x-y%29%28x-y%29 (since 1x%5E2-2xy%2B1y%5E2 is equivalent to 1x%5E2-1xy-1xy%2B1y%5E2)


note: %28x-y%29%28x-y%29 is equivalent to %28x-y%29%5E2 since the term x-y occurs twice. So 1x%5E2-2xy%2B1y%5E2 also factors to %28x-y%29%5E2



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So our expression goes from x%28x%5E2-2xy%2By%5E2%29 and factors further to x%28x-y%29%5E2




So x%5E3-2x%5E2y%2Bxy%5E2 factors to x%28x-y%29%5E2




21%2821-19%29%5E2 Plug in x= 21 and y = 19


21%282%29%5E2 Subtract


21%284%29 Square 2 to get 4


84 Multiply




So x%5E3+-+2x%5E2+y+%2B+x+y%5E2=84 when x= 21 and y = 19