Question 146160


{{{x^10y^3-4x^9y^2-21x^8y}}} Start with the given expression



{{{x^8y(x^2y^2-4xy-21)}}} Factor out the GCF {{{x^8y}}}



Now let's focus on the inner expression {{{x^2y^2-4xy-21}}}





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Looking at {{{1x^2y^2-4xy-21}}} we can see that the first term is {{{1x^2y^2}}} and the last term is {{{-21}}} where the coefficients are 1 and -21 respectively.


Now multiply the first coefficient 1 and the last coefficient -21 to get -21. Now what two numbers multiply to -21 and add to the  middle coefficient -4? Let's list all of the factors of -21:




Factors of -21:

1,3,7,21


-1,-3,-7,-21 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to -21

(1)*(-21)

(3)*(-7)

(-1)*(21)

(-3)*(7)


note: remember, the product of a negative and a positive number is a negative number



Now which of these pairs add to -4? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -4


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">-21</td><td>1+(-21)=-20</td></tr><tr><td align="center">3</td><td align="center">-7</td><td>3+(-7)=-4</td></tr><tr><td align="center">-1</td><td align="center">21</td><td>-1+21=20</td></tr><tr><td align="center">-3</td><td align="center">7</td><td>-3+7=4</td></tr></table>



From this list we can see that 3 and -7 add up to -4 and multiply to -21



Now looking at the expression {{{1x^2y^2-4xy-21}}}, replace {{{-4xy}}} with {{{3xy+-7xy}}} (notice {{{3xy+-7xy}}} adds up to {{{-4xy}}}. So it is equivalent to {{{-4xy}}})


{{{1x^2y^2+highlight(3xy+-7xy)+-21}}}



Now let's factor {{{1x^2y^2+3xy-7xy-21}}} by grouping:



{{{(1x^2y^2+3xy)+(-7xy-21)}}} Group like terms



{{{xy(xy+3)-7(xy+3)}}} Factor out the GCF of {{{xy}}} out of the first group. Factor out the GCF of {{{-7}}} out of the second group



{{{(xy-7)(xy+3)}}} Since we have a common term of {{{xy+3}}}, we can combine like terms


So {{{1x^2y^2+3xy-7xy-21}}} factors to {{{(xy-7)(xy+3)}}}



So this also means that {{{1x^2y^2-4xy-21}}} factors to {{{(xy-7)(xy+3)}}} (since {{{1x^2y^2-4xy-21}}} is equivalent to {{{1x^2y^2+3xy-7xy-21}}})




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So our expression goes from {{{x^8y(x^2y^2-4xy-21)}}} and factors further to {{{x^8y(xy-7)(xy+3)}}}



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Answer:


So {{{x^10y^3-4x^9y^2-21x^8y}}} factors to {{{x^8y(xy-7)(xy+3)}}}