Question 191475


Looking at {{{x^2y^2-6xy+9}}} we can see that the first term is {{{x^2y^2}}} and the last term is {{{9}}} where the coefficients are 1 and 9 respectively.


Now multiply the first coefficient 1 and the last coefficient 9 to get 9. Now what two numbers multiply to 9 and add to the  middle coefficient -6? Let's list all of the factors of 9:




Factors of 9:

1,3


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


These factors pair up and multiply to 9

1*9

3*3

(-1)*(-9)

(-3)*(-3)


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



Now which of these pairs add to -6? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -6


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">9</td><td>1+9=10</td></tr><tr><td align="center">3</td><td align="center">3</td><td>3+3=6</td></tr><tr><td align="center">-1</td><td align="center">-9</td><td>-1+(-9)=-10</td></tr><tr><td align="center">-3</td><td align="center">-3</td><td>-3+(-3)=-6</td></tr></table>



From this list we can see that -3 and -3 add up to -6 and multiply to 9



Now looking at the expression {{{x^2y^2-6xy+9}}}, replace {{{-6xy}}} with {{{-3xy-3xy}}} (notice {{{-3xy+-3xy}}} adds up to {{{-6xy}}}. So it is equivalent to {{{-6xy}}})


{{{x^2y^2+highlight(-3xy-3xy)+9}}}



Now let's factor {{{x^2y^2-3xy-3xy+9}}} by grouping:



{{{(x^2y^2-3xy)+(-3xy+9)}}} Group like terms



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



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


So {{{x^2y^2-3xy-3xy+9}}} factors to {{{(xy-3)(xy-3)}}}



So this also means that {{{x^2y^2-6xy+9}}} factors to {{{(xy-3)(xy-3)}}} (since {{{x^2y^2-6xy+9}}} is equivalent to {{{x^2y^2-3xy-3xy+9}}})



note:  {{{(xy-3)(xy-3)}}} is equivalent to  {{{(xy-3)^2}}} since the term {{{xy-3}}} occurs twice. So {{{x^2y^2-6xy+9}}} also factors to {{{(xy-3)^2}}}




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     Answer:

So {{{x^2y^2-6xy+9}}} factors to {{{(xy-3)^2}}}