```Question 407310

{{{9(9x^2-6xy+y^2)}}} Factor out the GCF {{{9}}}

Now let's focus on the inner expression {{{9x^2-6xy+y^2}}}

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

Now multiply the first coefficient 9 and the last coefficient 1 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 {{{9x^2-6xy+y^2}}}, replace {{{-6xy}}} with {{{-3xy-3xy}}} (notice {{{-3xy-3xy}}} combines back to {{{-6xy}}}. So it is equivalent to {{{-6xy}}})

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

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

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

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

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

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

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

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

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So our expression goes from {{{9(9x^2-6xy+y^2)}}} and factors further to {{{9(3x-y)^2}}}

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