Question 112854
{{{8u^3-24u^2v+18uv^2}}} Start with the given expression



{{{2u(4u^2-12uv+9v^2)}}} Factor out the GCF {{{2u}}}



Now let's focus on the inner expression {{{4u^2-12uv+9v^2}}}




Looking at {{{4u^2-12uv+9v^2}}} we can see that the first term is {{{4u^2}}} and the last term is {{{9v^2}}} where the coefficients are 4 and 9 respectively.


Now multiply the first coefficient 4 and the last coefficient 9 to get 36. Now what two numbers multiply to 36 and add to the  middle coefficient -12? Let's list all of the factors of 36:




Factors of 36:

1,2,3,4,6,9,12,18


-1,-2,-3,-4,-6,-9,-12,-18 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to 36

1*36

2*18

3*12

4*9

6*6

(-1)*(-36)

(-2)*(-18)

(-3)*(-12)

(-4)*(-9)

(-6)*(-6)


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



Now which of these pairs add to -12? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -12


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">36</td><td>1+36=37</td></tr><tr><td align="center">2</td><td align="center">18</td><td>2+18=20</td></tr><tr><td align="center">3</td><td align="center">12</td><td>3+12=15</td></tr><tr><td align="center">4</td><td align="center">9</td><td>4+9=13</td></tr><tr><td align="center">6</td><td align="center">6</td><td>6+6=12</td></tr><tr><td align="center">-1</td><td align="center">-36</td><td>-1+(-36)=-37</td></tr><tr><td align="center">-2</td><td align="center">-18</td><td>-2+(-18)=-20</td></tr><tr><td align="center">-3</td><td align="center">-12</td><td>-3+(-12)=-15</td></tr><tr><td align="center">-4</td><td align="center">-9</td><td>-4+(-9)=-13</td></tr><tr><td align="center">-6</td><td align="center">-6</td><td>-6+(-6)=-12</td></tr></table>



From this list we can see that -6 and -6 add up to -12 and multiply to 36



Now looking at the expression {{{4u^2-12uv+9v^2}}}, replace {{{-12uv}}} with {{{-6uv+-6uv}}} (notice {{{-6uv+-6uv}}} adds up to {{{-12uv}}}. So it is equivalent to {{{-12uv}}})


{{{4u^2+highlight(-6uv+-6uv)+9v^2}}}



Now let's factor {{{4u^2-6uv-6uv+9v^2}}} by grouping:



{{{(4u^2-6uv)+(-6uv+9v^2)}}} Group like terms



{{{2u(2u-3v)-3v(2u-3v)}}} Factor out the GCF of {{{2u}}} out of the first group. Factor out the GCF of {{{-3v}}} out of the second group



{{{(2u-3v)(2u-3v)}}} Since we have a common term of {{{2u-3v}}}, we can combine like terms


So {{{4u^2-6uv-6uv+9v^2}}} factors to {{{(2u-3v)(2u-3v)}}}



So this also means that {{{4u^2-12uv+9v^2}}} factors to {{{(2u-3v)(2u-3v)}}} (since {{{4u^2-12uv+9v^2}}} is equivalent to {{{4u^2-6uv-6uv+9v^2}}})


{{{2u(2u-3v)(2u-3v)}}} Now replace the 2u


So that means the original expression {{{8u^3-24u^2v+18uv^2}}} factors to 


{{{2u(2u-3v)(2u-3v)}}}