Question 115668
{{{m^3-9m^2n+18n^2m}}} Start with the given expression



{{{m(m^2-9mn+18n^2)}}} Factor out the GCF {{{m}}}



Now let's focus on the inner expression {{{m^2-9mn+18n^2}}}





Looking at {{{m^2-9mn+18n^2}}} we can see that the first term is {{{m^2}}} and the last term is {{{18n^2}}} where the coefficients are 1 and 18 respectively.


Now multiply the first coefficient 1 and the last coefficient 18 to get 18. Now what two numbers multiply to 18 and add to the  middle coefficient -9? Let's list all of the factors of 18:




Factors of 18:

1,2,3,6,9,18


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


These factors pair up and multiply to 18

1*18

2*9

3*6

(-1)*(-18)

(-2)*(-9)

(-3)*(-6)


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



Now which of these pairs add to -9? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -9


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">18</td><td>1+18=19</td></tr><tr><td align="center">2</td><td align="center">9</td><td>2+9=11</td></tr><tr><td align="center">3</td><td align="center">6</td><td>3+6=9</td></tr><tr><td align="center">-1</td><td align="center">-18</td><td>-1+(-18)=-19</td></tr><tr><td align="center">-2</td><td align="center">-9</td><td>-2+(-9)=-11</td></tr><tr><td align="center">-3</td><td align="center">-6</td><td>-3+(-6)=-9</td></tr></table>



From this list we can see that -3 and -6 add up to -9 and multiply to 18



Now looking at the expression {{{m^2-9mn+18n^2}}}, replace {{{-9mn}}} with {{{-3mn+-6mn}}} (notice {{{-3mn+-6mn}}} adds up to {{{-9mn}}}. So it is equivalent to {{{-9mn}}})


{{{m^2+highlight(-3mn+-6mn)+18n^2}}}



Now let's factor {{{m^2-3mn-6mn+18n^2}}} by grouping:



{{{(m^2-3mn)+(-6mn+18n^2)}}} Group like terms



{{{m(m-3n)-6n(m-3n)}}} Factor out the GCF of {{{m}}} out of the first group. Factor out the GCF of {{{-6n}}} out of the second group



{{{(m-6n)(m-3n)}}} Since we have a common term of {{{m-3n}}}, we can combine like terms


So {{{m^2-3mn-6mn+18n^2}}} factors to {{{(m-6n)(m-3n)}}}



So this also means that {{{m^2-9mn+18n^2}}} factors to {{{(m-6n)(m-3n)}}} (since {{{m^2-9mn+18n^2}}} is equivalent to {{{m^2-3mn-6mn+18n^2}}})



So {{{m^2-9mn+18n^2}}} factors to {{{(m-6n)(m-3n)}}}



{{{m(m-6n)(m-3n)}}} Now reintroduce the GCF


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Answer:



So {{{m^3-9m^2n+18n^2m}}} factors to {{{m(m-6n)(m-3n)}}}