Question 183160
I assume that you want to factor right? Please post full instructions.




{{{6g^3-24g^2+24g}}} Start with the given expression



{{{6g(g^2-4g+4)}}} Factor out the GCF {{{6g}}}



Now let's focus on the inner expression {{{g^2-4g+4}}}





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


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




Factors of 4:

1,2


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


These factors pair up and multiply to 4

1*4

2*2

(-1)*(-4)

(-2)*(-2)


note: remember two negative numbers multiplied together make a positive 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">4</td><td>1+4=5</td></tr><tr><td align="center">2</td><td align="center">2</td><td>2+2=4</td></tr><tr><td align="center">-1</td><td align="center">-4</td><td>-1+(-4)=-5</td></tr><tr><td align="center">-2</td><td align="center">-2</td><td>-2+(-2)=-4</td></tr></table>



From this list we can see that -2 and -2 add up to -4 and multiply to 4



Now looking at the expression {{{1g^2-4g+4}}}, replace {{{-4g}}} with {{{-2g+-2g}}} (notice {{{-2g+-2g}}} adds up to {{{-4g}}}. So it is equivalent to {{{-4g}}})


{{{1g^2+highlight(-2g+-2g)+4}}}



Now let's factor {{{1g^2-2g-2g+4}}} by grouping:



{{{(1g^2-2g)+(-2g+4)}}} Group like terms



{{{g(g-2)-2(g-2)}}} Factor out the GCF of {{{g}}} out of the first group. Factor out the GCF of {{{-2}}} out of the second group



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


So {{{1g^2-2g-2g+4}}} factors to {{{(g-2)(g-2)}}}



So this also means that {{{1g^2-4g+4}}} factors to {{{(g-2)(g-2)}}} (since {{{1g^2-4g+4}}} is equivalent to {{{1g^2-2g-2g+4}}})



note:  {{{(g-2)(g-2)}}} is equivalent to  {{{(g-2)^2}}} since the term {{{g-2}}} occurs twice. So {{{1g^2-4g+4}}} also factors to {{{(g-2)^2}}}




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So our expression goes from {{{6g(g^2-4g+4)}}} and factors further to {{{6g(g-2)^2}}}



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Answer:


So {{{6g^3-24g^2+24g}}} factors to {{{6g(g-2)^2}}}