```Question 190288

{{{2t^3(t^2-7t+12)}}} Factor out the GCF {{{2t^3}}}

Now let's focus on the inner expression {{{t^2-7t+12}}}

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Looking at the expression {{{t^2-7t+12}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{-7}}}, and the last term is {{{12}}}.

Now multiply the first coefficient {{{1}}} by the last term {{{12}}} to get {{{(1)(12)=12}}}.

Now the question is: what two whole numbers multiply to {{{12}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{-7}}}?

To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of {{{12}}} (the previous product).

Factors of {{{12}}}:

1,2,3,4,6,12

-1,-2,-3,-4,-6,-12

Note: list the negative of each factor. This will allow us to find all possible combinations.

These factors pair up and multiply to {{{12}}}.

1*12
2*6
3*4
(-1)*(-12)
(-2)*(-6)
(-3)*(-4)

Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{-7}}}:

<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td  align="center"><font color=black>1</font></td><td  align="center"><font color=black>12</font></td><td  align="center"><font color=black>1+12=13</font></td></tr><tr><td  align="center"><font color=black>2</font></td><td  align="center"><font color=black>6</font></td><td  align="center"><font color=black>2+6=8</font></td></tr><tr><td  align="center"><font color=black>3</font></td><td  align="center"><font color=black>4</font></td><td  align="center"><font color=black>3+4=7</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>-12</font></td><td  align="center"><font color=black>-1+(-12)=-13</font></td></tr><tr><td  align="center"><font color=black>-2</font></td><td  align="center"><font color=black>-6</font></td><td  align="center"><font color=black>-2+(-6)=-8</font></td></tr><tr><td  align="center"><font color=red>-3</font></td><td  align="center"><font color=red>-4</font></td><td  align="center"><font color=red>-3+(-4)=-7</font></td></tr></table>

From the table, we can see that the two numbers {{{-3}}} and {{{-4}}} add to {{{-7}}} (the middle coefficient).

So the two numbers {{{-3}}} and {{{-4}}} both multiply to {{{12}}} <font size=4><b>and</b></font> add to {{{-7}}}

Now replace the middle term {{{-7t}}} with {{{-3t-4t}}}. Remember, {{{-3}}} and {{{-4}}} add to {{{-7}}}. So this shows us that {{{-3t-4t=-7t}}}.

{{{t^2+highlight(-3t-4t)+12}}} Replace the second term {{{-7t}}} with {{{-3t-4t}}}.

{{{(t^2-3t)+(-4t+12)}}} Group the terms into two pairs.

{{{t(t-3)+(-4t+12)}}} Factor out the GCF {{{t}}} from the first group.

{{{t(t-3)-4(t-3)}}} Factor out {{{4}}} from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.

{{{(t-4)(t-3)}}} Combine like terms. Or factor out the common term {{{t-3}}}

So {{{t^2-7t+12}}} factors to {{{(t-4)(t-3)}}}.

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So our expression goes from {{{2t^3(t^2-7t+12)}}} and factors further to {{{2t^3(t-4)(t-3)}}}

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