```Question 388541

{{{4(x^2-7x+12)}}} Factor out the GCF {{{4}}}.

Now let's try to factor the inner expression {{{x^2-7x+12}}}

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Looking at the expression {{{x^2-7x+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 = 12
2*6 = 12
3*4 = 12
(-1)*(-12) = 12
(-2)*(-6) = 12
(-3)*(-4) = 12

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 {{{-7x}}} with {{{-3x-4x}}}. Remember, {{{-3}}} and {{{-4}}} add to {{{-7}}}. So this shows us that {{{-3x-4x=-7x}}}.

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

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

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

{{{x(x-3)-4(x-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.

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

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So {{{4(x^2-7x+12)}}} then factors further to {{{4(x-4)(x-3)}}}

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So {{{4x^2-28x+48}}} completely factors to {{{4(x-4)(x-3)}}}.

In other words, {{{4x^2-28x+48=4(x-4)(x-3)}}}.

Note: you can check the answer by expanding {{{4(x-4)(x-3)}}} to get {{{4x^2-28x+48}}} or by graphing the original expression and the answer (the two graphs should be identical).

If you need more help, email me at <a href="mailto:jim_thompson5910@hotmail.com?Subject=Algebra%20Help">jim_thompson5910@hotmail.com</a>

Also, feel free to check out my <a href="http://www.freewebs.com/jimthompson5910/home.html">tutoring website</a>

Jim```