<PRE><B>Question 648273</B>


{{{12x^3-27x^2-27x}}} Start with the given expression.



{{{3x(4x^2-9x-9)}}} Factor out the GCF {{{3x}}}.



Now let&#39;s try to factor the inner expression {{{4x^2-9x-9}}}



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Looking at the expression {{{4x^2-9x-9}}}, we can see that the first coefficient is {{{4}}}, the second coefficient is {{{-9}}}, and the last term is {{{-9}}}.



Now multiply the first coefficient {{{4}}} by the last term {{{-9}}} to get {{{(4)(-9)=-36}}}.



Now the question is: what two whole numbers multiply to {{{-36}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{-9}}}?



To find these two numbers, we need to list &lt;font size=4&gt;&lt;b&gt;all&lt;/b&gt;&lt;/font&gt; of the factors of {{{-36}}} (the previous product).



Factors of {{{-36}}}:

1,2,3,4,6,9,12,18,36

-1,-2,-3,-4,-6,-9,-12,-18,-36



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{-36}}}.

1*(-36) = -36
2*(-18) = -36
3*(-12) = -36
4*(-9) = -36
6*(-6) = -36
(-1)*(36) = -36
(-2)*(18) = -36
(-3)*(12) = -36
(-4)*(9) = -36
(-6)*(6) = -36


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{-9}}}:



&lt;table border=&quot;1&quot;&gt;&lt;th&gt;First Number&lt;/th&gt;&lt;th&gt;Second Number&lt;/th&gt;&lt;th&gt;Sum&lt;/th&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-36&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+(-36)=-35&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;2&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-18&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;2+(-18)=-16&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;3&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-12&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;3+(-12)=-9&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;4&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-9&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;4+(-9)=-5&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;6&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-6&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;6+(-6)=0&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;36&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+36=35&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-2&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;18&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-2+18=16&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-3&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;12&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-3+12=9&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-4&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;9&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-4+9=5&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-6&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;6&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-6+6=0&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{3}}} and {{{-12}}} add to {{{-9}}} (the middle coefficient).



So the two numbers {{{3}}} and {{{-12}}} both multiply to {{{-36}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{-9}}}



Now replace the middle term {{{-9x}}} with {{{3x-12x}}}. Remember, {{{3}}} and {{{-12}}} add to {{{-9}}}. So this shows us that {{{3x-12x=-9x}}}.



{{{4x^2+highlight(3x-12x)-9}}} Replace the second term {{{-9x}}} with {{{3x-12x}}}.



{{{(4x^2+3x)+(-12x-9)}}} Group the terms into two pairs.



{{{x(4x+3)+(-12x-9)}}} Factor out the GCF {{{x}}} from the first group.



{{{x(4x+3)-3(4x+3)}}} Factor out {{{3}}} 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-3)(4x+3)}}} Combine like terms. Or factor out the common term {{{4x+3}}}



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So {{{3x(4x^2-9x-9)}}} then factors further to {{{3x(x-3)(4x+3)}}}



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Answer:



So {{{12x^3-27x^2-27x}}} completely factors to {{{3x(x-3)(4x+3)}}}.



In other words, {{{12x^3-27x^2-27x=3x(x-3)(4x+3)}}}.



Note: you can check the answer by expanding {{{3x(x-3)(4x+3)}}} to get {{{12x^3-27x^2-27x}}} or by graphing the original expression and the answer (the two graphs should be identical).


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