<PRE><B>Question 653914</B>


{{{48x^3-40x^2-48x}}} Start with the given expression.



{{{8x(6x^2-5x-6)}}} Factor out the GCF {{{8x}}}.



Now let&#39;s try to factor the inner expression {{{6x^2-5x-6}}}



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Looking at the expression {{{6x^2-5x-6}}}, we can see that the first coefficient is {{{6}}}, the second coefficient is {{{-5}}}, and the last term is {{{-6}}}.



Now multiply the first coefficient {{{6}}} by the last term {{{-6}}} to get {{{(6)(-6)=-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 {{{-5}}}?



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 {{{-5}}}:



&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=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=red&gt;4&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-9&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&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 {{{4}}} and {{{-9}}} add to {{{-5}}} (the middle coefficient).



So the two numbers {{{4}}} and {{{-9}}} both multiply to {{{-36}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{-5}}}



Now replace the middle term {{{-5x}}} with {{{4x-9x}}}. Remember, {{{4}}} and {{{-9}}} add to {{{-5}}}. So this shows us that {{{4x-9x=-5x}}}.



{{{6x^2+highlight(4x-9x)-6}}} Replace the second term {{{-5x}}} with {{{4x-9x}}}.



{{{(6x^2+4x)+(-9x-6)}}} Group the terms into two pairs.



{{{2x(3x+2)+(-9x-6)}}} Factor out the GCF {{{2x}}} from the first group.



{{{2x(3x+2)-3(3x+2)}}} 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.



{{{(2x-3)(3x+2)}}} Combine like terms. Or factor out the common term {{{3x+2}}}



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So {{{8x(6x^2-5x-6)}}} then factors further to {{{8x(2x-3)(3x+2)}}}



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Answer:



So {{{48x^3-40x^2-48x}}} completely factors to {{{8x(2x-3)(3x+2)}}}.



In other words, {{{48x^3-40x^2-48x=8x(2x-3)(3x+2)}}}.



Note: you can check the answer by expanding {{{8x(2x-3)(3x+2)}}} to get {{{48x^3-40x^2-48x}}} or by graphing the original expression and the answer (the two graphs should be identical).


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