<PRE><B>Question 216205</B>
{{{7a^3b-35a^2b^2+42ab^3}}} Start with the given expression.



{{{7ab(a^2-5ab+6b^2)}}} Factor out the GCF {{{7ab}}}.



Now let&#39;s try to factor the inner expression {{{a^2-5ab+6b^2}}}



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Looking at the expression {{{a^2-5ab+6b^2}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{-5}}}, and the last coefficient is {{{6}}}.



Now multiply the first coefficient {{{1}}} by the last coefficient {{{6}}} to get {{{(1)(6)=6}}}.



Now the question is: what two whole numbers multiply to {{{6}}} (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 {{{6}}} (the previous product).



Factors of {{{6}}}:

1,2,3,6

-1,-2,-3,-6



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{6}}}.

1*6 = 6
2*3 = 6
(-1)*(-6) = 6
(-2)*(-3) = 6


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;6&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+6=7&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;3&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;2+3=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;-1&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;-1+(-6)=-7&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-2&lt;/font&gt;&lt;/td&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;-2+(-3)=-5&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{-2}}} and {{{-3}}} add to {{{-5}}} (the middle coefficient).



So the two numbers {{{-2}}} and {{{-3}}} both multiply to {{{6}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{-5}}}



Now replace the middle term {{{-5ab}}} with {{{-2ab-3ab}}}. Remember, {{{-2}}} and {{{-3}}} add to {{{-5}}}. So this shows us that {{{-2ab-3ab=-5ab}}}.



{{{a^2+highlight(-2ab-3ab)+6b^2}}} Replace the second term {{{-5ab}}} with {{{-2ab-3ab}}}.



{{{(a^2-2ab)+(-3ab+6b^2)}}} Group the terms into two pairs.



{{{a(a-2b)+(-3ab+6b^2)}}} Factor out the GCF {{{a}}} from the first group.



{{{a(a-2b)-3b(a-2b)}}} Factor out the GCF {{{-3b}}} from the second group.



{{{(a-3b)(a-2b)}}} Combine like terms.



So this means that {{{a^2-5ab+6b^2}}} factors to {{{(a-3b)(a-2b)}}}


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So {{{7ab(a^2-5ab+6b^2)}}} factors to {{{7ab(a-3b)(a-2b)}}}



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Answer:



So {{{7a^3b-35a^2b^2+42ab^3}}} completely factors to {{{7ab(a-3b)(a-2b)}}}</PRE>