<PRE><B>Question 550941</B>


{{{70w^3-125w^2+30w}}} Start with the given expression.



{{{5w(14w^2-25w+6)}}} Factor out the GCF {{{5w}}}.



Now let&#39;s try to factor the inner expression {{{14w^2-25w+6}}}



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Looking at the expression {{{14w^2-25w+6}}}, we can see that the first coefficient is {{{14}}}, the second coefficient is {{{-25}}}, and the last term is {{{6}}}.



Now multiply the first coefficient {{{14}}} by the last term {{{6}}} to get {{{(14)(6)=84}}}.



Now the question is: what two whole numbers multiply to {{{84}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{-25}}}?



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 {{{84}}} (the previous product).



Factors of {{{84}}}:

1,2,3,4,6,7,12,14,21,28,42,84

-1,-2,-3,-4,-6,-7,-12,-14,-21,-28,-42,-84



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{84}}}.

1*84 = 84
2*42 = 84
3*28 = 84
4*21 = 84
6*14 = 84
7*12 = 84
(-1)*(-84) = 84
(-2)*(-42) = 84
(-3)*(-28) = 84
(-4)*(-21) = 84
(-6)*(-14) = 84
(-7)*(-12) = 84


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{-25}}}:



&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;84&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+84=85&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;42&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;2+42=44&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;28&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;3+28=31&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;21&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;4+21=25&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;14&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;6+14=20&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;7&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;7+12=19&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;-84&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+(-84)=-85&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;-42&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-2+(-42)=-44&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;-28&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-3+(-28)=-31&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;-21&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-4+(-21)=-25&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;-14&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-6+(-14)=-20&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-7&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;-7+(-12)=-19&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{-4}}} and {{{-21}}} add to {{{-25}}} (the middle coefficient).



So the two numbers {{{-4}}} and {{{-21}}} both multiply to {{{84}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{-25}}}



Now replace the middle term {{{-25w}}} with {{{-4w-21w}}}. Remember, {{{-4}}} and {{{-21}}} add to {{{-25}}}. So this shows us that {{{-4w-21w=-25w}}}.



{{{14w^2+highlight(-4w-21w)+6}}} Replace the second term {{{-25w}}} with {{{-4w-21w}}}.



{{{(14w^2-4w)+(-21w+6)}}} Group the terms into two pairs.



{{{2w(7w-2)+(-21w+6)}}} Factor out the GCF {{{2w}}} from the first group.



{{{2w(7w-2)-3(7w-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.



{{{(2w-3)(7w-2)}}} Combine like terms. Or factor out the common term {{{7w-2}}}



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So {{{5w(14w^2-25w+6)}}} then factors further to {{{5w(2w-3)(7w-2)}}}



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Answer:



So {{{70w^3-125w^2+30w}}} completely factors to {{{5w(2w-3)(7w-2)}}}.



In other words, {{{70w^3-125w^2+30w=5w(2w-3)(7w-2)}}}.



Note: you can check the answer by expanding {{{5w(2w-3)(7w-2)}}} to get {{{70w^3-125w^2+30w}}} or by graphing the original expression and the answer (the two graphs should be identical).
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