<PRE><B>Question 338529</B>


{{{-x^3+x^2+42x}}} Start with the given expression.



{{{-x(x^2-x-42)}}} Factor out the GCF {{{-x}}}.



Now let&#39;s try to factor the inner expression {{{x^2-x-42}}}



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Looking at the expression {{{x^2-x-42}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{-1}}}, and the last term is {{{-42}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{-42}}} to get {{{(1)(-42)=-42}}}.



Now the question is: what two whole numbers multiply to {{{-42}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{-1}}}?



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 {{{-42}}} (the previous product).



Factors of {{{-42}}}:

1,2,3,6,7,14,21,42

-1,-2,-3,-6,-7,-14,-21,-42



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{-42}}}.

1*(-42) = -42
2*(-21) = -42
3*(-14) = -42
6*(-7) = -42
(-1)*(42) = -42
(-2)*(21) = -42
(-3)*(14) = -42
(-6)*(7) = -42


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{-1}}}:



&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;-42&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+(-42)=-41&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;-21&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;2+(-21)=-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;3&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;3+(-14)=-11&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;6&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-7&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;6+(-7)=-1&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;42&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+42=41&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;21&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-2+21=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;-3&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;-3+14=11&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;7&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-6+7=1&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{6}}} and {{{-7}}} add to {{{-1}}} (the middle coefficient).



So the two numbers {{{6}}} and {{{-7}}} both multiply to {{{-42}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{-1}}}



Now replace the middle term {{{-1x}}} with {{{6x-7x}}}. Remember, {{{6}}} and {{{-7}}} add to {{{-1}}}. So this shows us that {{{6x-7x=-1x}}}.



{{{x^2+highlight(6x-7x)-42}}} Replace the second term {{{-1x}}} with {{{6x-7x}}}.



{{{(x^2+6x)+(-7x-42)}}} Group the terms into two pairs.



{{{x(x+6)+(-7x-42)}}} Factor out the GCF {{{x}}} from the first group.



{{{x(x+6)-7(x+6)}}} Factor out {{{7}}} 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-7)(x+6)}}} Combine like terms. Or factor out the common term {{{x+6}}}



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So {{{-1x(x^2-x-42)}}} then factors further to {{{-(x-7)(x+6)}}}



===============================================================



Answer:



So {{{-x^3+x^2+42x}}} completely factors to {{{-(x-7)(x+6)}}}.



In other words, {{{-x^3+x^2+42x=-x(x-7)(x+6)}}}.



Note: you can check the answer by expanding {{{-x(x-7)(x+6)}}} to get {{{-x^3+x^2+42x}}} or by graphing the original expression and the answer (the two graphs should be identical).



If you need more help, email me at &lt;a href=&quot;mailto:jim_thompson5910@hotmail.com?Subject=I%20Need%20Algebra%20Help&quot;&gt;jim_thompson5910@hotmail.com&lt;/a&gt;


Also, feel free to check out my tutoring &lt;a href=&quot;http://www.freewebs.com/jimthompson5910/home.html&quot;&gt;website&lt;/a&gt;. 


Jim</PRE>