<PRE><B>Question 207387</B>
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{{{-w^2-8w+105}}} Start with the given expression.



{{{-(w^2+8w-105)}}} Factor out the GCF {{{-1}}}.



Now let&#39;s try to factor the inner expression {{{w^2+8w-105}}}



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Looking at the expression {{{w^2+8w-105}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{8}}}, and the last term is {{{-105}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{-105}}} to get {{{(1)(-105)=-105}}}.



Now the question is: what two whole numbers multiply to {{{-105}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{8}}}?



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 {{{-105}}} (the previous product).



Factors of {{{-105}}}:

1,3,5,7,15,21,35,105

-1,-3,-5,-7,-15,-21,-35,-105



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{-105}}}.

1*(-105) = -105
3*(-35) = -105
5*(-21) = -105
7*(-15) = -105
(-1)*(105) = -105
(-3)*(35) = -105
(-5)*(21) = -105
(-7)*(15) = -105


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{8}}}:



&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;-105&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+(-105)=-104&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;-35&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;3+(-35)=-32&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;5&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;5+(-21)=-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;7&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-15&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;7+(-15)=-8&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;105&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+105=104&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;35&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-3+35=32&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-5&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;-5+21=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;-7&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;15&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-7+15=8&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{-7}}} and {{{15}}} add to {{{8}}} (the middle coefficient).



So the two numbers {{{-7}}} and {{{15}}} both multiply to {{{-105}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{8}}}



Now replace the middle term {{{8w}}} with {{{-7w+15w}}}. Remember, {{{-7}}} and {{{15}}} add to {{{8}}}. So this shows us that {{{-7w+15w=8w}}}.



{{{w^2+highlight(-7w+15w)-105}}} Replace the second term {{{8w}}} with {{{-7w+15w}}}.



{{{(w^2-7w)+(15w-105)}}} Group the terms into two pairs.



{{{w(w-7)+(15w-105)}}} Factor out the GCF {{{w}}} from the first group.



{{{w(w-7)+15(w-7)}}} Factor out {{{15}}} 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.



{{{(w+15)(w-7)}}} Combine like terms. Or factor out the common term {{{w-7}}}



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So {{{-(w^2+8w-105)}}} then factors further to {{{-(w+15)(w-7)}}}



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Answer:



So {{{-w^2-8w+105}}} completely factors to {{{-(w+15)(w-7)}}}.



In other words, {{{-w^2-8w+105=-(w+15)(w-7)}}}.



Note: you can check the answer by expanding {{{-(w+15)(w-7)}}} to get {{{-w^2-8w+105}}} or by graphing the original expression and the answer (the two graphs should be identical).


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