<PRE><B>Question 181956</B>


Looking at the expression {{{a^2-2a-35}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{-2}}}, and the last term is {{{-35}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{-35}}} to get {{{(1)(-35)=-35}}}.



Now the question is: what two whole numbers multiply to {{{-35}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{-2}}}?



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 {{{-35}}} (the previous product).



Factors of {{{-35}}}:

1,5,7,35

-1,-5,-7,-35



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{-35}}}.

1*(-35)
5*(-7)
(-1)*(35)
(-5)*(7)


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{-2}}}:



&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;-35&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+(-35)=-34&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;5&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;5+(-7)=-2&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;35&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+35=34&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;7&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-5+7=2&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{5}}} and {{{-7}}} add to {{{-2}}} (the middle coefficient).



So the two numbers {{{5}}} and {{{-7}}} both multiply to {{{-35}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{-2}}}



Now replace the middle term {{{-2a}}} with {{{5a-7a}}}. Remember, {{{5}}} and {{{-7}}} add to {{{-2}}}. So this shows us that {{{5a-7a=-2a}}}.



{{{a^2+highlight(5a-7a)-35}}} Replace the second term {{{-2a}}} with {{{5a-7a}}}.



{{{(a^2+5a)+(-7a-35)}}} Group the terms into two pairs.



{{{a(a+5)+(-7a-35)}}} Factor out the GCF {{{a}}} from the first group.



{{{a(a+5)-7(a+5)}}} 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.



{{{(a-7)(a+5)}}} Combine like terms. Or factor out the common term {{{a+5}}}


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Answer:



So {{{a^2-2a-35}}} factors to {{{(a-7)(a+5)}}}.



Note: you can check the answer by FOILing {{{(a-7)(a+5)}}} to get {{{a^2-2a-35}}} or by graphing the original expression and the answer (the two graphs should be identical).
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