<PRE><B>Question 620953</B>


Looking at the expression {{{x^2-5x-6}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{-5}}}, and the last term is {{{-6}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{-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=red&gt;1&lt;/font&gt;&lt;/td&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;1+(-6)=-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;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)=-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;6&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+6=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;-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=1&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{1}}} and {{{-6}}} add to {{{-5}}} (the middle coefficient).



So the two numbers {{{1}}} and {{{-6}}} 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 {{{-5x}}} with {{{x-6x}}}. Remember, {{{1}}} and {{{-6}}} add to {{{-5}}}. So this shows us that {{{x-6x=-5x}}}.



{{{x^2+highlight(x-6x)-6}}} Replace the second term {{{-5x}}} with {{{x-6x}}}.



{{{(x^2+x)+(-6x-6)}}} Group the terms into two pairs.



{{{x(x+1)+(-6x-6)}}} Factor out the GCF {{{x}}} from the first group.



{{{x(x+1)-6(x+1)}}} Factor out {{{6}}} 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-6)(x+1)}}} Combine like terms. Or factor out the common term {{{x+1}}}



===============================================================



Answer:



So {{{x^2-5x-6}}} factors to {{{(x-6)(x+1)}}}.



In other words, {{{x^2-5x-6=(x-6)(x+1)}}}.



Note: you can check the answer by expanding {{{(x-6)(x+1)}}} to get {{{x^2-5x-6}}} or by graphing the original expression and the answer (the two graphs should be identical).
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