<PRE><B>Question 560713</B>


Looking at the expression {{{x^2+6x-7}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{6}}}, and the last term is {{{-7}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{-7}}} to get {{{(1)(-7)=-7}}}.



Now the question is: what two whole numbers multiply to {{{-7}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{6}}}?



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 {{{-7}}} (the previous product).



Factors of {{{-7}}}:

1,7

-1,-7



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{-7}}}.

1*(-7) = -7
(-1)*(7) = -7


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{6}}}:



&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;-7&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+(-7)=-6&lt;/font&gt;&lt;/td&gt;&lt;/tr&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;7&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-1+7=6&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{-1}}} and {{{7}}} add to {{{6}}} (the middle coefficient).



So the two numbers {{{-1}}} and {{{7}}} both multiply to {{{-7}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{6}}}



Now replace the middle term {{{6x}}} with {{{-x+7x}}}. Remember, {{{-1}}} and {{{7}}} add to {{{6}}}. So this shows us that {{{-x+7x=6x}}}.



{{{x^2+highlight(-x+7x)-7}}} Replace the second term {{{6x}}} with {{{-x+7x}}}.



{{{(x^2-x)+(7x-7)}}} Group the terms into two pairs.



{{{x(x-1)+(7x-7)}}} Factor out the GCF {{{x}}} from the first group.



{{{x(x-1)+7(x-1)}}} 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-1)}}} Combine like terms. Or factor out the common term {{{x-1}}}



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Answer:



So {{{x^2+6x-7}}} factors to {{{(x+7)(x-1)}}}.



In other words, {{{x^2+6x-7=(x+7)(x-1)}}}.



Note: you can check the answer by expanding {{{(x+7)(x-1)}}} to get {{{x^2+6x-7}}} or by graphing the original expression and the answer (the two graphs should be identical).

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