<PRE><B>Question 423063</B>


Looking at the expression {{{7x^2-4x-11}}}, we can see that the first coefficient is {{{7}}}, the second coefficient is {{{-4}}}, and the last term is {{{-11}}}.



Now multiply the first coefficient {{{7}}} by the last term {{{-11}}} to get {{{(7)(-11)=-77}}}.



Now the question is: what two whole numbers multiply to {{{-77}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{-4}}}?



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 {{{-77}}} (the previous product).



Factors of {{{-77}}}:

1,7,11,77

-1,-7,-11,-77



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{-77}}}.

1*(-77) = -77
7*(-11) = -77
(-1)*(77) = -77
(-7)*(11) = -77


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{-4}}}:



&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;-77&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+(-77)=-76&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;-11&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;7+(-11)=-4&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;77&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+77=76&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;11&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-7+11=4&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{7}}} and {{{-11}}} add to {{{-4}}} (the middle coefficient).



So the two numbers {{{7}}} and {{{-11}}} both multiply to {{{-77}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{-4}}}



Now replace the middle term {{{-4x}}} with {{{7x-11x}}}. Remember, {{{7}}} and {{{-11}}} add to {{{-4}}}. So this shows us that {{{7x-11x=-4x}}}.



{{{7x^2+highlight(7x-11x)-11}}} Replace the second term {{{-4x}}} with {{{7x-11x}}}.



{{{(7x^2+7x)+(-11x-11)}}} Group the terms into two pairs.



{{{7x(x+1)+(-11x-11)}}} Factor out the GCF {{{7x}}} from the first group.



{{{7x(x+1)-11(x+1)}}} Factor out {{{11}}} 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.



{{{(7x-11)(x+1)}}} Combine like terms. Or factor out the common term {{{x+1}}}



===============================================================



Answer:



So {{{7x^2-4x-11}}} factors to {{{(7x-11)(x+1)}}}.



In other words, {{{7x^2-4x-11=(7x-11)(x+1)}}}.



Note: you can check the answer by expanding {{{(7x-11)(x+1)}}} to get {{{7x^2-4x-11}}} or by graphing the original expression and the answer (the two graphs should be identical).



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Jim</PRE>