<PRE><B>Question 197636</B>


Looking at the expression {{{x^2+11x+28}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{11}}}, and the last term is {{{28}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{28}}} to get {{{(1)(28)=28}}}.



Now the question is: what two whole numbers multiply to {{{28}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{11}}}?



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 {{{28}}} (the previous product).



Factors of {{{28}}}:

1,2,4,7,14,28

-1,-2,-4,-7,-14,-28



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{28}}}.

1*28
2*14
4*7
(-1)*(-28)
(-2)*(-14)
(-4)*(-7)


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{11}}}:



&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;28&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+28=29&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;14&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;2+14=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;4&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;4+7=11&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;-28&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+(-28)=-29&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;-14&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-2+(-14)=-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;-4&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;-4+(-7)=-11&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{4}}} and {{{7}}} add to {{{11}}} (the middle coefficient).



So the two numbers {{{4}}} and {{{7}}} both multiply to {{{28}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{11}}}



Now replace the middle term {{{11x}}} with {{{4x+7x}}}. Remember, {{{4}}} and {{{7}}} add to {{{11}}}. So this shows us that {{{4x+7x=11x}}}.



{{{x^2+highlight(4x+7x)+28}}} Replace the second term {{{11x}}} with {{{4x+7x}}}.



{{{(x^2+4x)+(7x+28)}}} Group the terms into two pairs.



{{{x(x+4)+(7x+28)}}} Factor out the GCF {{{x}}} from the first group.



{{{x(x+4)+7(x+4)}}} 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+4)}}} Combine like terms. Or factor out the common term {{{x+4}}}


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Answer:



So {{{x^2+11x+28}}} factors to {{{(x+7)(x+4)}}}.



Note: you can check the answer by FOILing {{{(x+7)(x+4)}}} to get {{{x^2+11x+28}}} or by graphing the original expression and the answer (the two graphs should be identical).
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