<PRE><B>Question 150652</B>


Looking at the expression {{{x^2+10x+21}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{10}}}, and the last term is {{{21}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{21}}} to get {{{(1)(21)=21}}}.



Now the question is: what two whole numbers multiply to {{{21}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{10}}}?



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 {{{21}}} (the previous product).



Factors of {{{21}}}:

1,3,7,21

-1,-3,-7,-21



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{21}}}.

1*21
3*7
(-1)*(-21)
(-3)*(-7)


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{10}}}:



&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;21&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+21=22&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;3&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;3+7=10&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;-21&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+(-21)=-22&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&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;-7&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-3+(-7)=-10&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{3}}} and {{{7}}} add to {{{10}}} (the middle coefficient).



So the two numbers {{{3}}} and {{{7}}} both multiply to {{{21}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{10}}}



Now replace the middle term {{{10x}}} with {{{3x+7x}}}. Remember, {{{3}}} and {{{7}}} add to {{{10}}}. So this shows us that {{{3x+7x=10x}}}.



{{{x^2+highlight(3x+7x)+21}}} Replace the second term {{{10x}}} with {{{3x+7x}}}.



{{{(x^2+3x)+(7x+21)}}} Group the terms into two pairs.



{{{x(x+3)+(7x+21)}}} Factor out the GCF {{{x}}} from the first group.



{{{x(x+3)+7(x+3)}}} 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+3)}}} Combine like terms. Or factor out the common term {{{x+3}}}


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Answer:



So {{{x^2+10x+21}}} factors to {{{(x+7)(x+3)}}}.



Note: you can check the answer by FOILing {{{(x+7)(x+3)}}} to get {{{x^2+10x+21}}} or by graphing the original expression and the answer (the two graphs should be identical).
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