<PRE><B>Question 551105</B>


Looking at the expression {{{x^2+18x+81}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{18}}}, and the last term is {{{81}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{81}}} to get {{{(1)(81)=81}}}.



Now the question is: what two whole numbers multiply to {{{81}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{18}}}?



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 {{{81}}} (the previous product).



Factors of {{{81}}}:

1,3,9,27,81

-1,-3,-9,-27,-81



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{81}}}.

1*81 = 81
3*27 = 81
9*9 = 81
(-1)*(-81) = 81
(-3)*(-27) = 81
(-9)*(-9) = 81


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{18}}}:



&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;81&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+81=82&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;27&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;3+27=30&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;9&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;9&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;9+9=18&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;-81&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+(-81)=-82&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;-27&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-3+(-27)=-30&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-9&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-9&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-9+(-9)=-18&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{9}}} and {{{9}}} add to {{{18}}} (the middle coefficient).



So the two numbers {{{9}}} and {{{9}}} both multiply to {{{81}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{18}}}



Now replace the middle term {{{18x}}} with {{{9x+9x}}}. Remember, {{{9}}} and {{{9}}} add to {{{18}}}. So this shows us that {{{9x+9x=18x}}}.



{{{x^2+highlight(9x+9x)+81}}} Replace the second term {{{18x}}} with {{{9x+9x}}}.



{{{(x^2+9x)+(9x+81)}}} Group the terms into two pairs.



{{{x(x+9)+(9x+81)}}} Factor out the GCF {{{x}}} from the first group.



{{{x(x+9)+9(x+9)}}} Factor out {{{9}}} 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+9)(x+9)}}} Combine like terms. Or factor out the common term {{{x+9}}}



{{{(x+9)^2}}} Condense the terms.



===============================================================



Answer:



So {{{x^2+18x+81}}} factors to {{{(x+9)^2}}}.



In other words, {{{x^2+18x+81=(x+9)^2}}}.



Note: you can check the answer by expanding {{{(x+9)^2}}} to get {{{x^2+18x+81}}} or by graphing the original expression and the answer (the two graphs should be identical).
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