<PRE><B>Question 960515</B>


Looking at the expression {{{w^2+30w+81}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{30}}}, 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 {{{30}}}?



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 {{{30}}}:



&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=red&gt;3&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;27&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&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;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 {{{3}}} and {{{27}}} add to {{{30}}} (the middle coefficient).



So the two numbers {{{3}}} and {{{27}}} both multiply to {{{81}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{30}}}



Now replace the middle term {{{30w}}} with {{{3w+27w}}}. Remember, {{{3}}} and {{{27}}} add to {{{30}}}. So this shows us that {{{3w+27w=30w}}}.



{{{w^2+highlight(3w+27w)+81}}} Replace the second term {{{30w}}} with {{{3w+27w}}}.



{{{(w^2+3w)+(27w+81)}}} Group the terms into two pairs.



{{{w(w+3)+(27w+81)}}} Factor out the GCF {{{w}}} from the first group.



{{{w(w+3)+27(w+3)}}} Factor out {{{27}}} 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.



{{{(w+27)(w+3)}}} Combine like terms. Or factor out the common term {{{w+3}}}



===============================================================



Answer:



So {{{w^2+30w+81}}} factors to {{{(w+27)(w+3)}}}.



In other words, {{{w^2+30w+81=(w+27)(w+3)}}}.



Note: you can check the answer by expanding {{{(w+27)(w+3)}}} to get {{{w^2+30w+81}}} or by graphing the original expression and the answer (the two graphs should be identical).


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