<PRE><B>Question 599740</B>


Looking at the expression {{{b^2+16b+63}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{16}}}, and the last term is {{{63}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{63}}} to get {{{(1)(63)=63}}}.



Now the question is: what two whole numbers multiply to {{{63}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{16}}}?



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 {{{63}}} (the previous product).



Factors of {{{63}}}:

1,3,7,9,21,63

-1,-3,-7,-9,-21,-63



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{63}}}.

1*63 = 63
3*21 = 63
7*9 = 63
(-1)*(-63) = 63
(-3)*(-21) = 63
(-7)*(-9) = 63


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{16}}}:



&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;63&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+63=64&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;21&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;3+21=24&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;9&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;7+9=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;-1&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-63&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+(-63)=-64&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;-21&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-3+(-21)=-24&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;-9&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-7+(-9)=-16&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{7}}} and {{{9}}} add to {{{16}}} (the middle coefficient).



So the two numbers {{{7}}} and {{{9}}} both multiply to {{{63}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{16}}}



Now replace the middle term {{{16b}}} with {{{7b+9b}}}. Remember, {{{7}}} and {{{9}}} add to {{{16}}}. So this shows us that {{{7b+9b=16b}}}.



{{{b^2+highlight(7b+9b)+63}}} Replace the second term {{{16b}}} with {{{7b+9b}}}.



{{{(b^2+7b)+(9b+63)}}} Group the terms into two pairs.



{{{b(b+7)+(9b+63)}}} Factor out the GCF {{{b}}} from the first group.



{{{b(b+7)+9(b+7)}}} 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.



{{{(b+9)(b+7)}}} Combine like terms. Or factor out the common term {{{b+7}}}



===============================================================



Answer:



So {{{b^2+16b+63}}} factors to {{{(b+9)(b+7)}}}.



In other words, {{{b^2+16b+63=(b+9)(b+7)}}}.



Note: you can check the answer by expanding {{{(b+9)(b+7)}}} to get {{{b^2+16b+63}}} or by graphing the original expression and the answer (the two graphs should be identical).
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