<PRE><B>Question 581992</B>


{{{63b^2+294bf+343f^2}}} Start with the given expression.



{{{7(9b^2+42bf+49f^2)}}} Factor out the GCF {{{7}}}.



Now let&#39;s try to factor the inner expression {{{9b^2+42bf+49f^2}}}



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Looking at the expression {{{9b^2+42bf+49f^2}}}, we can see that the first coefficient is {{{9}}}, the second coefficient is {{{42}}}, and the last coefficient is {{{49}}}.



Now multiply the first coefficient {{{9}}} by the last coefficient {{{49}}} to get {{{(9)(49)=441}}}.



Now the question is: what two whole numbers multiply to {{{441}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{42}}}?



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 {{{441}}} (the previous product).



Factors of {{{441}}}:

1,3,7,9,21,49,63,147,441

-1,-3,-7,-9,-21,-49,-63,-147,-441



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{441}}}.

1*441 = 441
3*147 = 441
7*63 = 441
9*49 = 441
21*21 = 441
(-1)*(-441) = 441
(-3)*(-147) = 441
(-7)*(-63) = 441
(-9)*(-49) = 441
(-21)*(-21) = 441


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{42}}}:



&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;441&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+441=442&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;147&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;3+147=150&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;63&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;7+63=70&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;49&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;9+49=58&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;21&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;21&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;21+21=42&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;-441&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+(-441)=-442&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;-147&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-3+(-147)=-150&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;-63&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-7+(-63)=-70&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;-49&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-9+(-49)=-58&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&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;-21&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-21+(-21)=-42&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{21}}} and {{{21}}} add to {{{42}}} (the middle coefficient).



So the two numbers {{{21}}} and {{{21}}} both multiply to {{{441}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{42}}}



Now replace the middle term {{{42bf}}} with {{{21bf+21bf}}}. Remember, {{{21}}} and {{{21}}} add to {{{42}}}. So this shows us that {{{21bf+21bf=42bf}}}.



{{{9b^2+highlight(21bf+21bf)+49f^2}}} Replace the second term {{{42bf}}} with {{{21bf+21bf}}}.



{{{(9b^2+21bf)+(21bf+49f^2)}}} Group the terms into two pairs.



{{{3b(3b+7f)+(21bf+49f^2)}}} Factor out the GCF {{{3b}}} from the first group.



{{{3b(3b+7f)+7f(3b+7f)}}} Factor out {{{7f}}} 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.



{{{(3b+7f)(3b+7f)}}} Combine like terms. Or factor out the common term {{{3b+7f}}}



{{{(3b+7f)^2}}} Condense the terms.



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So {{{7(9b^2+42bf+49f^2)}}} then factors further to {{{7(3b+7f)^2}}}



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Answer:



So {{{63b^2+294bf+343f^2}}} completely factors to {{{7(3b+7f)^2}}}.



In other words, {{{63b^2+294bf+343f^2=7(3b+7f)^2}}}.



Note: you can check the answer by expanding {{{7(3b+7f)^2}}} to get {{{63b^2+294bf+343f^2}}} or by graphing the original expression and the answer (the two graphs should be identical).
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