<PRE><B>Question 167669</B>


Looking at the expression {{{6a^2-29a+9}}}, we can see that the first coefficient is {{{6}}}, the second coefficient is {{{-29}}}, and the last term is {{{9}}}.



Now multiply the first coefficient {{{6}}} by the last term {{{9}}} to get {{{(6)(9)=54}}}.



Now the question is: what two whole numbers multiply to {{{54}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{-29}}}?



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 {{{54}}} (the previous product).



Factors of {{{54}}}:

1,2,3,6,9,18,27,54

-1,-2,-3,-6,-9,-18,-27,-54



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{54}}}.

1*54
2*27
3*18
6*9
(-1)*(-54)
(-2)*(-27)
(-3)*(-18)
(-6)*(-9)


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{-29}}}:



&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;54&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+54=55&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;2&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;2+27=29&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;18&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;3+18=21&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;6&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;6+9=15&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;-54&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+(-54)=-55&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-2&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;-2+(-27)=-29&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;-18&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-3+(-18)=-21&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-6&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;-6+(-9)=-15&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{-2}}} and {{{-27}}} add to {{{-29}}} (the middle coefficient).



So the two numbers {{{-2}}} and {{{-27}}} both multiply to {{{54}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{-29}}}



Now replace the middle term {{{-29a}}} with {{{-2a-27a}}}. Remember, {{{-2}}} and {{{-27}}} add to {{{-29}}}. So this shows us that {{{-2a-27a=-29a}}}.



{{{6a^2+highlight(-2a-27a)+9}}} Replace the second term {{{-29a}}} with {{{-2a-27a}}}.



{{{(6a^2-2a)+(-27a+9)}}} Group the terms into two pairs.



{{{2a(3a-1)+(-27a+9)}}} Factor out the GCF {{{2a}}} from the first group.



{{{2a(3a-1)-9(3a-1)}}} 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.



{{{(2a-9)(3a-1)}}} Combine like terms. Or factor out the common term {{{3a-1}}}


---------------------------------------------



Answer:



So {{{6a^2-29a+9}}} factors to {{{(2a-9)(3a-1)}}}.



Note: you can check the answer by FOILing {{{(2a-9)(3a-1)}}} to get {{{6a^2-29a+9}}} or by graphing the original expression and the answer (the two graphs should be identical).
</PRE>