<PRE><B>Question 221654</B>

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Looking at the expression {{{5w^2-18w+9}}}, we can see that the first coefficient is {{{5}}}, the second coefficient is {{{-18}}}, and the last term is {{{9}}}.



Now multiply the first coefficient {{{5}}} by the last term {{{9}}} to get {{{(5)(9)=45}}}.



Now the question is: what two whole numbers multiply to {{{45}}} (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 {{{45}}} (the previous product).



Factors of {{{45}}}:

1,3,5,9,15,45

-1,-3,-5,-9,-15,-45



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{45}}}.

1*45 = 45
3*15 = 45
5*9 = 45
(-1)*(-45) = 45
(-3)*(-15) = 45
(-5)*(-9) = 45


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;45&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+45=46&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;15&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;3+15=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;5&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;5+9=14&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;-45&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+(-45)=-46&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;-15&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-3+(-15)=-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;-5&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;-5+(-9)=-14&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{-3}}} and {{{-15}}} add to {{{-18}}} (the middle coefficient).



So the two numbers {{{-3}}} and {{{-15}}} both multiply to {{{45}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{-18}}}



Now replace the middle term {{{-18w}}} with {{{-3w-15w}}}. Remember, {{{-3}}} and {{{-15}}} add to {{{-18}}}. So this shows us that {{{-3w-15w=-18w}}}.



{{{5w^2+highlight(-3w-15w)+9}}} Replace the second term {{{-18w}}} with {{{-3w-15w}}}.



{{{(5w^2-3w)+(-15w+9)}}} Group the terms into two pairs.



{{{w(5w-3)+(-15w+9)}}} Factor out the GCF {{{w}}} from the first group.



{{{w(5w-3)-3(5w-3)}}} Factor out {{{3}}} 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-3)(5w-3)}}} Combine like terms. Or factor out the common term {{{5w-3}}}



===============================================================


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Answer:



So {{{5w^2-18w+9}}} factors to {{{(w-3)(5w-3)}}}.



In other words, {{{5w^2-18w+9=(w-3)(5w-3)}}}.



Note: you can check the answer by expanding {{{(w-3)(5w-3)}}} to get {{{5w^2-18w+9}}} or by graphing the original expression and the answer (the two graphs should be identical).


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