<PRE><B>Question 631225</B>


{{{y^3-15y^2+54y}}} Start with the given expression.



{{{y(y^2-15y+54)}}} Factor out the GCF {{{y}}}.



Now let&#39;s try to factor the inner expression {{{y^2-15y+54}}}



---------------------------------------------------------------



Looking at the expression {{{y^2-15y+54}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{-15}}}, and the last term is {{{54}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{54}}} to get {{{(1)(54)=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 {{{-15}}}?



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 = 54
2*27 = 54
3*18 = 54
6*9 = 54
(-1)*(-54) = 54
(-2)*(-27) = 54
(-3)*(-18) = 54
(-6)*(-9) = 54


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{-15}}}:



&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=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=red&gt;-6&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;-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 {{{-6}}} and {{{-9}}} add to {{{-15}}} (the middle coefficient).



So the two numbers {{{-6}}} and {{{-9}}} both multiply to {{{54}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{-15}}}



Now replace the middle term {{{-15y}}} with {{{-6y-9y}}}. Remember, {{{-6}}} and {{{-9}}} add to {{{-15}}}. So this shows us that {{{-6y-9y=-15y}}}.



{{{y^2+highlight(-6y-9y)+54}}} Replace the second term {{{-15y}}} with {{{-6y-9y}}}.



{{{(y^2-6y)+(-9y+54)}}} Group the terms into two pairs.



{{{y(y-6)+(-9y+54)}}} Factor out the GCF {{{y}}} from the first group.



{{{y(y-6)-9(y-6)}}} 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.



{{{(y-9)(y-6)}}} Combine like terms. Or factor out the common term {{{y-6}}}



--------------------------------------------------



So {{{y(y^2-15y+54)}}} then factors further to {{{y(y-9)(y-6)}}}



===============================================================



Answer:



So {{{y^3-15y^2+54y}}} completely factors to {{{y(y-9)(y-6)}}}.



In other words, {{{y^3-15y^2+54y=y(y-9)(y-6)}}}.



Note: you can check the answer by expanding {{{y(y-9)(y-6)}}} to get {{{y^3-15y^2+54y}}} or by graphing the original expression and the answer (the two graphs should be identical).


&lt;font color=&quot;red&quot;&gt;--------------------------------------------------------------------------------------------------------------&lt;/font&gt;
If you need more help, email me at &lt;a href=&quot;mailto:jim_thompson5910@hotmail.com?Subject=I%20Need%20Algebra%20Help&quot;&gt;jim_thompson5910@hotmail.com&lt;/a&gt;


Also, please consider visiting my website: &lt;a href=&quot;http://www.freewebs.com/jimthompson5910/home.html&quot;&gt;http://www.freewebs.com/jimthompson5910/home.html&lt;/a&gt; and making a donation. Thank you


Jim
&lt;font color=&quot;red&quot;&gt;--------------------------------------------------------------------------------------------------------------&lt;/font&gt;</PRE>