<PRE><B>Question 548530</B>


Looking at the expression {{{t^2-24t+135}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{-24}}}, and the last term is {{{135}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{135}}} to get {{{(1)(135)=135}}}.



Now the question is: what two whole numbers multiply to {{{135}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{-24}}}?



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 {{{135}}} (the previous product).



Factors of {{{135}}}:

1,3,5,9,15,27,45,135

-1,-3,-5,-9,-15,-27,-45,-135



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{135}}}.

1*135 = 135
3*45 = 135
5*27 = 135
9*15 = 135
(-1)*(-135) = 135
(-3)*(-45) = 135
(-5)*(-27) = 135
(-9)*(-15) = 135


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{-24}}}:



&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;135&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+135=136&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;45&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;3+45=48&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;27&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;5+27=32&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;15&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;9+15=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;-1&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-135&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+(-135)=-136&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;-45&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-3+(-45)=-48&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;-27&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-5+(-27)=-32&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&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;-15&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-9+(-15)=-24&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{-9}}} and {{{-15}}} add to {{{-24}}} (the middle coefficient).



So the two numbers {{{-9}}} and {{{-15}}} both multiply to {{{135}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{-24}}}



Now replace the middle term {{{-24t}}} with {{{-9t-15t}}}. Remember, {{{-9}}} and {{{-15}}} add to {{{-24}}}. So this shows us that {{{-9t-15t=-24t}}}.



{{{t^2+highlight(-9t-15t)+135}}} Replace the second term {{{-24t}}} with {{{-9t-15t}}}.



{{{(t^2-9t)+(-15t+135)}}} Group the terms into two pairs.



{{{t(t-9)+(-15t+135)}}} Factor out the GCF {{{t}}} from the first group.



{{{t(t-9)-15(t-9)}}} Factor out {{{15}}} 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.



{{{(t-15)(t-9)}}} Combine like terms. Or factor out the common term {{{t-9}}}



===============================================================



Answer:



So {{{t^2-24t+135}}} factors to {{{(t-15)(t-9)}}}.



In other words, {{{t^2-24t+135=(t-15)(t-9)}}}.



Note: you can check the answer by expanding {{{(t-15)(t-9)}}} to get {{{t^2-24t+135}}} or by graphing the original expression and the answer (the two graphs should be identical).
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