<PRE><B>Question 908042</B>


{{{-r^2+11r-18}}} Start with the given expression.



{{{-(r^2-11r+18)}}} Factor out the GCF {{{-1}}}.



Now let&#39;s try to factor the inner expression {{{r^2-11r+18}}}



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Looking at the expression {{{r^2-11r+18}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{-11}}}, and the last term is {{{18}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{18}}} to get {{{(1)(18)=18}}}.



Now the question is: what two whole numbers multiply to {{{18}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{-11}}}?



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 {{{18}}} (the previous product).



Factors of {{{18}}}:

1,2,3,6,9,18

-1,-2,-3,-6,-9,-18



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{18}}}.

1*18 = 18
2*9 = 18
3*6 = 18
(-1)*(-18) = 18
(-2)*(-9) = 18
(-3)*(-6) = 18


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{-11}}}:



&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;18&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+18=19&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;9&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;2+9=11&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;6&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;3+6=9&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;-18&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+(-18)=-19&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;-9&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-2+(-9)=-11&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;-6&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-3+(-6)=-9&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{-2}}} and {{{-9}}} add to {{{-11}}} (the middle coefficient).



So the two numbers {{{-2}}} and {{{-9}}} both multiply to {{{18}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{-11}}}



Now replace the middle term {{{-11r}}} with {{{-2r-9r}}}. Remember, {{{-2}}} and {{{-9}}} add to {{{-11}}}. So this shows us that {{{-2r-9r=-11r}}}.



{{{r^2+highlight(-2r-9r)+18}}} Replace the second term {{{-11r}}} with {{{-2r-9r}}}.



{{{(r^2-2r)+(-9r+18)}}} Group the terms into two pairs.



{{{r(r-2)+(-9r+18)}}} Factor out the GCF {{{r}}} from the first group.



{{{r(r-2)-9(r-2)}}} 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.



{{{(r-9)(r-2)}}} Combine like terms. Or factor out the common term {{{r-2}}}



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So {{{-1(r^2-11r+18)}}} then factors further to {{{-(r-9)(r-2)}}}



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Answer:



So {{{-r^2+11r-18}}} completely factors to {{{-(r-9)(r-2)}}}.



In other words, {{{-r^2+11r-18=-(r-9)(r-2)}}}.



Note: you can check the answer by expanding {{{-(r-9)(r-2)}}} to get {{{-r^2+11r-18}}} or by graphing the original expression and the answer (the two graphs should be identical).
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