<PRE><B>Question 600724</B>
It looks like you want to factor here.




Looking at the expression {{{2x^2-21x+40}}}, we can see that the first coefficient is {{{2}}}, the second coefficient is {{{-21}}}, and the last term is {{{40}}}.



Now multiply the first coefficient {{{2}}} by the last term {{{40}}} to get {{{(2)(40)=80}}}.



Now the question is: what two whole numbers multiply to {{{80}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{-21}}}?



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 {{{80}}} (the previous product).



Factors of {{{80}}}:

1,2,4,5,8,10,16,20,40,80

-1,-2,-4,-5,-8,-10,-16,-20,-40,-80



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{80}}}.

1*80 = 80
2*40 = 80
4*20 = 80
5*16 = 80
8*10 = 80
(-1)*(-80) = 80
(-2)*(-40) = 80
(-4)*(-20) = 80
(-5)*(-16) = 80
(-8)*(-10) = 80


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{-21}}}:



&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;80&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+80=81&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;40&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;2+40=42&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;4&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;20&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;4+20=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;5&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;16&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;5+16=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;8&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;10&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;8+10=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;-1&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-80&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+(-80)=-81&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;-40&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-2+(-40)=-42&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-4&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-20&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-4+(-20)=-24&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-5&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-16&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-5+(-16)=-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;-8&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-10&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-8+(-10)=-18&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{-5}}} and {{{-16}}} add to {{{-21}}} (the middle coefficient).



So the two numbers {{{-5}}} and {{{-16}}} both multiply to {{{80}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{-21}}}



Now replace the middle term {{{-21x}}} with {{{-5x-16x}}}. Remember, {{{-5}}} and {{{-16}}} add to {{{-21}}}. So this shows us that {{{-5x-16x=-21x}}}.



{{{2x^2+highlight(-5x-16x)+40}}} Replace the second term {{{-21x}}} with {{{-5x-16x}}}.



{{{(2x^2-5x)+(-16x+40)}}} Group the terms into two pairs.



{{{x(2x-5)+(-16x+40)}}} Factor out the GCF {{{x}}} from the first group.



{{{x(2x-5)-8(2x-5)}}} Factor out {{{8}}} 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.



{{{(x-8)(2x-5)}}} Combine like terms. Or factor out the common term {{{2x-5}}}



===============================================================



Answer:



So {{{2x^2-21x+40}}} factors to {{{(x-8)(2x-5)}}}.



In other words, {{{2x^2-21x+40=(x-8)(2x-5)}}}.



Note: you can check the answer by expanding {{{(x-8)(2x-5)}}} to get {{{2x^2-21x+40}}} or by graphing the original expression and the answer (the two graphs should be identical).
</PRE>