<PRE><B>Question 610032</B>


{{{40x^3+34x^2-20x}}} Start with the given expression.



{{{2x(20x^2+17x-10)}}} Factor out the GCF {{{2x}}}.



Now let&#39;s try to factor the inner expression {{{20x^2+17x-10}}}



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Looking at the expression {{{20x^2+17x-10}}}, we can see that the first coefficient is {{{20}}}, the second coefficient is {{{17}}}, and the last term is {{{-10}}}.



Now multiply the first coefficient {{{20}}} by the last term {{{-10}}} to get {{{(20)(-10)=-200}}}.



Now the question is: what two whole numbers multiply to {{{-200}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{17}}}?



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 {{{-200}}} (the previous product).



Factors of {{{-200}}}:

1,2,4,5,8,10,20,25,40,50,100,200

-1,-2,-4,-5,-8,-10,-20,-25,-40,-50,-100,-200



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{-200}}}.

1*(-200) = -200
2*(-100) = -200
4*(-50) = -200
5*(-40) = -200
8*(-25) = -200
10*(-20) = -200
(-1)*(200) = -200
(-2)*(100) = -200
(-4)*(50) = -200
(-5)*(40) = -200
(-8)*(25) = -200
(-10)*(20) = -200


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{17}}}:



&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;-200&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+(-200)=-199&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;-100&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;2+(-100)=-98&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;-50&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;4+(-50)=-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;5&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;5+(-40)=-35&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;-25&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;8+(-25)=-17&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&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;-20&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;10+(-20)=-10&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;200&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+200=199&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;100&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-2+100=98&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;50&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-4+50=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;-5&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;-5+40=35&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-8&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;25&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-8+25=17&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&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;20&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-10+20=10&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{-8}}} and {{{25}}} add to {{{17}}} (the middle coefficient).



So the two numbers {{{-8}}} and {{{25}}} both multiply to {{{-200}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{17}}}



Now replace the middle term {{{17x}}} with {{{-8x+25x}}}. Remember, {{{-8}}} and {{{25}}} add to {{{17}}}. So this shows us that {{{-8x+25x=17x}}}.



{{{20x^2+highlight(-8x+25x)-10}}} Replace the second term {{{17x}}} with {{{-8x+25x}}}.



{{{(20x^2-8x)+(25x-10)}}} Group the terms into two pairs.



{{{4x(5x-2)+(25x-10)}}} Factor out the GCF {{{4x}}} from the first group.



{{{4x(5x-2)+5(5x-2)}}} Factor out {{{5}}} 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.



{{{(4x+5)(5x-2)}}} Combine like terms. Or factor out the common term {{{5x-2}}}



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So {{{2x(20x^2+17x-10)}}} then factors further to {{{2x(4x+5)(5x-2)}}}



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Answer:



So {{{40x^3+34x^2-20x}}} completely factors to {{{2x(4x+5)(5x-2)}}}.



In other words, {{{40x^3+34x^2-20x=2x(4x+5)(5x-2)}}}.



Note: you can check the answer by expanding {{{2x(4x+5)(5x-2)}}} to get {{{40x^3+34x^2-20x}}} or by graphing the original expression and the answer (the two graphs should be identical).


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Jim
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