<PRE><B>Question 168141</B>


Looking at the expression {{{2x^2-4x+2}}}, we can see that the first coefficient is {{{2}}}, the second coefficient is {{{-4}}}, and the last term is {{{2}}}.



Now multiply the first coefficient {{{2}}} by the last term {{{2}}} to get {{{(2)(2)=4}}}.



Now the question is: what two whole numbers multiply to {{{4}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{-4}}}?



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 {{{4}}} (the previous product).



Factors of {{{4}}}:

1,2,4

-1,-2,-4



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{4}}}.

1*4
2*2
(-1)*(-4)
(-2)*(-2)


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{-4}}}:



&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;4&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+4=5&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;2&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;2+2=4&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;-4&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+(-4)=-5&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;-2&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-2+(-2)=-4&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{-2}}} and {{{-2}}} add to {{{-4}}} (the middle coefficient).



So the two numbers {{{-2}}} and {{{-2}}} both multiply to {{{4}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{-4}}}



Now replace the middle term {{{-4x}}} with {{{-2x-2x}}}. Remember, {{{-2}}} and {{{-2}}} add to {{{-4}}}. So this shows us that {{{-2x-2x=-4x}}}.



{{{2x^2+highlight(-2x-2x)+2}}} Replace the second term {{{-4x}}} with {{{-2x-2x}}}.



{{{(2x^2-2x)+(-2x+2)}}} Group the terms into two pairs.



{{{2x(x-1)+(-2x+2)}}} Factor out the GCF {{{2x}}} from the first group.



{{{2x(x-1)-2(x-1)}}} Factor out {{{2}}} 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.



{{{(2x-2)(x-1)}}} Combine like terms. Or factor out the common term {{{x-1}}}


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Answer:



So {{{2x^2-4x+2}}} factors to {{{(2x-2)(x-1)}}}.



Note: you can check the answer by FOILing {{{(2x-2)(x-1)}}} to get {{{2x^2-4x+2}}} or by graphing the original expression and the answer (the two graphs should be identical).
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