<PRE><B>Question 185548</B>


Looking at the expression {{{2x^2-7x+3}}}, we can see that the first coefficient is {{{2}}}, the second coefficient is {{{-7}}}, and the last term is {{{3}}}.



Now multiply the first coefficient {{{2}}} by the last term {{{3}}} to get {{{(2)(3)=6}}}.



Now the question is: what two whole numbers multiply to {{{6}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{-7}}}?



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 {{{6}}} (the previous product).



Factors of {{{6}}}:

1,2,3,6

-1,-2,-3,-6



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{6}}}.

1*6
2*3
(-1)*(-6)
(-2)*(-3)


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{-7}}}:



&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;6&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+6=7&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;3&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;2+3=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;-1&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-6&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-1+(-6)=-7&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;-3&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-2+(-3)=-5&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{-1}}} and {{{-6}}} add to {{{-7}}} (the middle coefficient).



So the two numbers {{{-1}}} and {{{-6}}} both multiply to {{{6}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{-7}}}



Now replace the middle term {{{-7x}}} with {{{-x-6x}}}. Remember, {{{-1}}} and {{{-6}}} add to {{{-7}}}. So this shows us that {{{-x-6x=-7x}}}.



{{{2x^2+highlight(-x-6x)+3}}} Replace the second term {{{-7x}}} with {{{-x-6x}}}.



{{{(2x^2-x)+(-6x+3)}}} Group the terms into two pairs.



{{{x(2x-1)+(-6x+3)}}} Factor out the GCF {{{x}}} from the first group.



{{{x(2x-1)-3(2x-1)}}} Factor out {{{3}}} 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-3)(2x-1)}}} Combine like terms. Or factor out the common term {{{2x-1}}}


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Answer:



So {{{2x^2-7x+3}}} factors to {{{(x-3)(2x-1)}}}.



So {{{x-3}}} and {{{2x-1}}} are factors of {{{2x^2-7x+3}}}</PRE>