<PRE><B>Question 214534</B>

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Looking at the expression {{{x^2+2x-3}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{2}}}, and the last term is {{{-3}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{-3}}} to get {{{(1)(-3)=-3}}}.



Now the question is: what two whole numbers multiply to {{{-3}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{2}}}?



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 {{{-3}}} (the previous product).



Factors of {{{-3}}}:

1,3

-1,-3



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{-3}}}.

1*(-3) = -3
(-1)*(3) = -3


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{2}}}:



&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;-3&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+(-3)=-2&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;3&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-1+3=2&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{-1}}} and {{{3}}} add to {{{2}}} (the middle coefficient).



So the two numbers {{{-1}}} and {{{3}}} both multiply to {{{-3}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{2}}}



Now replace the middle term {{{2x}}} with {{{-x+3x}}}. Remember, {{{-1}}} and {{{3}}} add to {{{2}}}. So this shows us that {{{-x+3x=2x}}}.



{{{x^2+highlight(-x+3x)-3}}} Replace the second term {{{2x}}} with {{{-x+3x}}}.



{{{(x^2-x)+(3x-3)}}} Group the terms into two pairs.



{{{x(x-1)+(3x-3)}}} Factor out the GCF {{{x}}} from the first group.



{{{x(x-1)+3(x-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)(x-1)}}} Combine like terms. Or factor out the common term {{{x-1}}}



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Answer:



So {{{x^2+2x-3}}} factors to {{{(x+3)(x-1)}}}.



In other words, {{{x^2+2x-3=(x+3)(x-1)}}}.



Note: you can check the answer by expanding {{{(x+3)(x-1)}}} to get {{{x^2+2x-3}}} or by graphing the original expression and the answer (the two graphs should be identical).


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