<PRE><B>Question 191755</B>


Looking at the expression {{{x^2+2x-15}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{2}}}, and the last term is {{{-15}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{-15}}} to get {{{(1)(-15)=-15}}}.



Now the question is: what two whole numbers multiply to {{{-15}}} (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 {{{-15}}} (the previous product).



Factors of {{{-15}}}:

1,3,5,15

-1,-3,-5,-15



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{-15}}}.

1*(-15)
3*(-5)
(-1)*(15)
(-3)*(5)


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;-15&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+(-15)=-14&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&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;-5&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;3+(-5)=-2&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;15&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+15=14&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&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;5&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-3+5=2&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{-3}}} and {{{5}}} add to {{{2}}} (the middle coefficient).



So the two numbers {{{-3}}} and {{{5}}} both multiply to {{{-15}}} &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 {{{-3x+5x}}}. Remember, {{{-3}}} and {{{5}}} add to {{{2}}}. So this shows us that {{{-3x+5x=2x}}}.



{{{x^2+highlight(-3x+5x)-15}}} Replace the second term {{{2x}}} with {{{-3x+5x}}}.



{{{(x^2-3x)+(5x-15)}}} Group the terms into two pairs.



{{{x(x-3)+(5x-15)}}} Factor out the GCF {{{x}}} from the first group.



{{{x(x-3)+5(x-3)}}} 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.



{{{(x+5)(x-3)}}} Combine like terms. Or factor out the common term {{{x-3}}}


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Answer:



So {{{x^2+2x-15}}} factors to {{{(x+5)(x-3)}}}.



Note: you can check the answer by FOILing {{{(x+5)(x-3)}}} to get {{{x^2+2x-15}}} or by graphing the original expression and the answer (the two graphs should be identical).
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