<PRE><B>Question 200486</B>

Looking at the expression {{{z^2-5z-14}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{-5}}}, and the last term is {{{-14}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{-14}}} to get {{{(1)(-14)=-14}}}.



Now the question is: what two whole numbers multiply to {{{-14}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{-5}}}?



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 {{{-14}}} (the previous product).



Factors of {{{-14}}}:

1,2,7,14

-1,-2,-7,-14



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{-14}}}.

1*(-14)
2*(-7)
(-1)*(14)
(-2)*(7)


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{-5}}}:



&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;-14&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+(-14)=-13&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;-7&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;2+(-7)=-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;-1&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;14&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+14=13&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;7&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-2+7=5&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{2}}} and {{{-7}}} add to {{{-5}}} (the middle coefficient).



So the two numbers {{{2}}} and {{{-7}}} both multiply to {{{-14}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{-5}}}



Now replace the middle term {{{-5z}}} with {{{2z-7z}}}. Remember, {{{2}}} and {{{-7}}} add to {{{-5}}}. So this shows us that {{{2z-7z=-5z}}}.



{{{z^2+highlight(2z-7z)-14}}} Replace the second term {{{-5z}}} with {{{2z-7z}}}.



{{{(z^2+2z)+(-7z-14)}}} Group the terms into two pairs.



{{{z(z+2)+(-7z-14)}}} Factor out the GCF {{{z}}} from the first group.



{{{z(z+2)-7(z+2)}}} Factor out {{{7}}} 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.



{{{(z-7)(z+2)}}} Combine like terms. Or factor out the common term {{{z+2}}}


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Answer:



So {{{z^2-5z-14}}} factors to {{{(z-7)(z+2)}}}.



Note: you can check the answer by FOILing {{{(z-7)(z+2)}}} to get {{{z^2-5z-14}}} or by graphing the original expression and the answer (the two graphs should be identical).
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