<PRE><B>Question 790014</B>
I&#39;m assuming you want to factor.




Looking at the expression {{{5z^2-17z+14}}}, we can see that the first coefficient is {{{5}}}, the second coefficient is {{{-17}}}, and the last term is {{{14}}}.



Now multiply the first coefficient {{{5}}} by the last term {{{14}}} to get {{{(5)(14)=70}}}.



Now the question is: what two whole numbers multiply to {{{70}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{-17}}}?



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 {{{70}}} (the previous product).



Factors of {{{70}}}:

1,2,5,7,10,14,35,70

-1,-2,-5,-7,-10,-14,-35,-70



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{70}}}.

1*70 = 70
2*35 = 70
5*14 = 70
7*10 = 70
(-1)*(-70) = 70
(-2)*(-35) = 70
(-5)*(-14) = 70
(-7)*(-10) = 70


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{-17}}}:



&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;70&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+70=71&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;35&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;2+35=37&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&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;14&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;5+14=19&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&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;10&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;7+10=17&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;-70&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+(-70)=-71&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;-35&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-2+(-35)=-37&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&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;-14&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-5+(-14)=-19&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&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;-10&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-7+(-10)=-17&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{-7}}} and {{{-10}}} add to {{{-17}}} (the middle coefficient).



So the two numbers {{{-7}}} and {{{-10}}} both multiply to {{{70}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{-17}}}



Now replace the middle term {{{-17z}}} with {{{-7z-10z}}}. Remember, {{{-7}}} and {{{-10}}} add to {{{-17}}}. So this shows us that {{{-7z-10z=-17z}}}.



{{{5z^2+highlight(-7z-10z)+14}}} Replace the second term {{{-17z}}} with {{{-7z-10z}}}.



{{{(5z^2-7z)+(-10z+14)}}} Group the terms into two pairs.



{{{z(5z-7)+(-10z+14)}}} Factor out the GCF {{{z}}} from the first group.



{{{z(5z-7)-2(5z-7)}}} 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.



{{{(z-2)(5z-7)}}} Combine like terms. Or factor out the common term {{{5z-7}}}



===============================================================



Answer:



So {{{5z^2-17z+14}}} factors to {{{(z-2)(5z-7)}}}.



In other words, {{{5z^2-17z+14=(z-2)(5z-7)}}}.



Note: you can check the answer by expanding {{{(z-2)(5z-7)}}} to get {{{5z^2-17z+14}}} or by graphing the original expression and the answer (the two graphs should be identical).
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