<PRE><B>Question 248456</B>
{{{56-15z+z^2}}} Start with the given expression.



{{{z^2-15z+56}}} Rearrange the terms in descending degree.



Looking at the expression {{{z^2-15z+56}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{-15}}}, and the last term is {{{56}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{56}}} to get {{{(1)(56)=56}}}.



Now the question is: what two whole numbers multiply to {{{56}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{-15}}}?



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 {{{56}}} (the previous product).



Factors of {{{56}}}:

1,2,4,7,8,14,28,56

-1,-2,-4,-7,-8,-14,-28,-56



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{56}}}.

1*56 = 56
2*28 = 56
4*14 = 56
7*8 = 56
(-1)*(-56) = 56
(-2)*(-28) = 56
(-4)*(-14) = 56
(-7)*(-8) = 56


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{-15}}}:



&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;56&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+56=57&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;28&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;2+28=30&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;4&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;4+14=18&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;8&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;7+8=15&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;-56&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+(-56)=-57&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;-28&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-2+(-28)=-30&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-4&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;-4+(-14)=-18&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;-8&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-7+(-8)=-15&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{-7}}} and {{{-8}}} add to {{{-15}}} (the middle coefficient).



So the two numbers {{{-7}}} and {{{-8}}} both multiply to {{{56}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{-15}}}



Now replace the middle term {{{-15z}}} with {{{-7z-8z}}}. Remember, {{{-7}}} and {{{-8}}} add to {{{-15}}}. So this shows us that {{{-7z-8z=-15z}}}.



{{{z^2+highlight(-7z-8z)+56}}} Replace the second term {{{-15z}}} with {{{-7z-8z}}}.



{{{(z^2-7z)+(-8z+56)}}} Group the terms into two pairs.



{{{z(z-7)+(-8z+56)}}} Factor out the GCF {{{z}}} from the first group.



{{{z(z-7)-8(z-7)}}} Factor out {{{8}}} 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-8)(z-7)}}} Combine like terms. Or factor out the common term {{{z-7}}}



===============================================================



Answer:



So {{{56-15z+z^2}}} factors to {{{(z-8)(z-7)}}}.



In other words, {{{56-15z+z^2=(z-8)(z-7)}}}.



Note: you can check the answer by expanding {{{(z-8)(z-7)}}} to get {{{z^2-15z+56}}} or by graphing the original expression and the answer (the two graphs should be identical).
</PRE>