<PRE><B>Question 174465</B>
&lt;font size = 7 color = &quot;red&quot;&gt;&lt;b&gt;Edwin&#39;s &lt;u&gt;complete&lt;/u&gt; solution:&lt;/font&gt;

Simplify: {{{(x^5-y^5)/(x-y)}}}
&lt;pre&gt;&lt;font size = 4 color = &quot;indigo&quot;&gt;&lt;b&gt;
Most people learn the factorization of {{{x^N-y^N)}}}.

But if you forget you can divide that problem out using long division:

              x&lt;sup&gt;4&lt;/sup&gt;  +  x&lt;sup&gt;3&lt;/sup&gt;y  +  x&lt;sup&gt;2&lt;/sup&gt;y&lt;sup&gt;2&lt;/sup&gt; +  xy&lt;sup&gt;3&lt;/sup&gt; + y&lt;sup&gt;4&lt;/sup&gt;
      ---------------------------------------
x - y ) x&lt;sup&gt;5&lt;/sup&gt; + 0x&lt;sup&gt;4&lt;/sup&gt;y + 0x&lt;sup&gt;3&lt;/sup&gt;y&lt;sup&gt;2&lt;/sup&gt; + 0x&lt;sup&gt;2&lt;/sup&gt;y&lt;sup&gt;3&lt;/sup&gt; + 0xy&lt;sup&gt;4&lt;/sup&gt; - y&lt;sup&gt;5&lt;/sup&gt;
        x&lt;sup&gt;5&lt;/sup&gt; -  x&lt;sup&gt;4&lt;/sup&gt;y
        ---------
              x&lt;sup&gt;4&lt;/sup&gt;y + 0x&lt;sup&gt;3&lt;/sup&gt;y&lt;sup&gt;2&lt;/sup&gt;
              x&lt;sup&gt;4&lt;/sup&gt;y -  x&lt;sup&gt;3&lt;/sup&gt;y&lt;sup&gt;2&lt;/sup&gt;
              -----------
                     x&lt;sup&gt;3&lt;/sup&gt;y&lt;sup&gt;2&lt;/sup&gt; + 0x&lt;sup&gt;2&lt;/sup&gt;y&lt;sup&gt;3&lt;/sup&gt;
                     x&lt;sup&gt;3&lt;/sup&gt;y&lt;sup&gt;2&lt;/sup&gt; -  x&lt;sup&gt;2&lt;/sup&gt;y&lt;sup&gt;3&lt;/sup&gt;
                     ------------
                             x&lt;sup&gt;2&lt;/sup&gt;y&lt;sup&gt;3&lt;/sup&gt; + 0xy&lt;sup&gt;4&lt;/sup&gt;
                             x&lt;sup&gt;2&lt;/sup&gt;y&lt;sup&gt;3&lt;/sup&gt; -  xy&lt;sup&gt;4&lt;/sup&gt;
                             -----------
                                     xy&lt;sup&gt;4&lt;/sup&gt; - y&lt;sup&gt;5&lt;/sup&gt;
                                     xy&lt;sup&gt;4&lt;/sup&gt; - y&lt;sup&gt;5&lt;/sup&gt;
                                     --------
                                            0

So the answer is 

x&lt;sup&gt;4&lt;/sup&gt;  +  x&lt;sup&gt;3&lt;/sup&gt;y  +  x&lt;sup&gt;2&lt;/sup&gt;y&lt;sup&gt;2&lt;/sup&gt; +  xy&lt;sup&gt;3&lt;/sup&gt; + y&lt;sup&gt;4&lt;/sup&gt;


However if you memorize how to factor {{{x^N - y^N}}}, i.e.,

{{{matrix(1,14,

x^N - y^N, &quot;=&quot;, (x - y),&quot;(&quot;, x^(N-1)y, &quot;+&quot;, x^(N-2)y^2, &quot;+ ... +&quot;, x^2y^(N-2),&quot;+&quot;,xy^(N-1),&quot;+&quot;,y^N,&quot;)&quot; )}}}

then you will know how to factor {{{x^5-y^5}}}. Then to simplify
the expression

{{{(x^5-y^5)/(x-y)}}}

you will just factor the numerator according to the rule above:

{{{((x-y)(x^4+x^3y+x^2y^2+xy^3+y^4))/(x-y)}}}

{{{((cross(x-y))(x^4+x^3y+x^2y^2+xy^3+y^4))/(cross(x-y))}}}

{{{x^4+x^3y+x^2y^2+xy^3+y^4)}}}.

One thing to remember though, is that although that factorization
method holds for all values of N, it only factors COMPLETELY when
N is an ODD positive integer.  When N is EVEN, you should always 
begin by factoring it as the difference of two squares.  

Edwin&lt;/pre&gt;   </PRE>