SOLUTION: <pre> Can you please help me factor this expression? a<sup>5</sup> + b<sup>5</sup>. Thank you so much for your help.</pre>

Algebra ->  Equations -> SOLUTION: <pre> Can you please help me factor this expression? a<sup>5</sup> + b<sup>5</sup>. Thank you so much for your help.</pre>      Log On


   

Question 86392: 
Can you please help me factor this expression?
a5 + b5.
Thank you so much for your help.
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!

Can you please help me factor this expression?
a5 plus b5.
Thank you so much for your help.

Divide a5 + b5 by a+b.

To do this you have to write a5 + b5 as

a5 + 0a4b + 0a3b2 + 0a2b3 + 0ab4 + b5

     _______a4  -  a3b  +  a2b2 -  ab3 + b4
a + b)a5 + 0a4b + 0a3b2 + 0a2b3 + 0ab4 + b5
      a5 +  a4b
           -a4b + 0a3b2
           -a4b -  a3b2
                   a3b2 + 0a2b3
                   a3b2 +  a2b3
                          -a2b3 + 0ab4
                          -a2b3 -  ab4
                                   ab4 + b5
                                   ab4 + b5
                                          0

That means the factorization is:

(a + b)(a4 - a3b + a2b2 - ab3 + b4) 

Instead of dividing it out each time, you can just learn the
pattern for odd n:

an - bn factors as (a - b)(sum of all terms aPbQ where P+Q = n)

an + bn factors as (a + b)(sum with alternating signs of all terms aPbQ where P+Q=n)

Edwin