Lesson OVERVIEW of Factoring the binomials x^n-a^n and x^n+a^n

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Overview of Factoring the binomials x%5En-a%5En and x%5En%2Ba%5En


This lesson is the overview of factoring the binomials   x%5En-a%5En   and   x%5En%2Ba%5En.
It is based on the two preceding lessons Factoring the binomials   x%5En-a%5En   and Factoring the binomials   x%5En%2Ba%5En for odd degrees   that are under the current topic in this site.
For all the proofs, examples, motivations and other details consult these two lessons.


Factoring the binomials x%5En-a%5En


1. For all real numbers a and b and for any integer index n greater than or equal to 2 the formula is valid

      a%5En+-b%5En  =  .


2. For any integer index n greater than or equal to 2 and for any real number a the binomial   x%5En+-a%5En   is divided by the linear binomial   x+-a.
      The formula is valid

      x%5En+-a%5En  =  .

      This formula is factoring the binomial   x%5En+-a%5En   into the product of the linear binomial   x+-a and the polynomial   .
      The quotient of division the binomial   x%5En+-a%5En   by the binomial   x+-a   is the polynomial   .


3. For any integer index n greater than or equal to 2 the binomial   x%5En-1   is divided by the linear binomial   x-1.
      The formula is valid

      x%5En-1  =  %28x-1%29%2A%28x%5E%28n-1%29+%2B+x%5E%28n-2%29+%2B+x%5E%28n-3%29+%2B+ellipsis+%2B+x+%2B+1%29.

      This formula is factoring the binomial   x%5En-1   into the product of the linear binomial   x-1 and the polynomial   x%5E%28n-1%29+%2B+x%5E%28n-2%29+%2B+x%5E%28n-3%29+%2B+ellipsis+%2B+x+%2B+1%29.
      The quotient of division the binomial   x%5En-1   by the binomial   x-1   is the polynomial   x%5E%28n-1%29+%2B+x%5E%28n-2%29+%2B+x%5E%28n-3%29+%2B+ellipsis+%2B+x+%2B+1%29:
      %28x%5En+-1%29%2F%28x-1%29  =  x%5E%28n-1%29+%2B+x%5E%28n-2%29+%2B+x%5E%28n-3%29+%2B+ellipsis+%2B+x+%2B+1.


4. The sum of the first n terms of the geometric progression   a, a%2Aq, a%2Aq%5E2, ..., a%2Aq%5E%28n-1%29    is equal to
      S%5Bn%5D  =  a+%2B+a%2Aq+%2B+a%2Aq%5E2+%2B+ellipsis+%2B+aq%5E%28n-1%29  =  %28a%2Aq%5En+-+a%29%2F%28q-1%29.

Factoring the binomials x%5En%2Ba%5En


1. The formula is valid

      a%5En+%2Bb%5En  =  

      for all real numbers a and b and for odd integer index n greater than or equal to 3.


2. For odd integer index n greater than or equal to 3 and for any real number a the binomial   x%5En+%2Ba%5En   is divided by the linear binomial   x+%2Ba.
      The formula is valid

      x%5En+%2Ba%5En  =  .

      This formula is factoring the binomial   x%5En+%2Ba%5En   into the product of the linear binomial   x+%2Ba and the polynomial   .
      The quotient of division the binomial   x%5En+%2Ba%5En   by the binomial   x+%2Ba   is the polynomial   .


3. For odd integer index n greater than or equal to 3 the binomial   x%5En%2B1   is divided by the linear binomial   x%2B1.
      The formula is valid

      x%5En%2B1  =  %28x%2B1%29%2A%28x%5E%28n-1%29+-+x%5E%28n-2%29+%2B+x%5E%28n-3%29+%2B+ellipsis+-+x+%2B+1%29.

      This formula is factoring the binomial   x%5En%2B1   into the product of the linear binomial   x%2B1 and the polynomial   x%5E%28n-1%29+-+x%5E%28n-2%29+%2B+x%5E%28n-3%29+-+ellipsis+-+x+%2B+1%29.
      The quotient of division the binomial   x%5En%2B1   by the binomial   x%2B1   is the polynomial   x%5E%28n-1%29+-+x%5E%28n-2%29+%2B+x%5E%28n-3%29+-+ellipsis+-+x+%2B+1%29:
      %28x%5En+%2B1%29%2F%28x%2B1%29  =  x%5E%28n-1%29+-+x%5E%28n-2%29+%2B+x%5E%28n-3%29+-+ellipsis+-+x+%2B+1.


4. The binomial   x%5En%2Ba%5En   is not divisible by the binomial   x%2Ba   for even integer index   n.

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