Lesson Factoring the binomials x^n-a^n

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


Probably, you are already familiar with the shortcut multiplication formula for the difference of squares

a%5E2-b%5E2 = %28a%2Bb%29%2A%28a%2Bb%29                           (1)

(see the lesson The difference of squares formula under the current topic in this site).
There is similar shortcut multiplication formula for the difference of cubes

a%5E3-b%5E3 = %28a-b%29%2A%28a%5E2+%2B+a%2Ab+%2B+b%5E2%29              (2)

(see the lesson The difference of cubes formula under the current topic in this site).
Similar shortcut multiplication formula is valid for the difference of fourth degrees too

a%5E4-b%5E4 = %28a-b%29%2A%28a%5E3+%2B+a%5E2%2Ab+%2B+a%2Ab%5E2+%2B+b%5E3%29   (3)

as it was shown in the lesson The difference of squares formula in the Note 1 after the Example 7.

The goal of this lesson is to prove the general formula

a%5En+-b%5En  =  .     (4)

This formula is valid for all real numbers a and b and for any integer index n greater than or equal to 2.
It generalizes the above formulas (1), (2) and (3). To prove it let us perform direct calculations:

=      (Now apply the distributive property of addition and multiplication operations.)

               (One of two pairs of brackets is opened and the result is presented in two lines.)
- =           (Now open the resting brackets and apply the commutative property of multiplication operation)

                       (Note that the co-indexed terms in the upper and the lover lines have the opposite signs.)
     =              (Therefore, you can cancel these like terms.)

              (The crossing lines show canceling the like terms.)

     = a%5En+-+b%5En.         (You got the final result.)

The formula is proved.
The distributive and commutative properties of addition and multiplication operations were used in the proof.

Formula (4) is valid not only to numbers. It is valid for the polynomials with real coefficients too. For example,

x%5En+-a%5En  =  .     (5)

for any real number a. You can check validity of the formula (5) directly by performing all relevant calculations: opening the brackets, multiplying the terms and canceling
the like terms. You will get the same result because addition and multiplication operations for the polynomials with real coefficients have the same distributive and
commutative properties as for real numbers.

Formula (5) provides factorization the binomial x%5En+-a%5En into the product of the linear binomial x+-a and the polynomial .
This explicit factorization shows that the binomial x%5En+-a%5En is divided by the linear binomial x+-a.

Note the important special case of the formula (5) when the value of a is equal to 1:

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.                                     (6)

Again, formula (6) shows that the binomial x%5En+-1 is divided by the linear binomial x+-1. The quotient 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.
Let us write it explicitly:

%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.                                               (7)

Formula (7) has an important application. It relates to the sum of a geometric progression.
The geometric progression is the sequence of real numbers such that the ratio of the next term to the preceding one is a constant value for the given progression.
This constant value is called the ratio of the progression.
So, if the sequence of real numbers a%5B1%5D, a%5B2%5D, a%5B3%5D, ..., a%5Bn%5D is the geometric progression, then the ratio a%5Bk%5D%2Fa%5Bk-1%5D is a constant value for any two consequent terms a%5Bk-1%5D, a%5Bk%5D.
If we denote this constant ratio as q, then a%5Bk%5D%2Fa%5Bk-1%5D=q, or a%5Bk%5D+=+a%5Bk-1%5D%2Aq.
Thus, if the first term of a geometric progression a%5B1%5D is equal to a,   a%5B1%5D=a, then the second term a%5B2%5D=a%2Aq, the third term a%5B3%5D=a%5B2%5D%2Aq=a%2Aq%5E2, the fourth term a%5B4%5D=a%5B3%5D%2Aq=a%2Aq%5E3, and, generally, the k-th term a%5Bk%5D=a%2Aq%5E%28k-1%29.

Hence, the sum of the first n terms of the geometric progression with the first term a and the ratio q is equal to

a+%2B+a%2Aq+%2B+a%2Aq%5E2+%2B+a%2Aq%5E3+%2B+ellipsis+%2B+a%2Aq%5E%28n-1%29  =  a%2A%281+%2B+q+%2B+q%5E2+%2B+q%5E3+%2B+ellipsis+%2B+q%5E%28n-1%29%29.

According to the formula (7),

1+%2B+q+%2B+q%5E2+%2B+q%5E3+%2B+ellipsis+%2B+q%5E%28n-1%29  =  %28q%5En-1%29%2F%28q-1%29.

Therefore, the sum of the first n terms of the geometric progression with the first term a and the ratio q is equal to

a+%2B+a%2Aq+%2B+a%2Aq%5E2+%2B+ellipsis+%2B+a%2Aq%5E%28n-1%29  =  %28a%2Aq%5En+-+a%29%2F%28q-1%29.                                                       (8)

Summary


1. The formula is valid

      a%5En+-b%5En  =  

      for all real numbers a and b and for any integer index n greater than or equal to 2.


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.


To complete this lesson, we present examples that show how everything works.

Example 1

Factor the binomial   x%5E5+-+32.

Solution
Note that   x%5E5+-+32 = x%5E5+-+2%5E5.
In accordance with the formula (5),   x%5E5+-+2%5E5%29 = %28x-2%29%2A%28x%5E4+%2B+2%2Ax%5E3+%2B+4%2Ax%5E2+%2B+8%2Ax+%2B16%29%29.

So, the answer is   x%5E5+-+32 = %28x-2%29%2A%28x%5E4+%2B+2%2Ax%5E3+%2B+4%2Ax%5E2+%2B+8%2Ax+%2B16%29%29.


Example 2

Simplify the rational expression 1%2F%281%2Bx%2Bx%5E2%2Bx%5E3%2Bx%5E4%2Bx%5E5%2Bx%5E6%2Bx%5E7%29.

Solution
Multiply the numerator and the denominator of this rational fraction by x-1 and apply the formula (6). You get

1%2F%281%2Bx%2Bx%5E2%2Bx%5E3%2Bx%5E4%2Bx%5E5%2Bx%5E6%2Bx%5E7%29 = %28x-1%29%2F%28%28x-1%29%2A%281%2Bx%2Bx%5E2%2Bx%5E3%2Bx%5E4%2Bx%5E5%2Bx%5E6%2Bx%5E7%29%29 = %28x-1%29%2F%28x%5E8-1%29.


Example 3

Rationalize the fraction by making its denominator free of roots.

Solution
Multiply the numerator and the denominator of the fraction by %28root%287%2C3%29%29-1 and apply the formula (6). You get

= = %28root%287%2C3%29-1%29%2F%28%28root%287%2C3%29%29%5E7-1%29 = %28root%287%2C3%29-1%29%2F%283-1%29 = %28root%287%2C3%29-1%29%2F2.


Example 4

Calculate the sum of the geometric progression 1%2F2%2B1%2F4%2B1%2F8%2B1%2F16%2B1%2F32%2B1%2F64%2B1%2F128%2B1%2F256%29.

Solution
This is the geometric progression with the first term 1%2F2 and the ratio 1%2F2. We have to calculate the sum of the first 8 terms of this progression.
In accordance with the formula (8), this sum is equal to %281%2F2%29%2A%28%281%2F2%29%5E8-1%29%2F%281%2F2-1%29 = %282%5E8-1%29%2F2%5E8 = %28256-1%29%2F256 = 255%2F256 = 1+-+1%2F256.

For similar lessons see Factoring the binomials   x%5En%2Ba%5En for odd degrees   under the current topic in this site.

For divisibility and factoring of a general polynomial  f%28x%29  by a binomial  x-a  see the lesson  Divisibility of polynomial  f%28x%29  by binomial  x-a  under the current topic in this site.

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