Binomial theorem

Binomial theorem

 We all know that a polynomial with two terms is called a Binomial.

 Now, Between these two terms, both may be variable or a combination of a variable and a constant number.

As for example, 3x-4y, 2x+6, ax+b, 2x² -3y², a+ x ,..etc.

Now,  the square or cube of the binomial (a+x) can be determined easily by the formulas or basic multiplication methods.

  But , it is very hard and labourious to find the n-th power(n is any positive integer) of the binomial (a+x) by basic multiplication. So, the multiplication method fails to find the value of (a+x)ⁿ.

 Then, how can we find the value of (a+x)ⁿ=?

  In this case algebra helps us. To solve these type of problems, algebra gives us a special formula. This formula is known as  Binomial theorem.

  Definition:

In algebra a general formula is used to expand a binomial with power n( any positive integer n) as a series of finite terms.This formula is called  binomial theorem.

 The binomial theorem was discovered by sir Issac Newton.

Statement of binomial theorem:

For any positive integer n and any real number a and x ; the expansion of the binomial (a+x) with power n, i.e, the expansion of (a+x)ⁿ is given by:

(a+x)ⁿ= aⁿ + C(n,1)×(aⁿ⁻¹)×x¹ + C(n,2)× (aⁿ⁻²)×x² +...+xⁿ= aⁿ+ n×aⁿ⁻¹×x + {n(n-1)/2! }×aⁿ⁻²×x² +...+xⁿ.

Here, C(n,1) , C(n,2),...C(n,n) are called the binomial coefficients.

 Note:

The number of terms in the expansion of (a+x)ⁿ are  always finite and equals to (n+1).

Pascal's triangle and binomial theorem:

In 1660 Pascal introduced an expansion of a binomial with power n.

He observed that,

  (a+x)⁰ = 1

  (a+x)¹= a¹+ x¹

  (a+x)² = a² + 2ax + x²

  (a+x)³ = a³ + 3a²x + 3ax² + x³

  (a+x)⁴ = a⁴ + 4a³x + 6a²x² +  4ax³ +x⁴

      and so on.

Pascal observed the special pattern or relation between the power(exponent) 'n' and binomial coefficients.

Power(exponent) of binomial     coefficients

            0                                                    1

            1                                                  1      1

            2                                              1      2     1

            3                                           1      3      3   1

     And so on.

Expansion of some important binomial expressions:

(a-x)ⁿ = aⁿ - C(n,1)×{aⁿ⁻¹ }x¹ + C(n,2)×{aⁿ⁻²} x² -...+(-1)ⁿ × xⁿ.

(1+x)ⁿ = 1+C(n,1)×x + C(n,2)×x² +...+xⁿ.

(1-x)ⁿ = 1 - C(n,1)×x + C(n,2)×x² -...+(-1)ⁿ ×xⁿ.

Note:

(1) The (r+1)-th term in the expansion of (a+x)ⁿ = t(r+1) ={C(n,r)×aⁿ⁻ʳ}×xʳ.

(2)The (r+1)-th term in the expansion of (a-x)ⁿ = t(r+1) = {C(n,r)×aⁿ⁻ʳ}×(-1)ʳ ×xʳ.

(3) If n is even the middle term of (a+x)ⁿ  will be the (n/2 +1)-th term of the expansion.

(4) If n is odd, there will be two middle terms and they will be {(n-1)/2 +1}-th term and {(n+1)/2 + 1}-th term.

(5) The sum of the all coefficients in the expansion of (a+x)ⁿ is equals to 2ⁿ.

If you find out any incorrect information or know anything more about this , please write it in the comment section!