We are very much familiar with the expansions like (a + b)² = a² + 2ab + b² and
(a + b)³ = a³ + 3a2b + 3ab² + b³ etc. Such expansions are called binomial expansions as there are two terms in the base of L.H.S. In this chapter we will learn general formula for a positive integral index, n were n  N, which is also known as Binomial Theorem for positive integral index.

Binomial Theorem for positive integral index:
Statement: If 'n' is a positive integer
then (x + a)ⁿ = ⁿC₀ xⁿ + ⁿC₁x^{n -1} a + _nC² x^{n -2} a² + ... + ⁿC_r x^{n -r} a^r + ... + ⁿC_n aⁿ.
this is known as Binomial Theorem and can be proved using Mathematical Induction.
Note: 1. No. of terms in the expansion is  of (x + a)ⁿ is (n + 1).
2. General Term of the expansion is  T_{r + 1} = ⁿC_r x_{n -r}. a^r
When Tr + 1 is expansion........... (r + 1)^th term in the
3. In the above expansion, ⁿC₀, ⁿC₁, ⁿC₂, .... ⁿC_n are called binominal coefficients,
which are also simply denoted as C₀, C₁, C₂, ..... C_n
4. In the expansion of (x + a)n, by replacing 'x' with 1 and 'a' with 'x' all get
(1 + x)ⁿ = C₀ + C₁x + C₂x² + ... + C_rx^r + .... C_nxⁿ. where C_r = ⁿC_r.
5. In the expansion of (x + a)n, by replacing 'a' with '-a' we get
(x - a)ⁿ  = ⁿC₀ - ⁿC₁x^{n -1} a + ⁿC₂x^{n -1} a² - ... + (-1)^rnC_r x^{n -r} a^r + ... + (-1)ⁿⁿC_n aⁿ
Here general term T _{r + 1} = (-1)^{r n}C_r x^{n - r}. a^r.
6. In the above expansion replacing 'x' with 1 and 'a' with x, we get
(1 - x)ⁿ = C₀ - C₁x + C₂x² - .... + (-1)ⁿC_nxⁿ.
were C_r= ⁿC_r.
Ex.1: Find the expansion of (2x + 3y)⁵.
(2x + 3y)⁵ = 5C₀(2x)⁵ + 5C₁(2x)⁴(3y) + 5C₂. (2x)³. (3y)²
+ 5C₃. (2x)⁴ (3y)³ + 5C₄. (2x). (3y)⁴ + 5C₅. (3y)⁵
= 25. x⁵ + 5 × 2⁴ × 3 × x⁴y + 10 × 2³ × 3². x³y² + 10 × 2² × 3³ x²y³ + 5 × 2
× 3⁴ xy⁴ + 3⁵y⁵
= 32x⁵ + 240 x⁴y + 720x³y² + 1080x²y³ + 810xy⁴ + 243y⁵

Problems for Practice:-
1. Find the 4^th term in the expansion of (3a - 4b)⁸


                                                      = T₆

                                                = T_{13+ 1}
                                               = ²⁵C₁₃ .(3x)^{25 - 13} . (5y)¹³
                                               = ²⁵C₁₃ .(3x)¹² . (5y)¹³

Numerically greatest terms:
         In the expansion of (1 + x)n, the value of any term depends on the values of 'n' and 'x'. If 'x' is negative alternates terms become negative and when we considered the absolute value (numerical value, without considering the sign) of all the terms, there can be one (or two) numerically greatest terms depending on the value of
           
         If the Value is non - integer in the term of p + f where 0 < f < 1 then ... T_{p +1} is the numerically greatest term.
        But if the value is integer say 'p' then T_p and T_{p + 1} are two numerically greatest terms and they are equal
          

Eg: Find the numerically greatest term in the expansion of (5x - 6y)¹⁴

                                           
Hence T_p and T_{p + 1} are numerically greatest terms where p = 9
 T₉ and T₁₀ are numerically greatest terms. 

   

Problems for Practice:
 1. S.T. 3. C₀+ 7. C₁ + 11.C₂ + ... + (4n+3) C_n = (2n+3) 2ⁿ

 3. S.T C₀² - C₁² + C₂² - C₃² + ... + (-1)ⁿ C_n² = {0 if n is odd (-1)^n/2. ⁿC_n/2 if n is even
 4. If (1 + 3x - 2x²)¹⁰ = a₀ + a₁x+a_2x2+... + a₂₀ x²⁰, then  S.T.
(1) a₀ + a₁ + a₂ +... + a₂₀ = 2¹⁰ (2) a₀ - a₁ + a₂ +..... + a₂₀ = 2²⁰

Binomial Theorem for rational index:
     So for we have learnt binomial expansion for positive integral index. Now we learn a general formula (without Proof, as it is beyond the scope of this standard) for any rational index, as follows.