Binomial Theorem Class 11 Notes Maths Chapter 8

CBSE Class 11 Maths Notes Chapter 8 Binomial Theorem

Binomial Expression
An expression consisting of two terms, connected by + or – sign is called binomial expression.

Binomial Theorem
If a and b are real numbers and n is a positive integer, then

The general term of (r + 1) ^th term in the expression is given by
T _r+1 = ⁿ C _r a ^n-r b ^r

Some Important Observations from the Binomial Theorem
The total number of terms in the binomial expansion of (a + b) ⁿ is n + 1.

The sum of the indices of a and b in each term is n.

The coefficient of terms equidistant from the beginning and the end are equal. These coefficients are known as the binomial coefficient and
ⁿ C _r = ⁿ C _n-r , r = 0, 1, 2, 3,…, n

The values of the binomial coefficient steadily increase to a maximum and then steadily decrease.

The coefficient of x ^r in the expansion of (1 + x) ⁿ is ⁿ C _r .

In the binomial expansion (a + b) ⁿ , the r ^th term from the end is (n – r + 2) ^th term from the beginning.

Middle Term in the Expansion of (a + b) ⁿ
If n is even, then in the expansion of (a + b) ⁿ , the middle term is (\(\frac { n }{ 2 }\) + 1) th term.

If n is odd, then in the expansion of (a + b) ⁿ , the middle terms are (\(\frac { n+1 }{ 2 }\))th term and (\(\frac { n+3 }{ 2 }\))th term.

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