Binomial Theorem: Maths Class 11 Chapter-8
Key Features of NCERT Material for Class 11 Maths Chapter 8 – Binomial Theorem
Quick revision notes
In Chapter 7 of Class 11 Maths: you must have learned about Permutations and Combinations. In chapter 8: Binomial Theorem of Class 11 Maths Chapter 8, you will learn about the Binomial Theorem.
Binomial Theorem for Positive Integer
nC0, nC1, nC2, … , nno are called binomial coefficients and
nCr = n! / r!(n – r)! for 0 ≤ r ≤ n.
Properties of Binomial Theorem for Positive Integer
- Total number of terms in the expansion of (x + a)n is (n + 1).
- The sum of the indices of x and a in each term is n.
- The above expansion is also true when x and a are complex numbers.
- The coefficient of terms equidistant from the beginning and the end are equal. These coefficients are known as the binomial coefficients and
nCr = nCn – r, r = 0,1,2,…,n.
- General term in the expansion of (x + c)n is given by Tr + 1 = nCrxn – r ar.
- The values of the binomial coefficients steadily increase to maximum and then steadily decrease .
Middle term in the expansion of (1 + x)n
- It n is even, then in the expansion of (x + a)n, the middle term is (n/2 + 1)th terms.
- If n is odd, then in the expansion of (x + a)n, the middle terms are (n + 1) / 2 th term and (n + 3) / 2 th term.
Greatest Coefficient
- If n is even, then in (x + a)n, the greatest coefficient is nCn / 2
- Ifn is odd, then in (x + a)n, the greatest coefficient is nCn – 1 / 2 or nCn + 1 / 2 both being equal.
Binomial Expression
An articulation comprising of two terms, associated by + or – sign is called binomial articulation.
Binomial Theorem
On the off chance that an and b are genuine numbers and n is a positive whole number, at that point
The overall term of (r + 1)th term in the articulation is given by
Tr+1 = nCr a r br
Some Important Observations from the Binomial Theorem
The absolute number of terms in the binomial expansion of (a + b)n is n + 1.
The sum of the records of an and b in each term is n.
The coefficient of terms equidistant from the earliest starting point and the end are equivalent. These coefficients are known as the binomial coefficient and
nCr = nCn-r, r = 0, 1, 2, 3,… , n
The values of the binomial coefficient consistently increment to a maximum and afterward consistently decline.
The coefficient of xr in the expansion of (1 + x)n is nCr.
In the binomial expansion (a + b)n, the rth term from the end is (n – r + 2)th term from the earliest starting point.
Middle Term in the Expansion of series (a + b)n
On the off chance that n is even, at that point in the expansion of (a + b)n, the middle term is
( + 1) th term.
On the off chance that n is odd, at that point in the expansion of (a + b)n, the middle terms are ()th term and ()th term.