According to the CBSE Syllabus 2023-24, this chapter has been renumbered as Chapter 7.
The binomial expression is an expression comprising two terms connected by the -ve or +ve sign. Equations like x + a, 2x – 3y,
Binomial Expression
[p + q]n = [ nC0 × pn ] + [ nC1 × (pn – 1) × q ] + [ nC2 × (pn – 2) × q2 ] + [ nC3 × (pn – 3 )× q3 ] + . . . . . . . . . . . . . + [ nCn – 1 × p × (qn – 1) ] + [ nCn × qn ]. Where p and q are real numbers, and n is a positive integer.\(\begin{array}{l}\Rightarrow\end{array} \) Binomial Coefficient
The coefficients nC0, nC1, nC2 . . . . . . . . . nCn occurring in the Binomial expression are called Binomial coefficients. Given below are some conclusions that can be derived using the Binomial Theorem.
(i) [x + y]n = [ nC0 × (xn) ] + [ nC1 × (xn – 1) × y ] + [ nC2 × (xn – 2) × y2 ] + [ nC3 × (xn – 3) × y3 ] + . . . . . . . . . . . . . . . . . . + [ nCn × yn ]
(ii) [x – y]n = [ nC0 × (xn) ] – [ nC1 × (xn – 1) × y ] + [ nC2 × (xn – 2) × y2 ] – [ nC3 × (xn – 3) × y3 ] + . . . . . . . . . . . . . . . . . . +(-1)n [ nCn × yn ]
(iii) [1 – x]n = [ nC0 ] – [ nC1 . x ] + [ nC2 . x2 ] – [ nC3 . x3 ] + . . . . . . . . . . . . . . . . . . + (-1)n [ nCn . xn ]
(iv) (a + b)n =
NOTE:
- nCr = \(\begin{array}{l}\frac{n!}{r!(n-r)!}\end{array} \)where, n is a non-negative integer and [0 ≤ r ≤ n]
- nC0 = nCn = 1
- There are total (n + 1) terms in the expansion of (a + b)n
Important Formulas
- The general term in the expansion of (a + b)n:
Tr + 1 = nCr × (a)n – r × br - The middle term in the expansion of (a + b)n :
Case 1:
If n is even: The middle term =
Case 2:
If n is odd: The middle term =
Also Read: Binomial Theorem For Positive Integral Indices
Binomial Theorem Class 11 Important Questions
- Determine the coefficient of \(\begin{array}{l}(x)^{n}\end{array} \)in the expansion of\(\begin{array}{l}(x^{3}+3x^{2}+4x-17)^{4}\end{array} \)
- If the coefficient of the 2nd, 3rd and 4th terms in the expansion of \(\begin{array}{l}(1+x)^{2n}\end{array} \)are in Arithmetic Progression. Show that\(\begin{array}{l}2n^{2}-9n+7=0\end{array} \).
- Find the greatest term in the expansion of \(\begin{array}{l}(2+3x)^{9}\end{array} \), where x = 3/2.
- Determine the 4th term from the end in the expansion of \(\begin{array}{l}\left [ \frac{x^{2}}{5}-\frac{x}{3} \right ]^{8}\end{array} \)
- Expand the following \(\begin{array}{l}\left [2x^{2}-5x+8\right ]^{4}\end{array} \)
To get more details on Binomial Theorem, visit here.
Related Links:
- Exponents and Powers
- The Laws Of Exponents
- Binomial Distribution Formula
- NCERT Solutions for Class 11 Maths Chapter 8
- NCERT Exemplar for Class 11 Maths Chapter 8
Frequently Asked Questions on CBSE Class 11 Maths Notes Chapter 8 Binomial Theorem
What is the Binomial theorem?
The Binomial Theorem states the algebraic expansion of exponents of a binomial, which means it is possible to expand a polynomial (a + b) n into multiple terms.
What is a Pascal triangle?
Pascal’s triangle, in algebra, is a triangular arrangement of numbers that gives the coefficients in the expansion of any binomial expression, such as (x + y), to the power n.
What is a Factorial?
The product of all positive integers less than or equal to a given positive integer and denoted by that integer and an exclamation point is called a ‘Factorial’.
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