Binomial Theorem Class 11 Notes - Chapter 8

The binomial expression is an expression comprising of two terms connected by -ve or +ve sign. Equations like x + a, 2x – 3y,

\(\begin{array}{l}\frac{1}{x}-\frac{1}{x^{3}}\end{array} \)
,
\(\begin{array}{l}7x-\frac{2}{4x^{3}}\end{array} \)
are examples of binomial expressions. The binomial expansion of
\(\begin{array}{l}(p+q)^{n}\end{array} \)
will have a total of (n + 1) terms. The coefficients in the binomial expansion follow a pattern called as Pascal’s triangle. The sum of exponents of ‘p’ and ‘q’ is always equal to n.

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 as 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 =

\(\begin{array}{l}\sum_{r\;=\;0}^{n}\end{array} \)
nCr (a)n – r × br

NOTE:

  1. nCr =
    \(\begin{array}{l}\frac{n!}{r!(n-r)!}\end{array} \)
    where, n is a non-negative integer and [0 ≤ r ≤ n]
  2. nC0 = nCn = 1
  3. 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 =

\(\begin{array}{l}\left ( \frac{n}{2}+1 \right )^{th}term\end{array} \)

Case 2:

If n is odd: The middle term =

\(\begin{array}{l}\left ( \frac{n+1}{2}\right )^{th}term\;\;and\;\;\left ( \frac{n+1}{2} +1 \right )^{th}term\end{array} \)

Also Read: Binomial Theorem For Positive Integral Indices

Binomial Theorem Class 11 Important Questions

  1. Determine the coefficient of
    \(\begin{array}{l}(x)^{n}\end{array} \)
    in the expansion of
    \(\begin{array}{l}(x^{3}+3x^{2}+4x-17)^{4}\end{array} \)
  2. If the coefficient of 2nd, 3rd and 4th terms in the expansion of
    \(\begin{array}{l}(1+x)^{2n}\end{array} \)
    are in Arithmetic Progression. Show that
    \(\begin{array}{l}2n^{2}-9n+7=0\end{array} \)
    .
  3. Find the greatest term in the expansion of
    \(\begin{array}{l}(2+3x)^{9}\end{array} \)
    , where x = 3/2.
  4. Determine the 4th term from the end in the expansion of
    \(\begin{array}{l}\left [ \frac{x^{2}}{5}-\frac{x}{3} \right ]^{8}\end{array} \)
  5. 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:

  1. Exponents and Powers
  2. The Laws Of Exponents
  3. Binomial Distribution Formula
  4. NCERT Solutions for Class 11 Maths Chapter 8
  5. NCERT Exemplar for Class 11 Maths Chapter 8

Frequently asked Questions on CBSE Class 11 Maths Notes Chapter 8: Binomial Theorem

What is ‘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 the multiple terms.

What is a ‘Pascal Triangle’?

Pascal’s triangle, in algebra, 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 as a ‘Factorial’.

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