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, \(\frac{1}{x}-\frac{1}{x^{3}}\), \(7x-\frac{2}{4x^{3}}\) are examples of binomial expressions. The binomial expansion of \((p+q)^{n}\) 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

\(\Rightarrow\) 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 = \(\sum_{r\;=\;0}^{n}\) nCr (a)n – r × br

NOTE:

  1. nCr = \(\frac{n!}{r!(n-r)!}\) 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 = \(\left ( \frac{n}{2}+1 \right )^{th}term\)

Case 2:

If n is odd: The middle term = \(\left ( \frac{n+1}{2}\right )^{th}term\;\;and\;\;\left ( \frac{n+1}{2} +1 \right )^{th}term\)

 

Binomial Theorem Class 11 Important Questions

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

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