Binomial Expression
An algebraic expression consisting of two terms with a positive and negative sign between them is called binomial expression.
Binomial Theorem
If n is any positive integer, then
(x + a)n = nCoxn + nC1xn-1a + nC2xn-2a2 + nC3xn-3a3 + …. + nCnan
i.e. (x + a)n = \sum_{i=1}^{n}{^nC_rx^{n-r}a^r} — Binomial Theorem
(x – a)n = nCoxn – nC1xn-1a + nC2xn-2a2 – nC3xn-3a3 + …. – nCnan
where, x and a are real numbers and nCo, nC1, nC2, …, nCn are called binomial coefficients.
Also, n!\over r! (n - r)! 0≤ r ≤ n
Properties of Binomial Theorem for Positive Integer
- The total number of terms in the expression of (x + a)n is (n + 1).
- The sum of the indices of x and an in each term in n.
- The above expansion is also true when x and a are complex numbers.
- The values of the binomial coefficients steadily increase to a maximum and then steadily decrease.
General Term in a Binomial Expansion
(i) General term in the expansion of (x + a)n is
Tr+1 = nCrxn-rar
(ii) General term in the expansion of (x – a)n is
Tr+1 = (-1)r nCrxn-rar
Middle Term in a Binomial Expansion
- If n is even in the expansion of (x + a)n or (x – a)n, then the middle term is \bigg({n\over2}+1\bigg) th term.
- If n is odd in the expansion of (x + a)n or (x – a)n, then the middle term is \bigg({n+1\over2}\bigg) th term and \bigg({n+1\over2}+1\bigg) th term
Note: Where there are two middle terms in the expansion, then their binomial coefficients are equal.
Important Results on Binomial Coefficient
If Co, C1, C2, …, Cn are the coefficients of (1 + x)n, then
(i) nCr + nCr – 1 = n + 1C r
(ii) { {^n}C_r \over{^{n-1}}C_{r-1}}={n\over r}
(iii) { {^n}C_r \over{^{n}}C_{r-1}}={n-r+1\over r}
(iv) nC0 + nC1 + nC2 + nC3 + … + nCn = 2n
(v) nC0 + nC2 + nC4 + … = nC1 + nC3 + nC5 + … = 2n-1
