Odisha State Board CHSE Odisha Class 11 Math Notes Chapter 9 Binomial Theorem will enable students to study smartly.

## CHSE Odisha 11th Class Math Notes Chapter 9 Binomial Theorem

**Binomial Theorem For Positive Integral Index:**

For any a,b ∈ R, and n ∈ N

(a + b)^{n} = ^{n}C_{0} a^{n} + ^{n}C_{1} a^{n-1}b + ….. ^{n}C_{n} b^{n}

Note:

(a) (a + b)^{n} = a^{n} + na^{n-1} b + \(\frac{n(n-1)}{2 !}\) a^{n-2}b^{2} ….. + b^{n}

(b) (1 + x)^{n} = ^{n}C_{0} + ^{n}C_{1} x + ^{n}C_{2} x^{2} + ….. + ^{n}C_{n} x^{n}

(c) (a – b)^{n} = ^{n}C_{0} a^{n} – ^{n}C_{1} a^{n-1} b + ^{n}C_{2} a^{n-2}b^{2} ….. + (-1)^{n} b^{n}

(d) (1 – x)^{n} = ^{n}C_{0} – ^{n}C_{1} x + ^{n}C_{2} x^{2} ….. + (-1)^{n} x^{n}

Some conclusions from the Binomial theorem:

- There are (n + 1) terms in the expansion of (a + b)
^{n} - We can write (a + b)
^{n }= \(\sum_{r=0}^n{ }^n \mathrm{C}_r a^{n-r} b^r\) and (a – b)^{n }= \(\sum_{r=0}^n(-1)^r{ }^n \mathrm{C}_r a^{n-r} b^r\) - The sum of powers of a and b in each term = n
- As
^{n}C_{r}=^{n}C_{n-r}(The coefficient of terms equidistant from the beginning and the end are equal). - (r + 1)
^{th}term (General term)

= t_{r+1}=^{n}C_{r}a^{n-r}b^{r} - (a + b)
^{n}+ (a – b)^{n}= 2[^{n}C_{0}a^{n}+^{n}C_{2}a^{n-2}b^{2}+ ….] - (a + b)
^{n}– (a – b)^{n}= 2[^{n}C_{1}a^{n-1}b +^{n}C_{3}a^{n-3}b^{3}+ ….] - (middle terms):

⇒ If n is even then the middle term = \(t_{\left(\frac{n+2}{2}\right)}=t_{\left(\frac{n}{2}+1\right)}\)

⇒ If n is odd there are two middle terms. They are = \(t_{\left(\frac{n+1}{2}\right)} \text { and } t_{\left(\frac{n+3}{2}\right)}\) - t
_{r+1}from the end in the expansion of (a + b)^{n}= t_{r+1}from the beginning in the expansion of (b + a)^{n}.

**Binomial Theorem For Any Rational Index:**

If n ∈ Q and x ∈ R such that |x| < 1 then (1 + x)^{n} = 1 + nx + \(\frac{n(n-1)}{2 !} x^2\) + \(\frac{n(n-1)(n-2)}{3 !} x^3+\ldots .\)

Note:

(1) (1 + x)^{-1} = 1 – x + x^{2} – x^{3} + …..

(2) (1 – x)^{-1} = 1 + x + x^{2} + …..

(3) (1 + x)^{-2} = 1 – 2x + 3x^{2} – 4x^{3} + …..

(4) (1 – x)^{-2} = 1 + 2x + 3x^{2} + 4x^{3} + …..