Binomial Theorem

Binomial theorem is a theorem developed for calculating any power of any binomial without actually multiplying at length.

General binomial expansion is denoted by

(a+b)ⁿ  =ⁿc₀aⁿ +ⁿc₁a^n-1b+ⁿc₂a^n-2b²+…………………..+ⁿc_n-1ab^n-1+ⁿc_nbⁿ  where n>=0 and

Example – (x+y)³=³c₀x³+³c₁x²y+³c₂xy²+³c₂y³

Example – Expand (p+1)³

(p+1)³=³c₀*p³+³c₁*p²*1+³c₂*p*1²+³c₂*1³

=  p³+3p²+3p+1

Some Facts about Binomial Theorem

• General term for an expansion (a+b)ⁿ is T_r+1 =ⁿc_ra^n-rb^r is , where 0 <= r<= n.
• Number of terms in expansion will be n + 1.

Check Point:

Expand the following binomials –

1) (a+b)³

2) (s+3)³

3) (y²+1)⁴

4) (3+x)⁴

5) (z+2)⁶

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