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^ris , 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

Answer Key

1) a³+3a²b+3b²a+b³

2) s³+9s²+27s+27

3) y⁸+4y⁶+6y⁴+4y²+1

4) 81+108x+54x²+12x³+x⁴

5) z⁶+12z⁵+60z⁴+160z³+240z²+192z+64

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