# Binomial Theorem – HSC Math

Posted In: Mathematics

In elementary algebra, the binomial theorem or binomial expansion describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial \$(x+y)^n\$ into a sum involving terms of the form a \$ax^by^c\$, where the exponents \$b\$ and \$c\$ are nonnegative integers with \$b+c=n\$, and the coefficient \$a\$ of each term is a specific positive integer depending on \$n\$ and \$b\$.

1. Finite Binomial Series,
i. \$(a+x)^n=a^n+^nC_1a^{n-1}x+^nC_2a^{n-2}x^2+…+^nC_ra^{n-r}x^r+…+x^n\$
2. ii. \$(a-x)^n=\$
\$a^n-^nC_1a^{n-1}x+^nC_2a^{n-2}x^2-…+(-1)^r\ ^nC_ra^{n-r}x^r+…+(-1)^nx^n\$

iii. \$(1+x)^n=1+^nC_1x+^nC_2x^2+…+^nC_rx^r+…+x^n\$

iv. \$(1-x)^n=1-^nC_1x+^nC_2x^2-…+(-1)^r\ ^nC_rx^r+…+x^n\$

3. In expansion of \$(a+x)^n,\$ the \$(r+1)^{th}\$ term,
\$T_{r+1}=^nC_ra^{n-r}x^r\$
4. In case of \$(a+x)^n\$,
\$i.\$if \$n\$ is even, middle term is \$(n/2+1)^{th}\$ term.
\$ii.\$if \$n\$ is odd, middle terms are \$({n+1}/2)^{th}\$ and \$({n+1}/2+1)^{th}\$ term.
5. If \$^nC_x=^nC_y\$ and \$x≠y\$ then \$n=x+y\$,
6. For the expansion of \$(a+x)^n\$
\$i.{T_{r+1}}/{T_r}={n-r+1}/r.x/a\$
\$ii.{^nC_1}/{^nC_{r-1}}={n-r+1}/r\$
7. Infinite Binomial Series,
\$i.(1+x)^{-n}=1-nx+{n(n+1)}/{2!}x^2+…+(-1)^r{n(n+1)…(n+r-1)}/{r!}x^r+……\$
\$ii.(1-x)^{-n}=1+nx+{n(n+1)}/{2!}x^2+…+{n(n+1)…(n+r-1)}/{r!}x^r+……\$
\$iii.(1+x)^{-1}=1-x+x^2+…+(-1)^rx^r+……\$
\$iv.(1-x)^{-1}=1+x+x^2+…+x^r+……\$
\$v.(1+x)^{-2}=1-2x+3x^2+…+(-1)^r(r+1)x^r+……\$
\$vi.(1-x)^{-2}=1+2x+3x^2+…+(r+1)x^r+……\$
\$vii.(1+x)^{-3}=1-3x+6x^2-10x^3…+(-1)^r{(r+1)(r+2)}/2x^r+……\$
\$viii.(1-x)^{-3}=1+3x+6x^2+10x^3…+{(r+1)(r+2)}/2x^r+……\$
8. In expansion of \$(1+x)^{-n}\$, general term,
\$T_{r+1}=(-1)^r{n(n+1)…(n+r-1)}/{r!}x^r\$
9. In expansion of \$(1-x)^{-n}\$, general term,
\$T_{r+1}={n(n+1)…(n+r-1)}/{r!}x^r\$
10. For the expansion of \$(1+x)^n=c_0+c_1x+c_2x^2+……+c_nx^n,\$
\$i.c_0+c_1+c_2+…+c_n=2^n\$
\$ii.c_0+c_2+c_4+…=c_1+c_3+c_5+… = 2^{n-1}\$
\$iii.c_0^2+c_1^2+c_2^2+…+c_n^2={(2n)!}/{n!}\$
\$iv.c_0c_n+c_1c_{n-1}+…+c_nc_0={(2n)!}/{n!n!}\$
11. \$ln(1+x)=x-x^2/2+x^3/3-…….\$
12. \$ln(1-x)=x+x^2/2+x^3/3+…….\$

Post Tags: Algebra | HSC