The Binomial Theorem

The binomial theorem primarily helps to find the symbolic value of the algebraic expression of the form \((x + y)^n\).

Related Topics

• How to Solve Infinite Geometric Series
• How to Solve Geometric Sequences
• How to Solve Arithmetic Sequences

A step-by-step guide to the binomial theorem

According to the binomial theorem, it is possible to expand any non-negative power of binomial \((x + y)\) into a sum of the form, \((x+y)^n=\begin{pmatrix}n\\ 0\end{pmatrix}x^n y^0+ \begin{pmatrix}n\\ 1\end{pmatrix}x^{n-1} y^1 + \begin{pmatrix}n\\ 2\end{pmatrix}x^{n-2} y^2+…+ \begin{pmatrix}n\\ n-1\end{pmatrix}x^1 y^{n-1}+ \begin{pmatrix}n\\ n\end{pmatrix}x^0 y^n\)

where \(n≥0\) is an integer and each \(\begin{pmatrix}n\\k\end{pmatrix}\) is a positive integer known as a binomial coafficient.

Note: When power is zero, the corresponding power expression is \(1\).

Using summation notation, the binomial theorem can be given as:

\(\color{blue}{(x+y)^n=\sum _{k=0}^n\: \begin{pmatrix}n\\k\end{pmatrix} x^{n-k}y^{k}}\) \(\color{blue}{= \sum _{k=0}^n\: \begin{pmatrix}n\\k\end{pmatrix} x^{k}y^{n-k}}\)

The binomial theorem formula

The binomial theorem formula helps to expand a binomial that has been increased to a certain power. The binomial theorem states: if \(x\) and \(y\) are real numbers, then for all \(n ∈ N\):

\(\color{blue}{(x+y)^n=\sum _{r=0}^n\: (^nC_r)x^{n-r}y^{r}}\)

where, \(\color{blue}{^nC_r}\)\(\color{blue}{=\frac{n!}{r!(n-r)!}}\)

Properties of the binomial theorem

• The number of coefficients in the binomial expansion of \((x+y)^n\) is equal to \((n+1)\).
• In the expansion of \((x+y)^n\),there are \((n+1)\) terms.
• The first and the last terms are \(x^n\) and \(y^n\) respectively.
• From the beginning of the expansion, the powers of \(x\) decrease from \(n\) to \(0\), and the powers of \(a\) increase from \(0\) to \(n\).
• The general term in the expansion of \((x + y)^n\) is the \((r +1)^{th}\) term that can be represented as \(T_{r+1}\), \(T_{r+1}=\)\(^nC_r\)\(x^{n-r}y^r\)
• The binomial coefficients in the expansion are arranged in an array called the Pascal triangle. This developed model can be summarized with a binomial theorem formula.
• In the binomial expansion of \((x+y)^n\), the \(r^{th}\) term from the end is \((n-r+2)^{th}\) term from the beginning.
•  If \(n\) is odd, then in \((x + y)^n\), the middle terms are \(\frac{(n+1)}{2}\) and \(\frac{(n+3)}{2}\).
• If \(n\) is even, then in \((x + y)^n\), the middle term \(=(\frac{n}{2})+1\)

The Binomial Theorem – Example 1:

Expand \((x+2)^5\) using the binomial theorem.

Use this formula to expand: \(\color{blue}{(x+y)^n=\sum _{k=0}^n\: \begin{pmatrix}n\\k\end{pmatrix} x^{n-k}y^{k}}\)

\(=\sum _{k=0}^5\: \begin{pmatrix}5\\k\end{pmatrix}x^{(5-k)}.2^k\)

\((x+2)^5=\frac{5!}{0!(5-0)!}x^5.2^0+ \frac{5!}{1!(5-1)!}x^4.2^1+ \frac{5!}{2!(5-2)!}x^3.2^2+ \frac{5!}{3!(5-3)!}x^2.2^3+ \frac{5!}{4!(5-4)!}x^1.2^4+ \frac{5!}{5!(5-5)!}x^0.2^5 \)

\(x^5+10x^4+40x^3+80x^2+80x+32\)

Exercise for the Binomial Theorem

Find binomial expansion by using the binomial theorem.

1. \(\color{blue}{\left(x^2+4\right)^4}\)
2. \(\color{blue}{\left(2x+3x^2\right)^5}\)
3. \(\color{blue}{\left(2x+5\right)^3}\)

1. \(\color{blue}{x^8+16x^6+96x^4+256x^2+256}\)
2. \(\color{blue}{32x^5+240x^6+720x^7+1080x^8+810x^9+243x^{10}}\)
3. \(\color{blue}{8x^3+60x^2+150x+125}\)