# The Binomial Theorem

The Binomial Theorem is a way of expanding an expression that has been raised to any finite power. In this post, you will learn more about 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.

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

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

## Related to This Article

### More math articles

- Geometry Puzzle – Critical Thinking 17
- How is the TASC Test Scored?
- 8 Easy Steps for Success Study for a Math Test
- Geometry Puzzle – Challenge 62
- Top 10 Free Websites for SSAT Math Preparation
- 4th Grade M-STEP Math Worksheets: FREE & Printable
- 4th Grade MCAS Math FREE Sample Practice Questions
- Trigonometric Ratios
- Reciprocals
- The Ultimate OAR Math Course (+FREE Worksheets)

## What people say about "The Binomial Theorem - Effortless Math: We Help Students Learn to LOVE Mathematics"?

No one replied yet.