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

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

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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:

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

Binomial expansion

Binomial expansion of \((x+y)^n\) by using the binomial theorem is as follows:

\((x+y)^n=\) \(^nC\)\( _0x^n+\) \(^nC\)\( _1x^{n-1}y +\) \(^nC\)\( _2x^{n-2}y^2 +…+\) \(^nC\)\( _rx^{n-r}y^r+…\) \(^nC\)\( _ny^n\)

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 \(=(\frac{n}{2})+1\)

Binomial theorem coefficients

The binomial coefficients are the numbers related to the variables \(x\), \(y\) in the expansion \((x + y)^n\). The binomial coefficients are displayed as \(^nC_0\), \(^nC_1\), \(^nC_2\). The binomial coefficients are obtained through the Pascal triangle or by using the formula of compounds.

Pascal’s triangle

The values of the binomial coefficients show a special trend that can be seen as a Pascal triangle. The Pascal triangle is an order of binomial coefficients in triangular form. The numbers in the Pascal triangle have all the border elements of \(1\), and the remaining numbers in the triangle are placed in such a way that each number is the sum of the two numbers just above the number.

Combinations

The formula for combinations is used to find the value of binomial coefficients in expansions using the binomial theorem. The combinations, in this case, there are different methods for selecting the \(r\) variable from the existing \(n\) variables. The formula to find the combinations of \(r\) objects taken from \(n\) different objects is:

\(^nC_r\) \(=\frac{n!}{r!(n-r)!}\)

The coefficients have the following properties:

  • \(^nC_0\) \(=\)\(^nC_n\)\(=0\)
  • \(^nC_1\) \(=\)\(^nC_{n-1}\)\(=n\)
  • \(^nC_r\) \(=\)\(^nC_{r-1}\)

The following properties of binomial expansion can be obtained by substituting the simple numerical values \(x = 1\) and \(y = 1\) in the binomial expansion \((x + y)^n\). The properties of binomial coefficients are as follows.

  • \(C_1+C_2+C_3+C_4+…C_n=2^n\)
  • \(C_0+C_2+C_4+…=C_1+C_3+C_5+…..=2^{n-1}\)
  • \(C_0-C_1+C_2-C_3+C_4-C_5+…(-1)^nC_n=0\)
  • \(C_1+2C_2+3C_3+4C_4….+nC_n=n2^{n-1}\)
  • \(C_1-2C_2+3C_3-4C_4….+(-1)^{n-1}nC_n=0\)
  • \(C_0^2+C_1^2+C_2^2+C_3^2+C_4^2+……C_n^2=\frac{(2n)!}{(n!)^2}\)

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. \(\color{blue}{\left(2x+5\right)^3}\)
This image has an empty alt attribute; its file name is answers.png
  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. \(\color{blue}{8x^3+60x^2+150x+125}\)

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