What is Binomial Theorem – Formula, Expansion & Examples

You know how to find find the squares and cubes of binomials like $(a + b)$ and $(a – b)$, and also used these algebraic identities to find the numerical values of numbers like $(99)^2 = (100 – 1)^2$, or $(103)^2 = (100 + 3)^2$. 

However, for higher powers like $(99)^8$, or $(107)^{12}$, etc., the calculations become difficult by using repeated multiplication. This difficulty was overcome by a theorem known as the binomial theorem. 

Let’s understand what is binomial theorem, its formula, and expansion, using examples.

What is Binomial Theorem?

The Binomial Theorem is a method of expanding an expression that has been raised to any finite power. A binomial theorem is a powerful tool of expansion, which has applications in algebra, probability, etc. The Binomial Theorem can be traced to the 4th century B.C. and Euclid where one finds the formula for $(a + b)^2$. In the 3rd century B.C., the Indian mathematician Pingala presented what is now known as “Pascal’s Triangle” giving binomial coefficients in a triangle.

The binomial theorem states the principle for expanding the algebraic expression $(a + b)^n$ and expresses it as a sum of the terms involving individual exponents of variables $a$ and $b$. Each term in a binomial expansion is associated with a numeric value which is called a coefficient.

According to the binomial theorem, any algebraic expression $(a + b)$ with non-negative power can be expanded into a sum of the form $(a+b)^n = ^n \text{C}_0 a^nb^0 + ^n \text{C}_1 a^{n – 1}b^1 + ^n \text{C}_2 a^{n – 2}b^2 + … + ^n \text{C}_{n – 1} a^{1}b^{n – 1} + ^n \text{C}_n a^0b^n$

where, $n \ge 0$ is an integer and each $^n \text{C}_r$ is a positive integer known as a binomial coefficient.

This formula is also referred to as the binomial formula or the binomial identity. Using summation notation, the binomial theorem can be given as $(a + b)^n = \sum ^n \text{C}_r a^{n – r} b^r$.

Binomial Theorem Expansion Proof

Let $x$, $a$, $n \in \text{N}$. 

We will prove the binomial theorem formula through the principle of mathematical induction. 

According to the principle of mathematical induction, It is enough to prove for $n = 1$, $n = 2$, for $n = k \ge 2$, and for $n = k+ 1$.

It is obvious that $(x +y)^1 = x +y$ and $(x +y)^2 = (x + y) (x +y) = x^2 + xy + xy + y^2$ (using distributive property)

$= x^2 + 2xy + y^2$

Thus the result is true for $n = 1$ and $n = 2$. 

Let $k$ be a positive integer. 

Let us prove the result is true for $k \ge 2$.

Assuming $(x + y)^n = \sum ^n \text{C}_r x^{n-r}y^r$, $(x + y)^k = \sum ^k \text{C}_r x^{k – r}y^r$

$=> (x + y)^k = ^k \text{C}_0 x^ky^0 + ^k \text{C}_1 x^{k – 1}y^1 + ^k \text{C}_2 x^{k – 2} y^2 + … + ^k \text{C}_r x^{k – r}y^r +….+ ^k \text{C}_k x^0 y^k$

$= (x + y)^k = x^k + ^k \text{C}_1 x^{k -1} y^1 + ^k \text{C}_2 x^{k – 2} y^2 + … + ^k \text{C}_r x^{k – r} y^r +….+ y^k$

Thus the result is true for $n = k \ge 2$.

Now consider the expansion for $n = k + 1$.

$(x + y)^{k+1} = (x + y) (x + y)^k$

$= (x + y) (x^k + ^k \text{C}_1 x^{k -1} y^1 + ^k \text{C}_2 x^{k – 2} y^2 + … + ^k \text{C}_r x^{k – r} y^r +….+ y^k)$

$= x^{k+1} + (1 + ^k \text{C}_1)x^k y + (^k \text{C}_1 + ^k \text{C}_2) x^{k – 1}y^2 + … + (^k \text{C}_{r – 1} + ^k \text{C}_r) x^{k – r + 1}y^r + … + (^k \text{C}_{k -1} + 1) xy^k + y^{k+1}$

$= x^{k+1} + ^{k+1} \text{C}_1 x^k y + ^{k+1} \text{C}_2 x^{k-1}y^2 + … + ^{k+1} \text{C}_r x^{k-r+1}y^r + … + ^{k+1} \text{C}_k xy^k + y^{k+1}$ [Because $^n \text{C}_r + ^n \text{C}_{r-1} = ^{n+1} \text{C}_r$]

Thus the result is true for $n = k+1$. 

By mathematical induction, this result is true for all positive integers $n$. 

Examples of Binomial Theorem

Example 1: Expand $(2x + 5y)^6$.

Here $a = 2x$, $b = 5y$, and $n = 6$

Therefore, $(2x + 5y)^6 = ^6 \text{C}_0 (2x)^6 (5y)^0 + ^6 \text{C}_1 (2x)^5 (5y)^1 + ^6 \text{C}_2 (2x)^4 (5y)^2 + ^6 \text{C}_3 (2x)^3 (5y)^3$ 

$+ ^6 \text{C}_4 (2x)^2 (5y)^4 + ^6 \text{C}_5 (2x)^1 (5y)^5 + ^6 \text{C}_6 (2x)^0 (5y)^6$

$= 1 \times 64x^6 \times 1 + 6 \times  32x^5 \times 5y + 15 \times  16x^4  \times 25y^2 + 20 \times 8x^3 \times 125y^3 $

$+ 15 \times 4x^2 \times 625y^4 + 6 \times 2x  \times 3125y^5 + 1 \times 1 \times 15625y^5$

$= 64x^6 + 960x^5 y + 6000x^4 y^2 + 20000 x^3 y^3 + 37500x^2 y^4 + 37500x y^5 + 15625y^5$

Example 2: Expand $(x – 3y)^4$.

Here $a = x$, $b = -3y$, and $n = 4$

Therefore, $(x – 3y)^4 = ^4 \text{C}_0 x^4 (-3y)^0 + ^4 \text{C}_1 x^3 (-3y)^1 + ^4 \text{C}_2 x^2 (-3y)^2 + ^4 \text{C}_3 x^1 (-3y)^3 + ^4 \text{C}_4 x^0 (-3y)^4$

$= 1 \times x^4 \times 1 + 4 \times x^3 \times (-3y) + 6 x^2 \times 9y^2 + 4 x \times (-27y^3) + 1 \times 1  \times 81y^4$

$= x^4 -12 x^3 y + 54 x^2 y^2 – 108 x y^3 + 81y^4$

Example 3: Evaluate $(105)^6$.

$(105)^6 = (100 + 5)^6$

Here $a = 100$, $b = 5$, and $n = 6$

Therefore, $(100 + 5)^6 = ^6 \text{C}_0 \times 100^6 \times 5^0 + ^6 \text{C}_1 \times 100^5 \times 5^1 + ^6 \text{C}_2 \times 100^4 \times 5^2 + ^6 \text{C}_3 \times 100^3 \times 5^3$

$ + ^6 \text{C}_4 \times 100^2 \times 5^4 + ^6 \text{C}_5 \times 100^1 \times 5^5 + ^6 \text{C}_6 \times 100^0 \times 5^6$

$=1 \times 1000000000000 \times 1 + 6 \times 10000000000 \times 5 + 15 \times 100000000 \times 25 $

$+ 20 \times 1000000 \times 125 + 15 \times 10000 \times 625 + 6 \times 100 \times 3125 + 1 \times 1 \times 15625$

$=1000000000000 + 300000000000 + 37500000000 + 2500000000 + 93750000 + 1875000 + 15625$

$ = 1340095640625$

Properties of Binomial Theorem

The following are the important properties of the Binomial Theorem.

• The number of coefficients in the binomial expansion of $(a + b)^n$ is equal to $(n + 1)$
• There are $(n + 1)$ terms in the expansion of $(a + b)^n$
• The first and the last terms are $a^n$ and $b^n$ respectively
• From the beginning of the expansion of $(a + b)^n$, the powers of $a$, decrease from $n$ up to $0$, and the powers of $b$, increase from $0$ up to $n$
• The general term in the expansion of $(a + b)^n$ is the $(r +1)^{th}$ term that can be represented as $\text{T}_{r+1}$, $\text{T}_{r+1} = ^n \text{C}_r a^{n – r} b^r$
• The binomial coefficients in the expansion are arranged in an array, which is called Pascal’s triangle. This pattern developed is summed up by the binomial theorem formula
• In the binomial expansion of $(a + b)^n$, the $r^{th}$ term from the end is $(n – r + 2)^{th}$ term from the beginning
• If $n$ is even, then in $(a + b)^n$ the middle term = $\frac{n}{2} +1$ and if $n$ is odd, then in $(a + b)^n$, the middle terms are $\frac{n + 1}{2}$ and $\frac{n + 3}{2}$

Practice Problems

1. Expand
   • $(2x – 4y)^7$
   • $(\frac{2}{3}x + y)^5$
   • $(\frac{1}{2}a – \frac{2}{3} b)^6$
   • $(0.01x + 0.25y)^5$
2. Find the coefficient of $y^3$ in the expansion of $(1 + 2y)^9$
3. Find the coefficient of the term corresponding to $x^3$ in the expansion of $(x – 2y)^7$

FAQs

Conclusion

The binomial theorem states the principle for expanding the algebraic expression $(a + b)^n$ and expresses it as a sum of the terms involving individual exponents of variables $a$ and $b$. The general form of the binomial theorem is $(a + b)^n$. The formula is $(a+b)^n = ^n \text{C}_0 a^nb^0 + ^n \text{C}_1 a^{n – 1}b^1 + ^n \text{C}_2 a^{n – 2}b^2 + … + ^n \text{C}_{n – 1} a^{1}b^{n – 1} + ^n \text{C}_n a^0b^n$.

Recommended Reading

• What is Combination – Definition, Types, Formulas & Examples
• What is Permutation – Definition, Types, Formulas & Examples
• What is Factorial of a Number – Definition, Formula & Examples
• Fundamental Principle of Counting – Rules, Formulas & Examples
• Why Is 0 Factorial 1?