Binomial Theorem Formula: Theorem, Expansion Form and Solved Examples

The binomial theorem expresses an expanded form of an algebraic expression in the form of (a+b)_n. This formula is really useful in simplifying complicated equations. 

• This theorem works with larger powers.
• The co-efficient of larger power terms can be found through this theorem.
• This expansion is an algebraic expansion for the powers of a binomial.
• The exponent value of the binomial theorem expansion might be a negative integer or a fraction.

Read More: Polynomial Formula

Key Terms: Binomial theorem, Binomial expansion, Pascal’s triangle expansion, Coefficients, Exponents.

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Binomial Theorem

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The Binomial Theorem is a mathematical expression made up of two components that include addition or subtraction operations. The algebraic equation (a + b)_n is expanded as a sum of individual terms with exponents of variables a and b.

• The coefficients in the binomial expansion of (a+b)_n are called binomial coefficients. For example,ⁿC₀, ⁿC₁, ⁿC₂ . . . . . . .ⁿC_n are some of the coefficients. 
• According to this theorem If n is any positive integer, then (a+b)_n = ∑(n/r)a_n-r.b∏ , where r = 0 to n for ∑.
• Simplifying it the obtained expression will be– (n/r) = ⁿC_r = n!/r!(n-r)! , called the binomial coefficient (ⁿC_r).

(a + b)ⁿ = \(\sum ^n _{k=0} \binom{n}{k} a ^{n-k} b^k\)

Read More: Sequence and Series

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Binomial Expansion

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The expanded form of the binomial theorem is called binomial expansion. It gives the mathematical form of the expanded exponential power of the binomial expression.

The expansion will take a form like this–

(a+b)_n = ⁿC₀ aⁿ b⁰ + ⁿC₁ a^n-1b¹ + ⁿC₂ a^n-2 b² + ... + ⁿC_n-1 a¹b^n-1 + ⁿC_n a⁰bⁿ

Here, ⁿC₀, ⁿC₁, ⁿC₂ . . . . . . .ⁿC_n are some of the coefficients.

Read More: Permutations and Combinations

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Binomial Theorem Formula

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The universal formula for the binomial theorem is (a+b)_n= ∑ⁿr = 0ⁿC_r a^n-rb^r, where n is a positive integer and a, b are real numbers, and 0 < r ≤ n.

The formula will be - (a+b)ⁿ = ⁿC₀ aⁿ b⁰ + ⁿC₁ a^n-1b¹ +……..+ ⁿC_r a^n-r b^r +………+ ⁿC_n a⁰bⁿ

Here ¹C₀ = 1 and ¹C₁ = 1

So it can be inferred that 

(a+b)^k = ^kC₀ a^k b⁰ + ^kC₁ a^k-1b¹ +……..+ ^kC_r a^k-r b^r +………+ ^kC_k a⁰b^k

Read More: Bayes Theorem

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Binomial Theorem Expansion Expression

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The expression of the binomial theorem can be expanded by the principle of mathematical induction. 

From previously obtained expression it was observed that (a +b)¹ = a + b and

(a +b)² = (a + b) (a +b)

= a² + ab + ab + b² 

= a² + 2ab + b²

This is true for n = 1 and n = 2. Let k be a positive integer. The result is true for k ≥ 2.

Assuming (a + b)ⁿ = ∑ⁿr=0ⁿC_r a^n-rb^r,

(a + b)^k = ∑^kr=0^kCr a^k-rb^r

⇒ (a+b)^k = ^kC₀ a^kb⁰ + ^kC₁ a^k-1b¹ + ^kC₂ a^k-2 b² + ... + ^kC_r a^k-rb^r +....+ ^kC_k a⁰b^k

⇒ (a+b)^k = a^k + ^kC₁ a^k-1b¹ + ^kC₂ a^k-2 b² + ... + ^kC_r a^k-rb^r +....+ b^k

Thus the result is true for n = k ≥ 2.

Now consider the expansion for n = k + 1.

(a + b) ^k+1 = (a + b) (a + b)^k

= (a + b) (a^k + ^kC₁ a^k-1b¹ + ^kC₂ a^k-2 b² + ... + ^kC_r a^k-rb^r +....+ b^k)

= a^k+1 + (1 + ^kC₁)a^kb + (^kC₁ + ^kC₂) a^k-1b² + ... + (^kCr-1 + ^kCr) a^k-r+1b^r + ... + (^kC_k-1 + 1) ab^k + b^k+1

= a^k+1 + ^k+1C₁a^kb + ^k+1C₂ a^k-1b² + ... + ^k+1C_r a^k-r+1b^r + ... + ^k+1C_k ab^k + b^k+1 [Because ⁿC_r + ⁿC_r-1 = ⁿ⁺¹C_r]

For n = k+1 for all positive integers 'n' this expression is true.

Read More: Geometric Progression

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Pascal’s Triangle Binomial Expansion

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The binomial coefficients can be obtained with the help of Pascal’s triangle. In this triangle, binomial coefficients are arranged in a triangular manner. All of the boundary components of Pascal's triangle are 1, and the remaining numbers within the triangle are arranged so that each number is the summed-up form of two numbers just above the number.

Obtained coefficients from this expansion are–

• ⁿC_n = ⁿC₀ = 1
• ⁿC₁ = ⁿC_n-1 = n
• ⁿC_r = ⁿC_r-1

Read More: Multiplication Theory on Probability

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Properties of Binomial Theorem

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A few properties of the binomial coefficients are given below– 

1. C₀ + C₁ + C₂ + … + C_n = 2_n
2. C₀ – C₁ + C₂ – … + (-1)nC_n = 0
3. C₀ + C₂ + C₄ + … = C₁ + C₃ + C₅ + … = 2_n-1
4. nC_r = nC_n-r
5. r(nC_r) = n^n-1 C_r-1
6. nC_r/ r+1 = (n + 1) C _r+1 / (n+1)
7. nC_r + nC_r-1 = (n+1) C_r

Where n ∈ N, r ∈ W and r ≤ n

Read More: Real-Valued Function

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Solved Examples

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Example 1: What is the binomial expansion of (x² + 1)⁵ using the binomial theorem?

Solution: Obtained formula from the Binomial Theorem is helpful to find the expansion.

(x + y)ⁿ = (x+y)ⁿ = ⁿC₀ xⁿy⁰ + ⁿC₁ x^n-1y¹ + ⁿC₂ x^n-2 y² + ... + ⁿCk x^n-ky^k +....+ ⁿC_n x⁰yⁿ

(x² + 1)⁵ = ⁵C₀ (x²)⁵1⁰ + ⁵C1 (x²)^5-11¹ + ⁵C₂ (x²)^5-2 1² + ⁵C₃ (x²)^5-3 1³ + ⁵C₄ (x²)^5-4 1⁴ + ⁵C₅ (x²)^5-5 1⁵

= x¹⁰ + 5 x⁸ + 10 x⁶ + 10 x⁴ + 5 x² + 1

Example 2: Find the coefficient of x² in (x +(1/x))⁸

Solution: By applying the binomial theorem formula in the expansion of (x +1/x)8, here x2 is considered as the fourth term.

⁸C₀ x⁸(1/x)⁰ + ⁸C₁ x⁷(1/x)¹ + ⁸C₂ x⁶(1/x)² + ⁸C₃ x⁵(1/x)³ + ⁸C₄ x⁴(1/x)⁴ + ⁸C₅ x³(1/x)⁵ + ⁸C₆ x²(1/x)⁶ + ⁸C₇ x¹(1/x)⁷ + ⁸C₈ x⁰(1/x)⁸ 

The coefficient of the fourth term is ⁸C₃= 56

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Things to Remember

• The binomial theorem is expressed as an expanded form of an algebraic expression in the mathematical form of (a+b)_n.
• This expansion is an algebraic expansion for the powers of a binomial.
• The exponent value of the binomial theorem expansion might be a negative integer or a fraction.
• The coefficients in the binomial expansion of (a+b)_n are called binomial coefficients. For example, ⁿC₀, ⁿC₁, ⁿC₂ . . . . . . .ⁿC_n are some of the coefficients. 
• The binomial expansion will take a form like this– (a+b)ⁿ = ⁿC₀ aⁿ b⁰ + ⁿC₁ a^n-1b¹ + ⁿC₂ a^n-2 b² + ... + ⁿC_n-1 a¹b^n-1 + ⁿC_n a⁰bⁿ
• The universal formula for the binomial theorem is (a+b)ⁿ= ∑ⁿr=0ⁿC^r a^n-rb^r, where n is a positive integer and a, b are real numbers, and 0 < r ≤ n.