Binomial Expansion

About Binomial Expansion

The binomial expansion, commonly known as the binomial theorem, is a formula for expanding the exponential power of a binomial expansion.

The following is the binomial expansion of (x + y)ⁿ using the binomial theorem:,

(x + y)ⁿ = ⁿC₀xⁿy⁰+ⁿC₁x^n-1y¹+.......+ⁿC_n-1x¹y^{n - 1}+ⁿC_nx⁰yⁿ

Binomial Theorem Formula

In order to expand any binomial power into a series, the binomial theorem formula is needed. The formula for the binomial theorem is (a+b)ⁿ = ⁿC_r a^n-rb^r, where n is a positive integer and a, b are real numbers, and 0 < r ≤ n. This formula can be used to expand binomial expressions like (x + a)¹⁰, (2x + 5)³, (x - (1/x))⁴, and so on. The formula for the binomial theorem aids in the expansion of a binomial raised to a particular power. If x and y are both real numbers, then the binomial theorem holds for all n ∈ N,

(x + y)ⁿ = ⁿc₀xⁿy⁰ + ⁿc₁x^n-1y¹ + ⁿc₂x^n-2y²-------+ⁿc_n-1x¹y^n-1 + ⁿc_nx⁰y⁰

⇒ (x + y)ⁿ = ⁿC_k x^{n - k}y ^k

where, ⁿC_r = n! / [r! (n - r)!]

Binomial Theorem Expansion Proof

Let x, a, n ∈ N. Let us use the idea of mathematical induction to show the binomial theorem formula. It is sufficient to demonstrate for n = 1, n = 2, and n = 3, for n = k ≥ 2, and for n = k+ 1.

obviously (x +y)¹ = x +y and (x +y)² = (x + y) (x +y) = x² + xy + xy + y² = x² + 2xy + y²

As a result, the result holds for both n = 1 and n = 2. Make k a positive number. Let’s prove that the result is true for k ≥ 2.

1. By assuming (x + y)n = ⁿC_r x^n-ry^r
2. (x + y)^k = ^kC_rx^n-ry^r
3. (x+y)k = ^kC₀ x^ky⁰ +^kC₁x^k-1y¹ + ^kC₂x^k-2y²+ ... + ^kC_rx^k-ry^r +....+ ^kC_kx^k-ky^k

Hence, this result is true for n = k ≥ 2.

Now by considering the expansion also for n = k + 1.

1. (x + y) ^k+1 = (x + y) (x + y)^k
2. =(x + y) (x^k + ^kC₁ x^k-1y¹ + ^kC₂ x^k-2y²+ ... + ^kC^r x^k-ry^r +....+ y^k)
3. = x^k+1 + ^k+1C₁x^ky + ^k+1C₂x^k-1y² + ... + ^k+1C_rx^k-ry^r + …... + ^k+1C_kxy^k + y^k+1 [Because ⁿC_r + ⁿC_r-1 = ⁿ⁺¹C_r]

Thus the result is true for n = k+1. By mathematical induction, this result is true for all positive integers 'n'. Hence proved.To get all the Maths formulas check out the main page. 

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