Binomial Theorem

Binomial theorem primarily helps to find the expanded value of the algebraic expression of the form (x + y)ⁿ. Finding the value of (x + y)², (x + y)³, (a + b + c)² is easy and can be obtained by algebraically multiplying the number of times based on the exponent value. But finding the expanded form of (x + y)¹⁷ or other such expressions with higher exponential values involves too much calculation. It can be made easier with the help of the binomial theorem.

The exponent value of this binomial theorem expansion can be a negative number or a fraction. Here we limit our explanations to only non-negative values. Let us learn more about the terms, formula and the properties of coefficients in this binomial expansion article.

The first mention of the binomial theorem was in the 4th century BC by a famous Greek mathematician by name of Euclids. The binomial theorem states the principle for expanding the algebraic expression (x + y)ⁿ and expresses it as a sum of the terms involving individual exponents of variables x and y. Each term in a binomial expansion is associated with a numeric value which is called coefficient.

Statement: 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 = ⁿC₀ xⁿy⁰ + ⁿC₁ x^n-1y¹ + ⁿC₂ x^n-2 y² + ... + ⁿC_n-1 x¹y^n-1 + ⁿC_n x⁰yⁿ

where, n ≥ 0 is an integer and each ⁿC_k is a positive integer known as a binomial coefficient.

Note: When an exponent is zero, the corresponding power expression is 1. This multiplicative factor is often omitted from the term, therefore often the right hand side is directly written as ⁿC₀ xⁿ + .... This formula is also referred to as the binomial formula or the binomial identity. Using summation notation, the binomial theorem can be given as,

(x+y)ⁿ = ∑ⁿ_k=0ⁿC_k x^n-ky^k = ∑ⁿ_k=0ⁿC_k x^ky^n-k

Example: Let us expand (x+3)⁵ using the binomial theorem. Here y = 3 and n = 5. Substituting and expanding, we get:

(x+3)⁵ = ⁵C₀ x⁵3⁰ + ⁵C₁ x^5-13¹ + ⁵C₂ x^5-2 3² + ⁵C₃ x^5-3 3³ + ⁵C₄ x^5-4 3⁴ + ⁵C₅ x^5-5 3⁵

= x⁵ + 5 x⁴. 3 + 10 x³ . 9 + 10 x² . 27 + 5x .81 + 3⁵

= x⁵ + 15 x⁴ + 90x³ + 270 x² + 405 x + 243

The binomial theorem is also known as the binomial expansion which gives the formula for the expansion of the exponential power of a binomial expression. Binomial expansion of (x + y)ⁿ by using the binomial theorem is as follows,

(x+y)ⁿ = ⁿC₀ xⁿy⁰ + ⁿC₁ x^n-1y¹ + ⁿC₂ x^n-2 y² + ... + ⁿC_n-1 x¹y^n-1 + ⁿC_n x⁰yⁿ

The binomial theorem formula is used in the expansion of any power of a binomial in the form of a series. The binomial theorem formula is (a+b)ⁿ= ∑ⁿ_r=0ⁿC_r a^n-rb^r, where n is a positive integer and a, b are real numbers, and 0 < r ≤ n. This formula helps to expand the binomial expressions such as (x + a)¹⁰, (2x + 5)³, (x - (1/x))⁴, and so on. The binomial theorem formula helps in the expansion of a binomial raised to a certain power. Let us understand the binomial theorem formula and its application in the following sections.

The binomial theorem states: If x and y are real numbers, then for all n ∈ N,

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

⇒ (x + y)ⁿ = ∑ⁿ_k=0ⁿC_k x^n-ky^k

where, ⁿC_r = n! / [r! (n - r)!]

Let x, a, n ∈ N. Let us prove the binomial theorem formula through the principle of mathematical induction. It is enough to prove for n = 1, n = 2, for n = k ≥ 2, and for n = k+ 1.

It is obvious that (x +y)¹ = x +y and

(x +y)² = (x + y) (x +y)

= x² + xy + xy + y² (using distributive property)

= x² + 2xy + y²

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 ≥ 2.

Assuming (x + y)ⁿ = ∑ⁿ_r=0ⁿC_r x^n-ry^r,

(x + y)^k = ∑^k_r=0^kC_r x^k-ry^r

⇒ (x+y)^k = ^kC₀ x^ky⁰ + ^kC₁ x^k-1y¹ + ^kC₂ x^k-2 y² + ... + ^kC_r x^k-ry^r +....+ ^kC_k x⁰y^k

⇒ (x+y)^k = x^k + ^kC₁ x^k-1y¹ + ^kC₂ x^k-2 y² + ... + ^kC_r x^k-ry^r +....+ y^k

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

Now consider the expansion for n = k + 1.

(x + y) ^k+1 = (x + y) (x + y)^k

= (x + y) (x^k + ^kC₁ x^k-1y¹ + ^kC₂ x^k-2 y² + ... + ^kC_r x^k-ry^r +....+ y^k)

= x^k+1 + (1 + ^kC₁)x^ky + (^kC₁ + ^kC₂) x^k-1y² + ... + (^kC_r-1 + ^kC_r) x^k-r+1y^r + ... + (^kC_k-1 + 1) xy^k + y^k+1

= x^k+1 + ^k+1C₁x^ky + ^k+1C₂ x^k-1y² + ... + ^k+1C_r x^k-r+1y^r + ... + ^k+1C_k xy^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.

• The number of coefficients in the binomial expansion of (x + y)ⁿ is equal to (n + 1).
• There are (n+1) terms in the expansion of (x+y)ⁿ.
• The first and the last terms are xⁿ and yⁿ respectively.
• From the beginning of the expansion of (x + a)ⁿ, the powers of x, decrease from n up to 0, and the powers of a, increase from 0 up 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-ry^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 (x + y)ⁿ, the r^th term from the end is (n – r + 2)^th term from the beginning.
• If n is even, then in (x + y)ⁿ the middle term = (n/2)+1 and if n is odd, then in (x + y)ⁿ, the middle terms are (n+1)/2 and (n+3)/2.

The binomial coefficients are the numbers associated with the variables x, and y in the expansion of (x + y)ⁿ. The binomial coefficients are represented as ⁿC₀, ⁿC₁, ⁿC₂ ..... The binomial coefficients are obtained through the pascal triangle or by using the combinations formula.

Binomial Theorem Coefficients

The values of the binomial coefficients exhibit a specific trend which can be observed in the form of Pascal's triangle. Pascal's triangle is an arrangement of binomial coefficients in triangular form. It is named after the French mathematician Blaise Pascal. The numbers in Pascal's triangle have all the border elements as 1 and the remaining numbers within the triangle are placed in such a way that each number is the sum of two numbers just above the number.

Combinations

The formula for combinations is used to find the value of the binomial coefficients in the expansions using the binomial theorem. The combinations in this case are the different ways of picking r variables from the available n variables. The formula to find the combinations of r objects taken from n different objects is ⁿC_r = n! / [r! (n - r)!]. Here the coefficients have the following properties.

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

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

• C₁ + C₂ + C₃ + C₄ + .......C_n = 2ⁿ
• C₀ + C₂ + C₄ + .... = C₁ + C₃ + C₅ + ....... = 2^n-1
• C₀ - C₁ + C₂ - C₃ + C₄ - C₅ + .... = 0
• C₁ + 2C₂ + 3C₃ + 4C₄ + .......nC_n = n2^n-1
• C₁ - 2C₂ + 3C₃ - 4C₄ + .......(-1)ⁿnC_n = 0
• C₁² + C₂² + C₃² + C₄² + .......C_n² = (2n)! / (n!)²

The following terms related to binomial expansion using the binomial theorem are helpful to find the terms. The details of each of the terms are as follows.

General Term: This term symbolizes all of the terms in the binomial expansion of (x + y)ⁿ. The general term in the binomial expansion of (x + y)ⁿ is T_r+1 = ⁿC_r x^n-ry^r. Here the r-value is one less than the number of the term of the binomial expansion. Also, ⁿC_r is the coefficient, and the sum of the exponents of the variables x and y is equal to n.

Middle Term: The total number of terms in the expansion of (x + y)ⁿ is equal to n + 1. The middle term in the binomial expansion depends on the value of n. The middle term and the number of middle terms depend on the value of n: n is even or odd. For an even value of n there is only one middle term and (n/2 + 1)^th term is the middle term. For an odd value of n, there are two middle terms, and the two middle terms are n/2, and n/2 + 1 terms.

Identifying a Particular Term: There are two simple steps to identify a particular term containing x^p. First, we need to find the general term in the expansion of (x + y)ⁿ. which is is T_r+1 = ⁿC_r x^n-ry^r. Secondly, we need to compare this with x^p to obtain the r-value. Here the r-value is helpful to find the particular term in the binomial expansion. Let us find the fifth term in the expansion of (2x + 3)⁹ using the binomial theorem.

The formula to find the n^th term in the binomial expansion of (x + y)ⁿ is T_r+1 = ⁿC_r x^n-ry^r.

Applying this to (2x + 3)⁹ , T₅ = T₄₊₁ = ⁹C₄ (2x)^9-43⁴. Thus the 5th term is = ⁹C₄ (2x)⁵3⁴

Term Independent of X: The steps to find the term independent of x is similar to finding a particular term in the binomial expansion. First, we need to find the general term in the expansion of (x + y)ⁿ. which is T_r+1 = = ⁿC_r x^n-ry^r. Here to find the term independent of x, we need to find and equalize the exponent of x in the general term to zero. From the obtained value of r, the term independent of x is (r + 1)^th term.

Numerically Greatest Term: The formula to find the numerically greatest term in the expansion of (1 + x)ⁿ is [(n+1) |x|] / (1 + |x|). There are two points to be remembered while using this formula to find the numerically greatest term. First, we need to convert any binomial expansion into the form of (1 + x)ⁿ. We can convert (2x + 3y)⁵ to (1 + 3y/2x)⁵ . Further for this expansion |x| is the numeric value and is equal to 3/2 in this given example. The final answer is rounded to the integral value to obtain the numerically greatest term.

The binomial theorem expansion also applies to exponents with negative values. The standard coefficient values of binomial expansion for positive exponents are the same for the expansion with the negative exponents. The terms and the coefficient values remain the same, but the algebraic relationship between the terms varies in the binomial expansion of negative exponents.

• (1 + x)^-1 = 1 - x + x² - x³ + x⁴ - x⁵ + .......
• (1 - x)^-1 = 1 + x + x² + x³ + x⁴ + x⁵ + .......
• (1 + x)^-2 = 1 - 2x + 3x² - 4x³ + ........
• (1 - x)^-2 = 1 + 2x + 3x² + 4x³ + ........
• (1 + x)^-3 = 1 - 3x + 6x² - 10x³ + 15x⁴ + ......
• (1 - x)^-3 = 1 + 3x + 6x² + 10x³ + 15x⁴ + ......

☛ Related Articles:

• Algebraic Identities
• Binomial Distribution Formula
• Binomial Distribution
• Probability and Statistics
• Geometric Distribution Formula

Important Notes on Binomial Theorem

The following key points would help in a better understanding of the binomial theorem.

• The number of terms in the binomial expansion of (x + y)ⁿ is equal to n + 1.
• In the expansion of (x + y)ⁿ, the sum of the powers of x and y in each term is equal to n.
• The value of the binomial coefficients from either side of the expansion is equal.
• The number of terms in the binomial expansion of (x + y + z)ⁿ is n(n + 1).

What Is the Binomial Theorem?

The binomial theorem is used for the expansion of the algebraic terms of the form(x + y)ⁿ . (x+y)ⁿ = ⁿC₀ xⁿy⁰ + ⁿC₁ x^n-1y¹ + ⁿC₂ x^n-2 y² + ... + ⁿC_n-1 x¹y^n-1 + ⁿC_n x⁰yⁿ. Here the number of terms in the binomial expansion having an exponent of n is n + 1. The exponent of the first term in the expansion is decreasing and the exponent of the second term in the expansion is increasing in a progressive manner. The coefficients of the binomial expansion can be found from the pascals triangle or using the combinations formula of ⁿC_r = n! / [r! (n - r)!].

What Is the Constant Term in the Binomial Theorem?

The constant term in the binomial expansion is a numeric value and is independent of the variables. For a binomial expansion of (x + y)ⁿ the term independent of x can be calculated by finding the term independent of x.

What Is the Coefficient in the Binomial Theorem?

The coefficients in the binomial expansion are ⁿC₀, ⁿC₁, ⁿC₂, ....ⁿC_r ......ⁿC_n. The coefficient value for n = 4 is equal to 1, 4, 6, 4,1, and the coefficient value for n = 5 is 1, 5, 10, 20, 10, 5, 1. The coefficient values can be found either from the pascals triangle or by using the combinations formula of ⁿC_r = n! / [r! (n - r)!].

Where Is Binomial Theorem Used?

The binomial theorem is useful to do the binomial expansion and find the expansions for the algebraic identities. Further, the binomial theorem is also used in probability for binomial expansion. A few of the algebraic identities derived using the binomial theorem are as follows.

• (a + b)² = a² + 2ab + b²
• (a - b)² = a² - 2ab + b²
• (a + b)(a - b) = a² - b²
• (a + b)³ = a³ +3a²b + 3ab² + b³
• (a - b)³ = a³ - 3a²b + 3ab² - b³
• (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ac

What Is the General Term in Binomial Theorem?

The general term of the binomial expansion of (x + y)ⁿ is T_r+1 = ⁿC_r x^n-ry^r.. The r-value for the term is one less than the number of the term. In the general term, the sum of the exponents of both the terms is equal to n. Further, the general term is helpful to derive the independent term, the numerically greatest term, and the term with particular power.

How to Find Terms in Binomial Theorem?

The number of terms in a binomial expansion with an exponent of n is equal to n + 1. Further to find a particular term in the expansion of (x + y)ⁿ we make use of the general term formula. The general term of the binomial expansion is T_r+1 = ⁿC_r x^n-ry^r. . Here the coefficient values are found from the pascals triangle or using the combinations formula, and the sum of the exponents of both the terms in the general term is equal to n.

How Do You Find the Number of Terms in the Expansion using the Binomial Theorem Formula?

The number of terms in the binomial expansion of (x+a)ⁿ is n+1 terms. i.e. add 1 to the power, the binomial coefficient is raised to. If we expand (2x+5)¹⁰, we will have 10 +1 = 11 terms on expansion.

How to Find Number of Terms in Binomial Theorem?

The number of terms in the binomial expansion is one more than the exponent value. The number of terms in the expansion of (x + y)ⁿ is equal to (n + 1).

What Is the Middle Term in Binomial Theorem?

The middle term in the binomial expansion depends on the value of n. The middle term and the number of middle terms depend on the value of n: n is even or odd. For an even value of n there is only one middle term and (n/2 + 1) term is the middle term. For an odd value of n, there are two middle terms, and the two middle terms are n/2, and n/2 + 1 terms.

What Is the General Form of Binomial Theorem Formula?

The general formula of binomial theorem formula is used in the expansion of binomial expression of the form:

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

• x, y, n are natural numbers, and,
• 0 < k ≤ n

Where Is Binomial Theorem Formula Used?

The binomial theorem formula is used to expand the binomial expressions. We apply the formula in finding probability, combinatorics, calculus, and in other important areas of math. For example, (101)⁵ =(100+1)⁵ = 100⁵+ 5 × 100⁴ + 10 × 100³+ 10 × 100²+ 5 × 100 + 1 = 10,000,000,000+ 500,000,000 + 10,000,000 + 100,000 + 500 +1 = 10,510,100,501

What Is n and r in The Binomial Theorem Formula?

In the binomial theorem formula of expansion (x+a)ⁿ, we use the combinatorics formula that is denoted as ⁿC_r, where n is the exponent in the expansion and r is the term number that ranges from 0 to n.