Binomial Theorem Formula: Explanation, Terms, Problems

Binomial Theorem Formula – As the exponent grows, computing the expansion becomes increasingly intricate and laborious. The Binomial Theorem offers a convenient method for efficiently calculating the expansion of a binomial expression raised to a substantial power. Here, you'll gain insight into the binomial theorem's definition and statement, explore binomial expansion formulas, examine the properties of the binomial theorem, discover techniques for determining binomial coefficients, delve into the components of the binomial expansion, and explore its various applications.

Binomial Theorem Explanation

The binomial theorem is a mathematical technique used for expanding expressions that have been raised to any positive integer power. This theorem serves as a versatile and valuable tool for expansion, finding applications across various fields, including algebra and probability.

Binomial Expressions Defined

A binomial expression, in algebra, is an expression characterized by having exactly two dissimilar terms. Examples of such expressions include a + b, a ³ + b ³ , and so forth.

Binomial Expansion

The total number of terms in the expansion of (x + y) ⁿ is always (n + 1).

The sum of the exponents of x and y in each term of the expansion is always equal to n.

The binomial coefficients, denoted as nC ₀ , nC ₁ , nC ₂ and so on (also represented as C ₀ , C ₁ , C _2, and so forth), play a significant role in the expansion.

Binomial coefficients that are symmetrically positioned in the expansion, both from the beginning and the end, are equal. For example, nC ₀ = nC _n , nC ₁ = nC _n-1 , nC ₂ = nC _n-2 , and so on.

Binomial Expansion Formula: Let n be a non-negative integer, and x, y be real numbers. The binomial expansion formula for (x + y) ⁿ is given by:

(x + y) ⁿ = ⁿ Σ _r=0 nC _r x ^{n – r} · y ^r

Where:

nCr represents the binomial coefficient, which is also expressed as Cn,r.

x and y are real numbers.

The sum is taken from r = 0 to n, covering all terms in the expansion.

Binomial Expansion Example

1. Expanding a Binomial Expression:

Expand (a + b)^4.

Using the Binomial Theorem, we get:

(a + b)^4 = C(4,0) * a^4 * b^0 + C(4,1) * a^3 * b^1 + C(4,2) * a^2 * b^2 + C(4,3) * a^1 * b^3 + C(4,4) * a^0 * b^4

Simplifying this expression gives the expanded form:

(a^4) + 4(a^3)(b) + 6(a^2)(b^2) + 4(a)(b^3) + (b^4)

2. Finding Specific Terms:

Determine the coefficient of the term containing x^3 in the expansion of (2x + 3)^5.

Using the Binomial Theorem, we can focus on the term with x^3:

C(5, k) * (2x)^(5-k) * (3)^k

To get x^3, we set (5 - k) equal to 3, so k = 2. Plug this into the expression:

C(5, 2) * (2x)^(5-2) * (3)^2 = C(5, 2) * (2^3x^3) * 9

C(5, 2) is the binomial coefficient, equal to 10:

10 * 8x^3 * 9 = 720x^3

So, the coefficient of the x^3 term is 720.

3. Finding the Middle Term:

Find the middle term in the expansion of (x + y)^6.

To find the middle term, we first calculate the total number of terms in the expansion using (n + 1), where n = 6:

Total terms = 6 + 1 = 7 terms

The middle term is the fourth term, as there are 3 terms on each side:

(x + y)^6 = C(6, 3) * x^3 * y^3

C(6, 3) is the binomial coefficient, equal to 20:

20 * x^3 * y^3

So, the middle term in the expansion is 20x^3y^3.

These examples illustrate how the Binomial Theorem can be used to expand expressions, find specific terms, and locate the middle term in a binomial expansion.

Also Check – Cubes Roots Formula

Binomial Expansion Formulas

The Binomial Expansion Formula allows you to expand a binomial expression raised to a positive integer power (n). Here is the general formula:

(x + y)^n = Σr=0 to n [nCr * x^(n-r) * y^r]

In this formula:

(x + y)^n represents the binomial expression to be expanded.

n is a non-negative integer, which is the exponent to which the binomial expression is raised.

nCr, also known as binomial coefficient or combination, is calculated as C(n, r) and represents the number of ways to choose r items from a set of n items. It can be computed as C(n, r) = n! / (r! * (n - r)!), where! denotes factorial.

x^(n-r) represents x raised to the power of (n - r).

y^r represents y raised to the power of r.

The summation Σr=0 to n indicates that you need to sum up terms for all values of r from 0 to n.

Using this formula, you can expand any binomial expression (x + y)^n to find the individual terms in the expansion. These terms are often referred to as "binomial coefficients" multiplied by powers of x and y.

Here are some common binomial expansion formulas for specific values of n:

Square of a Binomial (n = 2):

(x + y)^2 = x^2 + 2xy + y^2

Cubing a Binomial (n = 3):

(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3

Expanding a Binomial to the Fourth Power (n = 4):

(x + y)^4 = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4

These are some basic examples, and you can use the general binomial expansion formula to calculate expressions for any positive integer value of n. The coefficients in these formulas are determined by binomial coefficients (nCr), which describe the combinations of terms needed to expand the binomial expression.

Some Other Useful Expansions

Sum and Difference of Binomial Expansions:

The sum of (x + y)^n and (x - y)^n is equal to 2 times the series with coefficients C0 * x^n + C2 * x^(n-1) * y^2 + C4 * x^(n-4) * y^4 + ...

The difference between (x + y)^n and (x - y)^n is equal to 2 times the series with coefficients C1 * x^(n-1) * y + C3 * x^(n-3) * y^3 + C5 * x^(n-5) * y^5 + ...

Binomial Expansion of (1 + x)^n:

The expansion of (1 + x)^n is given by nΣr=0 [nCr * x^r], which can be expressed as C0 + C1 * x + C2 * x^2 + ... + Cn * x^n.

Sum and Difference of (1 + x)^n:

The sum of (1 + x)^n and (1 - x)^n is equal to 2 times the series with coefficients C0 + C2 * x^2 + C4 * x^4 + ...

The difference between (1 + x)^n and (1 - x)^n is equal to 2 times the series with coefficients C1 * x + C3 * x^3 + C5 * x^5 + ...

Number of Terms in (x + a)^n + (x - a)^n:

If "n" is even, the number of terms in the expansion of (x + a)^n + (x - a)^n is (n + 2)/2.

If "n" is odd, the number of terms in the expansion of (x + a)^n + (x - a)^n is (n + 1)/2.

Number of Terms in (x + a)^n - (x - a)^n:

If "n" is even, the number of terms in the expansion of (x + a)^n - (x - a)^n is n/2.

If "n" is odd, the number of terms in the expansion of (x + a)^n - (x - a)^n is (n + 1)/2.

These rephrased statements provide a clearer understanding of the properties and results related to binomial expansions.

Also Check – Rational Number Formula

Terms in the Binomial Expansion

General Term: The expression that represents any term within the binomial expansion.

In the binomial expansion of (x + y)^n, each term is represented by the general term Tr+1, which can be expressed as Tr+1 = nCr * x^(n-r) * y^r.

General Term in (1 + x)^n:

For the binomial expansion of (1 + x)^n, the general term is simply nCr * x^r.

Position of the rth Term in (x + y)^n:

In the binomial expansion of (x + y)^n, the term positioned as the rth term from the end corresponds to the (n - r + 2)th term in the expansion.

Middle Term: The term located at the center of the expansion, especially crucial when the number of terms is odd.

In cases where "n" represents an even number, the middle term corresponds to the term located at position (n/2 + 1) within the binomial expansion.

However, when "n" is an odd number, the middle terms are found at positions [(n+1)/2] and [(n+3)/2] within the expansion.

Independent Term: The term that does not have either of the variables (x or y) and often corresponds to the constant term.

The term that is independent of 'x' in the expansion of [ax^p + (b/x^q)]^n is represented by Tr+1 = nCr * a^(n-r) * b^r, where 'r' is an integer determined by the formula r = (np)/(p + q).

Determining a Particular Term: The process of finding a specific term in the expansion, often guided by the term's position or power.

In the expansion of (ax^p + b/x^q)^n, the coefficient of x^m is determined by the coefficient of Tr+1, where the value of r is calculated as [(np - m)/(p + q)].

In the expansion of (x + a)^n, the ratio of the coefficient of the (r+1)th term (Tr+1) to the coefficient of the rth term (Tr) is given by (n - r + 1)/r times the constant term 'a' divided by 'x'.

Numerically Greatest Term: The term with the highest absolute value or magnitude among all the terms in the expansion.

If [(n+1)|x|]/[|x|+1] = P, where P is a positive integer, then the Pth term and (P+1)th terms are the numerically greatest terms in the expansion of (1+x)^n.

However, if [(n+1)|x|]/[|x|+1] = P + F, where P is a positive integer, and 0 < F < 1, then only the (P+1)th term is numerically the greatest term in the expansion of (1+x)^n.

Ratio of Consecutive Terms/Coefficients: The relationship between consecutive terms, which can provide insights into the behavior and convergence of the expansion.

The coefficient of xr in the expansion is nCr, while the coefficient of xr + 1 is nCr + 1.

The relationship between these coefficients can be expressed as (nCr / nCr - 1) = (n - r + 1) / r.

These terms and concepts are essential when working with binomial expansions, allowing for precise calculations and a deeper understanding of the expansion's properties.

Applications of Binomial Theorem

The Binomial Theorem finds a wide array of applications in mathematics, including techniques for:

1. Finding Remainders Using Binomial Theorem:

For instance, to find the remainder when 7103 is divided by 25, you can utilize the Binomial Theorem to express 7103 as a binomial power of 7 and then simplify it to determine the remainder, which is 18.

Similarly, in the context of the fractional part of (2403 / 15), the Binomial Theorem aids in expressing the fraction in a simplified form, helping you find the value of K, which is 8 in this case.

2. Finding Digits of a Number:

The Binomial Theorem can be employed to extract the last two digits of a number, as demonstrated with (13)10, yielding the result 49.

3. Relation between Two Numbers:

In scenarios where you need to compare or establish a relationship between two numbers like 9950 + 10050 and 10150, the Binomial Theorem can be employed to demonstrate the inequality and provide a clear understanding of the relation.

4. Divisibility Tests:

To determine the divisibility of numbers, the Binomial Theorem can be applied. For example, it can be used to illustrate that 119 + 911 is divisible by 10.

Formulae:

The Binomial Theorem is accompanied by several formulae, such as the formula for the number of terms in the expansion of (x1 + x2 + … + xr)^n, which is (n + r - 1)Cr - 1. Additionally, the formula for finding the sum of the coefficients of (ax + by)^n is (a + b)^n.

Further, if you have a polynomial function f(x) = (a0 + a1x + a2x^2 + …. + amx^m)^n, the Binomial Theorem provides formulas for calculating the sum of coefficients and the sum of coefficients of even and odd powers of x.

These applications and formulae demonstrate the versatility and utility of the Binomial Theorem in solving various mathematical problems and exploring relationships between numbers and polynomials.

Also Check – Linear Equation Formula

Problems on Binomial Theorem

Question 1:

Find the positive value of λ for which the coefficient of x^2 in the expression x^2[√x + (λ/x^2)]^10 is 720.

Solution: In the expression x^2[√x + (λ/x^2)]^10, we aim to determine λ. By applying the Binomial Theorem and simplifying, we conclude that λ^2 = 16, yielding two possible solutions: λ = ±4. However, since we seek a positive value, λ = 4.

Question 2:

The sum of the real values of x for which the middle term in the binomial expansion of (x^3/3 + 3/x)^8 equals 5670 is?

Solution: To find the sum of real values of x for which the middle term equals 5670, we first identify the middle term (T5) and then solve for x. The solutions are x = ±√3. Therefore, the sum of these real values is zero.

Question 3:

Let (x + 10)^50 + (x – 10)^50 = a0 + a1x + a2x^2 + ... + a50x^50 for all x ∈R, then a2/a0 is equal to?

Solution: In the expression (x + 10)^50 + (x – 10)^50, we aim to find a2/a0. After calculation, we determine that a2/a0 = 12.25.

Question 4:

Find the coefficient of x^9 in the expansion of (1 + x)(1 + x^2)(1 + x^3) ... (1 + x^100).

Solution: To find the coefficient of x^9, we consider the ways x^9 can be formed. There are eight such ways, each contributing a coefficient of 1. Therefore, the coefficient of x^9 is 8.

Question 5:

The coefficients of three consecutive terms of (1 + x)^n+5 are in the ratio 5:10:14. Find n.

Solution: By considering the ratio of coefficients, we derive two equations involving 'n' and 'r.' Solving these equations, we find n = 6.

Question 6: Find the digit in the units place of the number 183! + 3183.

Solution: By analyzing the unit digits of 183! and 3183, we determine that the units digit of their sum is 7.

Question 7: Find the total number of terms in the expansion of (x + a)^100 + (x – a)^100.

Solution: The total number of terms in the expansion of (x + a)^100 + (x – a)^100 is 51.

Question 8: Find the coefficient of t^4 in the expansion of [(1-t^6)/(1 – t)].

Solution: To find the coefficient of t^4 in [(1-t^6)/(1 – t)], we expand the expression and extract the coefficient, which is 15.