Binomial Theorem: Formula, Expansion, Pascal Triangle, and Examples

Q: Why is the Binomial Theorem useful?

It simplifies expanding large powers, helps in finding specific terms efficiently, and is used in algebra, probability, calculus, and combinatorics to solve problems involving exponentiation and coefficients.

Q: What happens to the powers of ‘a’ and ‘b’ in each term of the expansion?

The exponent of a starts at n and decreases to 0, while the exponent of b starts at 0 and increases to n, maintaining a sum of n in each term.

Q: Why is the Binomial Theorem important in probability?

The binomial theorem is used in binomial probability distributions, helping calculate the likelihood of different outcomes in repeated independent trials.

Q: What is the difference between binomial and multinomial expansion?

The binomial expansion applies to two terms, while the multinomial expansion extends the idea to three or more terms, using multinomial coefficients for each term’s distribution.

where:

n is a positive integer (the exponent).
a, b are real numbers.
the binomial coefficient , defined as:

The expansion contains (n + 1) terms, where each term follows the pattern
The powers of a decrease from n to 0, while the powers of b increase from 0 to n.

Example: Expand (2x - 3)⁵ using the Binomial Theorem

Let’s use the binomial theorem to expand (2x - 3)⁵ .

Solution:

Using the formula: Here,

Expanding step by step:

Breaking it down term by term:

Using binomial coefficients:

= 1 (32 x^{5}) (1) + 5 (16 x^{4}) (- 3) + 10 (8 x^{3}) (9) + 10 (4 x^{2}) (- 27) + 5 (2 x) (81) + 1 (1) (- 243)

3

Thus, the expansion of (2x-3)⁵ is:

This method can be applied to any binomial expression raised to a positive integer power.

Binomial Theorem Properties

The Binomial Theorem follows some key properties that make it easier to expand and analyze binomial expressions. Here’s a simple breakdown of these properties:

Number of Terms: In the expansion of (x + y)ⁿ, the total number of terms is (n + 1).

Example: (x + y)³ expands to x³ + 3x²y + 3xy² + y³ (4 terms because n + 1 = 4). First and Last Terms: The first term is always xⁿ , and the last term is always yⁿ .

For example:

In (x + y)⁵, the first term is x⁵, and the last term is y⁵. General Term : Any term in the expansion is found using

For example:

In (x + y)⁴, the 3rd term (when r = 2) is:

Values for Small n: When n is small, the expansions follow simple patterns:

Binomial Coefficients Symmetry: The coefficients are symmetrical:

Example:

In (x + y)⁵ , the coefficients 1, 5, 10, 10, 5, 1 are symmetric.

Middle Term(s):

If n is even , the middle term is

If n is odd , there are two middle terms , found at

Example:

If is even , there is one middle term at position:

Example: In $x + y)^4$ , the middle term is:

If is odd , there are two middle terms at positions:

Example: In , the middle terms are:

Pascal’s Triangle

Pascal’s Triangle provides a simple way to determine the coefficients in the expansion of (a + b)ⁿ . Each row of Pascal’s Triangle corresponds to the coefficients of a binomial expansion, where each number is the sum of the two directly above it.

For example, in (x + y)⁴ , the coefficients are 1, 4, 6, 4, 1 , which match the 5th row of Pascal’s Triangle:

To expand (x + y)⁴ , these coefficients are used to determine the terms:

Each coefficient in Pascal’s triangle can also be found using the combination formula : For (x + y)⁴ , calculating the coefficients: This confirms that the coefficients 1, 4, 6, 4, 1 appear in both Pascal’s triangle and the combination formula. For smaller values of n , Pascal’s triangle is an easy way to determine coefficients, while for larger n , the combination formula is more efficient.

Key Terms in the Binomial Theorem

The Binomial Theorem involves several important terms that help in understanding and applying its expansion. Here’s a explanation of these terms:

General Terms of Expansion

The general term in a binomial expansion represents any term in the expansion and helps in writing the full expression without expanding everything manually. It is given by: where r is the term position, starting from 0 to n . By substituting different values of r , we can obtain each term in the expansion. For example, in (x + y)⁵ , setting r = 2 gives the third term :

2. Middle Term of Expansion

The middle term is the one that appears at the center of the expansion. The number of terms in (x + y)ⁿ is n + 1 , so the middle term depends on whether n is even or odd.

If n is even, there is one middle term at position (n/2) + 1 .
If n is odd, there are two middle terms at positions (n+1)/2 and (n+3)/2 .

For example, in (x + y)⁶ , the middle term is 4th term

For (x + y)⁷ , the middle terms are 4th and 5th terms .

3. Finding a Specific Term in Expansion

To find a specific term, we use the general term formula and substitute the required position.

Example: Find the 6th term in (3x + 4)⁸ .

Using the general formula:

For the 6th term , r = 5 :

Calculating,

4. Term Independent of x

A term is independent of x if it does not contain x . To find it, we set the power of x to zero in the general term formula.

Example: Find the term independent of x in (2x + 1)⁸ .

Using the general term formula: For x to disappear , solve 8 - r = 0 , which gives r = 8 . Thus, the term independent of x is 1 .

Numerically Largest Term in Expansion

The largest term in the binomial expansion can be found using the formula:

This formula is useful when we need to determine the term with the highest numerical value in the expansion. These terms help in understanding the structure of binomial expansion and solving problems related to it efficiently.

Applications of the Binomial Theorem

The binomial theorem is useful in various mathematical calculations, especially when dealing with large numbers. Some common applications include:

Finding the Remainder When Dividing Large Numbers

The binomial theorem helps determine the remainder when a large number is divided by another number. Instead of performing lengthy calculations, we rewrite the number in a convenient form and expand it using the binomial theorem.

Example: Find the remainder when is divided by 15.

Solution:

First, express 297 as a sum that includes 15:

Now raise it to the power of 24 :

Using the binomial theorem, every term in the expansion except for 1²⁴ includes 15 , which is a multiple of 15. These terms do not contribute to the remainder. The only term left is 1²⁴ = 1 . Thus, the remainder when 297²⁴ is divided by 15 is 1 .

2. Finding the Last Digit of a Large Number

The last digit of a number can be determined by rewriting it in a form that makes expansion easier.

Example: Find the last digit of

Solution:

We rewrite 7¹⁰ as:

Now, we expand using the binomial theorem: Each term containing 50 is a multiple of 10 , meaning these terms will always end in 0 . The only term that affects the last digit is (-1)⁵ = -1 . Thus, the last digit of 7¹⁰ is 9 .

3. Checking Divisibility

The binomial theorem helps check whether a number is divisible by another number by expanding it into terms that reveal patterns in divisibility. T This is especially useful in modular arithmetic and number theory. For example, numbers in the form (a + b)ⁿ can be expanded, and if every term except one contains the divisor as a factor, the remainder can be easily determined.

The binomial theorem makes expanding expressions simple and efficient. Instead of multiplying repeatedly, it provides a formula to find each term easily. It is widely used in algebra, probability, and real-world calculations. Understanding this theorem helps solve problems faster and makes learning algebra easier.

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Why is the Binomial Theorem useful?

It simplifies expanding large powers, helps in finding specific terms efficiently, and is used in algebra, probability, calculus, and combinatorics to solve problems involving exponentiation and coefficients.

What happens to the powers of ‘a’ and ‘b’ in each term of the expansion?

The exponent of a starts at n and decreases to 0, while the exponent of b starts at 0 and increases to n, maintaining a sum of n in each term.

Why is the Binomial Theorem important in probability?

The binomial theorem is used in binomial probability distributions, helping calculate the likelihood of different outcomes in repeated independent trials.

What is the difference between binomial and multinomial expansion?

The binomial expansion applies to two terms, while the multinomial expansion extends the idea to three or more terms, using multinomial coefficients for each term’s distribution.

This method can be applied to any binomial expression raised to a positive integer power.

Binomial Theorem Properties

The Binomial Theorem follows some key properties that make it easier to expand and analyze binomial expressions. Here’s a simple breakdown of these properties:

Number of Terms: In the expansion of (x + y)ⁿ, the total number of terms is (n + 1).

Example: (x + y)³ expands to x³ + 3x²y + 3xy² + y³ (4 terms because n + 1 = 4). First and Last Terms: The first term is always xⁿ , and the last term is always yⁿ .

For example:

In (x + y)⁵, the first term is x⁵, and the last term is y⁵. General Term : Any term in the expansion is found using

For example:

In (x + y)⁴, the 3rd term (when r = 2) is:

Values for Small n: When n is small, the expansions follow simple patterns:

Binomial Coefficients Symmetry: The coefficients are symmetrical:

Example:

In (x + y)⁵ , the coefficients 1, 5, 10, 10, 5, 1 are symmetric.

Middle Term(s):

If n is even , the middle term is

If n is odd , there are two middle terms , found at

Example:

If is even , there is one middle term at position:

Example: In $x + y)^4$ , the middle term is:

If is odd , there are two middle terms at positions:

Example: In , the middle terms are:

Pascal’s Triangle

For example, in (x + y)⁴ , the coefficients are 1, 4, 6, 4, 1 , which match the 5th row of Pascal’s Triangle:

To expand (x + y)⁴ , these coefficients are used to determine the terms:

Key Terms in the Binomial Theorem

The Binomial Theorem involves several important terms that help in understanding and applying its expansion. Here’s a explanation of these terms:

General Terms of Expansion

2. Middle Term of Expansion

The middle term is the one that appears at the center of the expansion. The number of terms in (x + y)ⁿ is n + 1 , so the middle term depends on whether n is even or odd.

If n is even, there is one middle term at position (n/2) + 1 .
If n is odd, there are two middle terms at positions (n+1)/2 and (n+3)/2 .

For example, in (x + y)⁶ , the middle term is 4th term

For (x + y)⁷ , the middle terms are 4th and 5th terms .

3. Finding a Specific Term in Expansion

To find a specific term, we use the general term formula and substitute the required position.

Example: Find the 6th term in (3x + 4)⁸ .

Using the general formula:

For the 6th term , r = 5 :

Calculating,

4. Term Independent of x

A term is independent of x if it does not contain x . To find it, we set the power of x to zero in the general term formula.

Example: Find the term independent of x in (2x + 1)⁸ .

Using the general term formula: For x to disappear , solve 8 - r = 0 , which gives r = 8 . Thus, the term independent of x is 1 .

Numerically Largest Term in Expansion

The largest term in the binomial expansion can be found using the formula:

Applications of the Binomial Theorem

The binomial theorem is useful in various mathematical calculations, especially when dealing with large numbers. Some common applications include:

Finding the Remainder When Dividing Large Numbers

Example: Find the remainder when is divided by 15.

Solution:

First, express 297 as a sum that includes 15:

Now raise it to the power of 24 :

2. Finding the Last Digit of a Large Number

The last digit of a number can be determined by rewriting it in a form that makes expansion easier.

Example: Find the last digit of

Solution:

We rewrite 7¹⁰ as:

3. Checking Divisibility

Related Articles
Median of Triangle	Additive Inverse
Hyperbola	Differentiation
Indian Numeral System	Surface Area of Cylinder

Binomial Theorem: Formula, Expansion, Pascal Triangle, and Examples

Binomial Theorem Statement

Solution:

Binomial Expansion Formula

Solution:

Binomial Theorem Properties

Pascal’s Triangle

Key Terms in the Binomial Theorem

General Terms of Expansion

2. Middle Term of Expansion

3. Finding a Specific Term in Expansion

4. Term Independent of x

Numerically Largest Term in Expansion

Applications of the Binomial Theorem

Finding the Remainder When Dividing Large Numbers

2. Finding the Last Digit of a Large Number

3. Checking Divisibility

<strong style=

Why is the Binomial Theorem useful?

What happens to the powers of ‘a’ and ‘b’ in each term of the expansion?

Why is the Binomial Theorem important in probability?

What is the difference between binomial and multinomial expansion?

<strong style=

Why is the Binomial Theorem useful?

What happens to the powers of ‘a’ and ‘b’ in each term of the expansion?

Why is the Binomial Theorem important in probability?

What is the difference between binomial and multinomial expansion?

Binomial Theorem: Formula, Expansion, Pascal Triangle, and Examples

Binomial Theorem Statement

Solution:

Binomial Expansion Formula

Solution:

Binomial Theorem Properties

Pascal’s Triangle

Key Terms in the Binomial Theorem

General Terms of Expansion

2. Middle Term of Expansion

3. Finding a Specific Term in Expansion

4. Term Independent of x

Numerically Largest Term in Expansion

Applications of the Binomial Theorem

Finding the Remainder When Dividing Large Numbers

2. Finding the Last Digit of a Large Number

3. Checking Divisibility