Binomial Expansion of \((a + b)^n\) and Binomial Coefficients

1. Binomial Expansion of \((a + b)^n\)

The binomial expansion is a method to expand expressions of the form \((a + b)^n\), where \(n\) is a non-negative integer.

The formula for the binomial expansion is given by:

\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]

where \(\binom{n}{k}\) is known as the binomial coefficient, defined as:

\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]

Example

Expand \((a + b)^4\):

Using the binomial expansion formula, we get:

\[ (a + b)^4 = \binom{4}{0} a^4 b^0 + \binom{4}{1} a^3 b^1 + \binom{4}{2} a^2 b^2 + \binom{4}{3} a^1 b^3 + \binom{4}{4} a^0 b^4 \]

Simplifying each term:

\[ = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4 \]

Thus, the expanded form of \((a + b)^4\) is \(a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4\).

2. Binomial Coefficients

Binomial coefficients are the coefficients in the expansion of \((a + b)^n\). They are represented as \(\binom{n}{k}\), where \(k\) ranges from 0 to \(n\).

The general formula for the binomial coefficient is:

\[ \binom{n}{k} = \frac{n!}{k!(n - k)!} \]

Properties of Binomial Coefficients

• Symmetry: \(\binom{n}{k} = \binom{n}{n-k}\)
• Recursive Relation: \(\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}\)

Example

Calculate the binomial coefficients for \((a + b)^5\):

We have \(n = 5\), so the coefficients are:

\[ \binom{5}{0} = 1, \quad \binom{5}{1} = 5, \quad \binom{5}{2} = 10, \quad \binom{5}{3} = 10, \quad \binom{5}{4} = 5, \quad \binom{5}{5} = 1 \]

These coefficients are used to expand \((a + b)^5\) as follows:

\[ (a + b)^5 = a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 + 5ab^4 + b^5 \]

Applications of Binomial Expansion

The binomial expansion is useful in various fields of mathematics, including probability, algebra, and calculus. It provides a way to express powers of binomials without directly multiplying them.

Pascal's Triangle

Introduction to Pascal's Triangle

Pascal's Triangle is a triangular array of numbers where each entry is the sum of the two entries directly above it. The triangle starts with a 1 at the top, and each row corresponds to the coefficients in the expansion of a binomial expression.

Structure of Pascal's Triangle

The first few rows of Pascal's Triangle are:

\[ \begin{array}{cccccccc} & & & & 1 & & & \\ & & & 1 & & 1 & & \\ & & 1 & & 2 & & 1 & \\ & 1 & & 3 & & 3 & & 1 \\ 1 & & 4 & & 6 & & 4 & & 1 \\ & \dots & \end{array} \]

Each row in Pascal's Triangle represents the coefficients of the binomial expansion for increasing powers of \( (a + b)^n \). For example, the fourth row (1, 4, 6, 4, 1) corresponds to the expansion of \( (a + b)^4 \).

Properties of Pascal's Triangle

• **Symmetry**: Pascal's Triangle is symmetric. Each row is a mirror image of itself.
• **Binomial Coefficients**: The entries in Pascal's Triangle are the binomial coefficients. The \( k \)-th entry in the \( n \)-th row is given by \(\binom{n}{k}\):

  \[ \binom{n}{k} = \frac{n!}{k!(n - k)!} \]

• **Sum of Rows**: The sum of the elements in the \( n \)-th row is \( 2^n \). For example, the sum of the elements in the third row (1, 3, 3, 1) is 8, which is \( 2^3 \).
• **Powers of 11**: Each row can represent the powers of 11. For example, the fourth row (1, 4, 6, 4, 1) corresponds to \( 11^4 = 14641 \).
• **Fibonacci Sequence**: The sums of the diagonals of Pascal's Triangle correspond to the Fibonacci sequence. For example:

  \[ 1, 1, 2, 3, 5, 8, 13, \dots \]

Using Pascal's Triangle for Binomial Expansion

Each row of Pascal's Triangle gives the coefficients for the expansion of \((a + b)^n\). For example, to expand \( (a + b)^3 \), use the fourth row of Pascal's Triangle (1, 3, 3, 1):

\[ (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \]

Example: Expanding \( (a + b)^5 \) Using Pascal's Triangle

To expand \( (a + b)^5 \), use the sixth row of Pascal's Triangle (1, 5, 10, 10, 5, 1):

\[ (a + b)^5 = a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 + 5ab^4 + b^5 \]

Solutions to Pascal's Triangle Expansion Problems

Question (a): Expand \( (3x + 2)^3 \)

Using Pascal's Triangle, the coefficients for expanding \( (a + b)^3 \) are 1, 3, 3, and 1.

The expansion of \( (3x + 2)^3 \) is:

\[ (3x + 2)^3 = 1 \cdot (3x)^3 + 3 \cdot (3x)^2 \cdot 2 + 3 \cdot (3x) \cdot 2^2 + 1 \cdot 2^3 \]

Simplifying each term:

\[ = 27x^3 + 54x^2 + 36x + 8 \]

Question (b): Expand \( (5 - 2x)^4 \)

Using Pascal's Triangle, the coefficients for expanding \( (a + b)^4 \) are 1, 4, 6, 4, and 1.

The expansion of \( (5 - 2x)^4 \) is:

\[ (5 - 2x)^4 = 1 \cdot 5^4 + 4 \cdot 5^3 \cdot (-2x) + 6 \cdot 5^2 \cdot (-2x)^2 + 4 \cdot 5 \cdot (-2x)^3 + 1 \cdot (-2x)^4 \]

Simplifying each term:

\[ = 625 - 500x + 150x^2 - 20x^3 + 16x^4 \]

Question (c): Expand \( (1 - 2x)^5 \)

Using Pascal's Triangle, the coefficients for expanding \( (a + b)^5 \) are 1, 5, 10, 10, 5, and 1.

The expansion of \( (1 - 2x)^5 \) is:

\[ (1 - 2x)^5 = 1 \cdot 1^5 + 5 \cdot 1^4 \cdot (-2x) + 10 \cdot 1^3 \cdot (-2x)^2 + 10 \cdot 1^2 \cdot (-2x)^3 + 5 \cdot 1 \cdot (-2x)^4 + 1 \cdot (-2x)^5 \]

Simplifying each term:

\[ = 1 - 10x + 40x^2 - 80x^3 + 80x^4 - 32x^5 \]

Question (d): Find the coefficient of \( x^3 \) in the expansion of \( (3 + 5x)(1 - 2x)^5 \)

First, expand \( (1 - 2x)^5 \) using Pascal's Triangle as done in part (c):

\[ (1 - 2x)^5 = 1 - 10x + 40x^2 - 80x^3 + 80x^4 - 32x^5 \]

Now, multiply by \( (3 + 5x) \):

\[ (3 + 5x)(1 - 10x + 40x^2 - 80x^3 + 80x^4 - 32x^5) \]

We are only interested in the terms that will result in \( x^3 \):

• From \( 3 \cdot (-80x^3) = -240x^3 \)
• From \( 5x \cdot 40x^2 = 200x^3 \)

Add these terms to find the coefficient of \( x^3 \):

\[ -240 + 200 = -40 \]

Thus, the coefficient of \( x^3 \) is \( -40 \).

Solutions to Pascal's Triangle Expansion Problems

1. Use Pascal's triangle to find the expansions of:

(a) \( (x + 2)^3 \)

Using Pascal's Triangle, the coefficients for expanding \( (a + b)^3 \) are 1, 3, 3, and 1.

The expansion of \( (x + 2)^3 \) is:

\[ (x + 2)^3 = 1 \cdot x^3 + 3 \cdot x^2 \cdot 2 + 3 \cdot x \cdot 2^2 + 1 \cdot 2^3 \]

Simplifying each term:

\[ = x^3 + 6x^2 + 12x + 8 \]

(b) \( (1 - x)^4 \)

Using Pascal's Triangle, the coefficients for expanding \( (a + b)^4 \) are 1, 4, 6, 4, and 1.

The expansion of \( (1 - x)^4 \) is:

\[ (1 - x)^4 = 1 \cdot 1^4 - 4 \cdot 1^3 \cdot x + 6 \cdot 1^2 \cdot x^2 - 4 \cdot 1 \cdot x^3 + x^4 \]

Simplifying each term:

\[ = 1 - 4x + 6x^2 - 4x^3 + x^4 \]

(c) \( (x + y)^3 \)

Using Pascal's Triangle, the coefficients for expanding \( (a + b)^3 \) are 1, 3, 3, and 1.

The expansion of \( (x + y)^3 \) is:

\[ (x + y)^3 = 1 \cdot x^3 + 3 \cdot x^2 \cdot y + 3 \cdot x \cdot y^2 + y^3 \]

Simplifying each term:

\[ = x^3 + 3x^2y + 3xy^2 + y^3 \]

(d) \( (2 - x)^3 \)

Using Pascal's Triangle, the coefficients for expanding \( (a + b)^3 \) are 1, 3, 3, and 1.

The expansion of \( (2 - x)^3 \) is:

\[ (2 - x)^3 = 1 \cdot 2^3 - 3 \cdot 2^2 \cdot x + 3 \cdot 2 \cdot x^2 - x^3 \]

Simplifying each term:

\[ = 8 - 12x + 6x^2 - x^3 \]

2. Find the coefficient of \( x^3 \) in the expansions of:

(a) \( (x + 3)^4 \)

The expansion of \( (x + 3)^4 \) is given by:

\[ (x + 3)^4 = x^4 + 4 \cdot x^3 \cdot 3 + 6 \cdot x^2 \cdot 3^2 + 4 \cdot x \cdot 3^3 + 3^4 \]

The term containing \( x^3 \) is:

\[ 4 \cdot x^3 \cdot 3 = 12x^3 \]

Therefore, the coefficient of \( x^3 \) is 12.

(b) \( (1 + x)^5 \)

The expansion of \( (1 + x)^5 \) is given by:

\[ (1 + x)^5 = 1 + 5x + 10x^2 + 10x^3 + 5x^4 + x^5 \]

The coefficient of \( x^3 \) is 10.

(c) \( (3 - x)^5 \)

The expansion of \( (3 - x)^5 \) is given by:

\[ (3 - x)^5 = 3^5 - 5 \cdot 3^4 \cdot x + 10 \cdot 3^3 \cdot x^2 - 10 \cdot 3^2 \cdot x^3 + 5 \cdot 3 \cdot x^4 - x^5 \]

The term containing \( x^3 \) is:

\[ -10 \cdot 3^2 \cdot x^3 = -90x^3 \]

Therefore, the coefficient of \( x^3 \) is -90.

(d) \( (4 + x)^4 \)

The expansion of \( (4 + x)^4 \) is given by:

\[ (4 + x)^4 = 4^4 + 4 \cdot 4^3 \cdot x + 6 \cdot 4^2 \cdot x^2 + 4 \cdot 4 \cdot x^3 + x^4 \]

The term containing \( x^3 \) is:

\[ 4 \cdot 4 \cdot x^3 = 64x^3 \]

Therefore, the coefficient of \( x^3 \) is 64.

3. Find the values of \( A \), \( B \), and \( C \) in the expression \( (3 + x)^5 + (3 - x)^5 \equiv A + Bx^2 + Cx^4 \)

First, expand \( (3 + x)^5 \) and \( (3 - x)^5 \):

\( (3 + x)^5 = 243 + 405x + 270x^2 + 90x^3 + 15x^4 + x^5 \)

\( (3 - x)^5 = 243 - 405x + 270x^2 - 90x^3 + 15x^4 - x^5 \)

Adding these two expansions:

\[ (3 + x)^5 + (3 - x)^5 = (243 + 243) + (270 + 270)x^2 + (15 + 15)x^4 \]

Simplifying each term:

\[ = 486 + 540x^2 + 30x^4 \]

Thus, \( A = 486 \), \( B = 540 \), and \( C = 30 \).