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Binomial Expressions and Binomial Expansion

Learning Objectives:

By the end of this lesson, students will be able to:
1. Define a binomial expression.
2. Use Pascal’s triangle to expand a binomial expression.
3. State the general binomial expansion of \((a + x)^n\) and \((1+ x)^n\).
4. Use the binomial series to expand expressions of the form \((a + x)^n\) for positive, negative, and fractional values of \(n\).
5. Determine the \(r\)-th term of a binomial expansion.
6. Apply the binomial expansion to practical problems.

1. Definition of a Binomial Expression

A binomial expression is an algebraic expression that contains exactly two terms. For example, \(a + x\) and \(2x – 3\) are binomial expressions.

2. Using Pascal’s Triangle to Expand a Binomial Expression

Pascal’s Triangle is a triangular array of numbers where each number is the sum of the two directly above it. The \(n\)-th row of Pascal’s Triangle provides the coefficients for the expansion of \((a + x)^n\).

Example:
Expand \((a + x)^3\) using Pascal’s Triangle.

Pascal’s Triangle up to the 4th row:
 1
1 1
1 2 1
1 3 3 1

The coefficients for \((a + x)^3\) are 1, 3, 3, 1.

Thus, \((a + x)^3 = 1a^3 + 3a^2x + 3ax^2 + 1x^3\)

3. General Binomial Expansion

Binomial Theorem:
The general binomial expansion of \((a + x)^n\) is given by:
\[
(a + x)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} x^r
\]
where \(\binom{n}{r}\) is a binomial coefficient, calculated as \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\).

For \((1 + x)^n\), the expansion simplifies to:
\[
(1 + x)^n = \sum_{r=0}^{n} \binom{n}{r} x^r
\]

4. Binomial Series for Any \(n\)

The binomial series can be used to expand \((a + x)^n\) for positive, negative, and fractional values of \(n\):

\[
(1 + x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \ldots
\]

For \(a \neq 1\):
\[
(a + x)^n = a^n \left(1 + \frac{x}{a}\right)^n
\]

Example:

Expand \((2 + x)^{-2}\):

\[
(2 + x)^{-2} = 2^{-2} \left(1 + \frac{x}{2}\right)^{-2} = \frac{1}{4} \left(1 – 2 \frac{x}{2} + 3 \frac{x^2}{4} – \ldots \right)
\]

5. Determining the \(r\)-th Term of a Binomial Expansion

The general term (or \(r\)-th term) in the expansion of \((a + x)^n\) is given by:
\[
T_{r+1} = \binom{n}{r} a^{n-r} x^r
\]

Example:
Find the 4th term of \((3 + x)^5\):

The 4th term is \(T_4\), so \(r = 3\):

\[
T_4 = \binom{5}{3} 3^{5-3} x^3 = \binom{5}{3} 3^2 x^3 = 10 \cdot 9 x^3 = 90 x^3
\]

6. Practical Applications of Binomial Expansion

Example: Probability
In probability, the binomial expansion can be used to calculate probabilities of different outcomes.

Suppose we want to find the probability of getting exactly 3 heads in 5 tosses of a fair coin. Using the binomial expansion:
\[
P(3 \text{ heads}) = \binom{5}{3} \left(\frac{1}{2}\right)^3 \left(\frac{1}{2}\right)^{5-3} = 10 \cdot \left(\frac{1}{2}\right)^5 = 10 \cdot \frac{1}{32} = \frac{10}{32} = \frac{5}{16}
\]

Example: Finance
Binomial expansions can be used in financial mathematics to model compound interest, option pricing, and other financial instruments.

Summary

1. Define a binomial expression.
2. Use Pascal’s Triangle for expansion.
3. State the General Binomial Expansion for \((a + x)^n\) and \((1 + x)^n\).
4. Expand Expressions for any \(n\) using the binomial series.
5. Determine the \(r\)-th Term of a binomial expansion.
6. Apply the Binomial Expansion to practical problems.