Problem 106 - Precalculus with Limits

Question

Another form of the Binomial Theorem is \( \left(x + y\right)^n = x^n + \dfrac{nx^{n - 1}y}{1!} + \dfrac{n\left(n - 1\right)x^{n-2}y^2}{2!} + \dfrac{n\left(n - 1\right)\left(n - 2\right)x^{n-3}y^3}{3!} + \cdots + y^n \). Use this form of the Binomial Theorem to expand and simplify each expression. (a) \( \left(2 + 3\right)^6 \) (b) \( \left(x + ay\right)^4 \) (c) \( \left(x - ay\right)^5 \) (d) \( \left(1 + x\right)^{12} \)

Step-by-Step Solution

Answer

a) 15625, b) \( x^4 + 4ax^3y + 6a^2x^2y^2 + 4a^3xy^3 + a^4y^4 \), c) \( x^5 - 5ax^4y + 10a^2x^3y^2 -10a^3x^2y^3 + 5a^4xy^4 - a^5y^5 \), d) \( 1 + 12x + 66x^2 + 220x^3 + 495x^4 + 792x^5 + 924x^6 + 792x^7 + 495x^8 + 220x^9 + 66x^{10} + 12x^{11} + x^{12} \)

Step 1: Expand (a) \( \left(2 + 3\right)^6 \)

By applying the binomial theorem and simplification, \( \left(2 + 3\right)^6 \) results in 15625.

Step 2: Expand (b) \( \left(x + ay\right)^4 \)

Using the binomial theorem, \( \left(x + ay\right)^4 \) expands to \( x^4 + 4ax^3y + 6a^2x^2y^2 + 4a^3xy^3 + a^4y^4 \)

Step 3: Expand (c) \( \left(x - ay\right)^5 \)

Applying the binomial theorem gives \( \left(x - ay\right)^5 \) as \( x^5 - 5ax^4y + 10a^2x^3y^2 -10a^3x^2y^3 + 5a^4xy^4 - a^5y^5 \)

Step 4: Expand (d) \( \left(1 + x\right)^{12} \)

Again using the binomial theorem, \( \left(1 + x\right)^{12} \) simplifies to \( 1 + 12x + 66x^2 + 220x^3 + 495x^4 + 792x^5 + 924x^6 + 792x^7 + 495x^8 + 220x^9 + 66x^{10} + 12x^{11} + x^{12} \)

Key Concepts

Polynomial Expansion

Polynomial expansion involves rewriting a binomial raised to a power as a sum of terms. Each term in the expansion is a product of coefficients, powers of the first term of the binomial, and powers of the second term. In the exercise given, you are asked to expand expressions using the Binomial Theorem. This theorem provides us with a formula that allows us to determine each term in the expansion without directly multiplying everything out, which is especially helpful with higher powers.

A typical term in the expansion of \((x + y)^n\) takes the form \(\binom{n}{k} x^{n-k} y^k\), where the binomial coefficient is computed as \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\). These coefficients correspond to specific values:

• The starting coefficient is always 1, when \(k = 0\).
• The second term involves the power of \(x\) decreasing by 1 and the power of \(y\) increasing from zero upwards.

By using this form of expansion, complex binomials become much easier to handle with accuracy and efficiency, making it a critical tool in both algebra and calculus.

Combinatorics

Combinatorics concerns itself with the arrangement, combination, and selection of objects within a set. Its principles underpin the Binomial Theorem's results, often providing the "why" behind how the coefficients in polynomial expansions behave. In these expansions, the coefficients or binomial coefficients play a significant role.

Here's where combinatoric principles come in:

• The coefficients in a binomial expansion are selected via combinatorial methods, specifically through choosing combinations: \(\binom{n}{k}\) which tells us how to select \(k\) items from \(n\) total items without regard to order.
• This concept of choosing is not just academic but practical, as it reflects real-world tasks like deciding committees or forming groups, where order does not matter.

The connections between combinatorics and polynomial expansion reinforce the power and flexibility of mathematics to describe both straightforward and complex scenarios.

Power Series

Power series are infinite series that expand functions into sums of power terms. Although the expansions covered in the exercise above are finite, understanding power series gives deeper insights into how expressions like \((x + y)^n\) can be expanded in general terms.

In mathematics, a power series typically takes the form \(\sum_{k=0}^{\infty} a_kx^k\). The Binomial Theorem in particular is a finite application that is very similar to this concept. It allows us to approximate functions over intervals or derive new functions from existing ones. For instance:

• The Binomial Theorem without bounds can approximate values of functions via Taylor series expansions, which are specific types of power series.
• This makes them invaluable in calculus, especially when considering function convergence and approximation within certain limits.

Understanding the fundamentals of power series gives a glimpse of the vast applications that these mathematical constructs have, linking algebraic expressions to deeper mathematical contexts.