Problem 16 - Precalculus: Mathematics for Calculus

Question

Pascal's Triangle Use Pascal's triangle to expand the expression. $$\left(2+\frac{x}{2}\right)^{5}$$

Step-by-Step Solution

Answer

The expanded expression is \(32 + 40x + 20x^2 + 5x^3 + \frac{5x^4}{8} + \frac{x^5}{32}\).

Step 1: Understand the Expression

We are given the expression \( \left(2 + \frac{x}{2} \right)^{5} \). The goal is to expand this expression using the binomial theorem.

Step 2: Write the General Formula of Binomial Theorem

The binomial theorem states that \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). For our expression, \(a = 2\), \(b = \frac{x}{2}\), and \(n = 5\).

Step 3: Identify Coefficients Using Pascal's Triangle

The coefficients for the expansion \((a + b)^5\) are given by the 5th row of Pascal's Triangle: 1, 5, 10, 10, 5, 1.

Step 4: Expand the Expression Using Coefficients

Substitute the coefficients and the terms into the binomial expansion formula: \[\left( 2 + \frac{x}{2} \right)^{5} = 1 \cdot 2^{5} \cdot \left(\frac{x}{2}\right)^{0} + 5 \cdot 2^{4} \cdot \left(\frac{x}{2}\right)^{1} + 10 \cdot 2^{3} \cdot \left(\frac{x}{2}\right)^{2} + 10 \cdot 2^{2} \cdot \left(\frac{x}{2}\right)^{3} + 5 \cdot 2^{1} \cdot \left(\frac{x}{2}\right)^{4} + 1 \cdot 2^{0} \cdot \left(\frac{x}{2}\right)^{5}\]

Step 5: Simplify Each Term

Calculate each term:- First term: \(1 \cdot 32 \cdot 1 = 32\)- Second term: \(5 \cdot 16 \cdot \frac{x}{2} = 5 \cdot 16 \cdot \frac{x}{2} = 40x\)- Third term: \(10 \cdot 8 \cdot \left(\frac{x}{2}\right)^2 = 10 \cdot 8 \cdot \frac{x^2}{4} = 20x^2\)- Fourth term: \(10 \cdot 4 \cdot \left(\frac{x}{2}\right)^3 = 10 \cdot 4 \cdot \frac{x^3}{8} = 5x^3\)- Fifth term: \(5 \cdot 2 \cdot \left(\frac{x}{2}\right)^4 = 5 \cdot 2 \cdot \frac{x^4}{16} = \frac{5x^4}{8}\)- Sixth term: \(1 \cdot 1 \cdot \left(\frac{x}{2}\right)^5 = \frac{x^5}{32}\)

Step 6: Write the Expanded Expression

Combine all simplified terms to write the final expanded expression:\[\left( 2 + \frac{x}{2} \right)^{5} = 32 + 40x + 20x^2 + 5x^3 + \frac{5x^4}{8} + \frac{x^5}{32}\]

Key Concepts

Binomial Theorem

The binomial theorem is a fundamental tool in algebra that provides a formula for expanding expressions raised to a power. It is expressed as \[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\].This formula allows us to find each term in the expansion without directly multiplying the expression repeatedly.
In this context, \(a\) and \(b\) represent the terms in the binomial expression, while \(n\) is the exponent or power to which the binomial is raised. The notation \(\binom{n}{k}\) denotes a binomial coefficient, calculated using factorials and gives the number of ways to choose \(k\) elements from \(n\) elements, thereby indicating how many times each term appears in the expansion.
Using the binomial theorem makes polynomial expansions efficient, especially when handling higher powers, as it outlines a structured approach involving coefficients related to Pascal's Triangle.

Polynomial Expansion

Polynomial expansion involves rewriting a binomial expression raised to a power as a sum of terms. Each term consists of coefficients, powers of the first term \(a\), and powers of the second term \(b\). This is incredibly useful for simplifying expressions and solving equations.
For example, expanding \(\left(2 + \frac{x}{2}\right)^{5}\) involves using each coefficient from Pascal's Triangle to multiply corresponding powers of \(2\) and \(\frac{x}{2}\). Each successive term decreases the power of 2 and increases the power of \( \frac{x}{2} \).

• First term: constant term, highest power of \(a\).
• Middle terms: blended powers of \(a\) and \(b\).
• Last term: highest power of \(b\), no power of \(a\).

Polynomial expansion thus provides an organized way to derive all components of a powered expression without simply multiplying everything out step-by-step.

Combinatorics

Combinatorics is the branch of mathematics dealing with counting combinations and permutations of sets. It plays a crucial role in the binomial theorem, as the coefficients in the expansion are binomial coefficients derived from combinatorial principles.
The binomial coefficient, often written as \(\binom{n}{k}\), represents the number of ways to choose \(k\) items from a total of \(n\) items without regard to the order. Mathematically, it is given by:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]where \(n!\) (n factorial) is the product of all positive integers up to \(n\).
This combinatorial insight allows the binomial theorem to structure polynomial expansions effectively, whereby each term's coefficient in the expansion is one such binomial coefficient from Pascal's Triangle. This linkage makes complex algebraic expansions manageable, illustrating how combinatorics supports broader mathematical applications.