Problem 25 - Precalculus: Mathematics for Calculus

Question

Use the Binomial Theorem to expand the expression. $$(x+2 y)^{4}$$

Step-by-Step Solution

Answer

\( (x+2y)^4 = x^4 + 8x^3y + 24x^2y^2 + 32xy^3 + 16y^4 \).

Step 1: Write the Binomial Theorem Formula

The Binomial Theorem states that for any positive integer \( n \), \( (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \) where \( \binom{n}{k} \) is the binomial coefficient calculated as \( \frac{n!}{k!(n-k)!} \).

Step 2: Identify Components

In the expression \((x+2y)^{4}\), we have \(a = x\), \(b = 2y\), and \(n = 4\). We'll use these components in the binomial formula to expand the expression.

Step 3: Apply the Binomial Theorem

Substitute \(a\), \(b\), and \(n\) into the Binomial Theorem. Calculate each term for \(k=0\) to \(k=4\): - For \(k=0\): \( \binom{4}{0}x^{4}(2y)^{0} = 1 \cdot x^4 \cdot 1 = x^4 \)- For \(k=1\): \( \binom{4}{1}x^{3}(2y)^{1} = 4 \cdot x^3 \cdot 2y = 8x^3y \)- For \(k=2\): \( \binom{4}{2}x^{2}(2y)^{2} = 6 \cdot x^2 \cdot 4y^2 = 24x^2y^2 \)- For \(k=3\): \( \binom{4}{3}x^{1}(2y)^{3} = 4 \cdot x \cdot 8y^3 = 32xy^3 \)- For \(k=4\): \( \binom{4}{4}x^{0}(2y)^{4} = 1 \cdot 1 \cdot 16y^4 = 16y^4 \)

Step 4: Combine the Terms

Write out the expanded form by combining all the terms calculated in Step 3: \( x^4 + 8x^3y + 24x^2y^2 + 32xy^3 + 16y^4 \).

Key Concepts

Binomial Expansion

The binomial expansion is a method to expand expressions raised to a power, using the Binomial Theorem. The expression \((a+b)^n\) is expanded as a series of terms. Each term involves a specific power of the two variables \(a\) and \(b\), multiplied by a coefficient.

To understand binomial expansion, consider the expression \((x+2y)^4\). Using the Binomial Theorem:
- You are expanding with two distinct terms \(x\) and \(2y\).- The expression is raised to the 4th power (\(n = 4\)).The formula for binomial expansion is: \[ (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \] Applying this formula allows you to break down the expression into a sum of terms, each term being a product of a binomial coefficient, a power of \(x\), and a power of \(2y\).

This series of operations simplifies the process of handling any binomial expression, even involving larger powers.

Binomial Coefficient

The binomial coefficient is a key component in binomial expansion. It determines the weight each term in the expansion carries, calculated based on the combination of different powers of \(a\) and \(b\). In the binomial formula, the binomial coefficient is represented as \(\binom{n}{k}\).The formula to find the binomial coefficient is: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \] - Here, \(n!\) ("n factorial") is the product of all positive integers up to \(n\).- \(k\) is the particular term in the sequence.- This coefficient tells you how many ways you can choose \(k\) elements from \(n\) elements, ignoring order.

Taking the example of \((x+2y)^4\): - When calculating \(\binom{4}{0}\), \(\binom{4}{1}\), \(\binom{4}{2}\), \(\binom{4}{3}\), and \(\binom{4}{4}\), these coefficients turn out to be 1, 4, 6, 4, and 1 respectively. - Each of these coefficients will multiply the respective term formed using the powers of \(x\) and \(2y\). This calculation makes the binomial expansion precise and orderly.

Polynomial Expressions

Polynomial expressions involve sums of terms that have variables raised to non-negative integer powers. They are foundational in algebra and can be simple or complex depending on the degree and number of variables. A polynomial expression takes the form of \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\),where each \(a\) represents a coefficient and each power of \(x\) indicates the order.Looking at the expanded form from our example:\[ x^4 + 8x^3y + 24x^2y^2 + 32xy^3 + 16y^4 \] You have a polynomial with five terms derived from expanding \((x + 2y)^4\).

Each term is a product of coefficients (like \(8\), \(24\)) and variables raised to powers (like \(x^3\), \(y^2\)).

- The powers of \(x\) decrease, and the powers of \(y\) increase, following a consistent pattern in line with binomial expansion.

- The resulting polynomial gives full insight into the behavior of the expression \((x + 2y)^4\) and displays how the interaction of \(x\) and \(y\) changes as they are raised to higher powers.