Problem 25 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The expanded form is \(x^4 + 8x^3y + 24x^2y^2 + 32xy^3 + 16y^4\).

Step 1: Identify Binomial Components

The expression \((x + 2y)^4\) is a binomial raised to a power. Here, the binomial components are \(a = x\) and \(b = 2y\), and the exponent \(n = 4\).

Step 2: Recall the Binomial Theorem Formula

The Binomial Theorem states: \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). For this problem, substitute \(n = 4\), \(a = x\), and \(b = 2y\) into the formula.

Step 3: Calculate Each Binomial Coefficient

Calculate the binomial coefficients \(\binom{4}{k}\) for each \(k\) from 0 to 4. These are: \(\binom{4}{0} = 1\), \(\binom{4}{1} = 4\), \(\binom{4}{2} = 6\), \(\binom{4}{3} = 4\), \(\binom{4}{4} = 1\).

Step 4: Expand Using the Binomial Coefficients

Substitute the binomial coefficients into the expansion: \((x+2y)^4 = \binom{4}{0}x^4(2y)^0 + \binom{4}{1}x^3(2y)^1 + \binom{4}{2}x^2(2y)^2 + \binom{4}{3}x^1(2y)^3 + \binom{4}{4}x^0(2y)^4\).

Step 5: Simplify Each Term

Evaluate and simplify each term from the expansion:- \(1 \, \cdot \, x^4 \, \cdot \, 1 = x^4\)- \(4 \, \cdot \, x^3 \, \cdot \, 2y = 8x^3y\)- \(6 \, \cdot \, x^2 \, \cdot \, (2y)^2 = 24x^2y^2\)- \(4 \, \cdot \, x \, \cdot \, (2y)^3 = 32xy^3\)- \(1 \, \cdot \, 1 \, \cdot \, (2y)^4 = 16y^4\).

Step 6: Write the Full Expansion

Combine the simplified terms to get the expanded form: \(x^4 + 8x^3y + 24x^2y^2 + 32xy^3 + 16y^4\).

Key Concepts

Binomial Coefficients

In mathematics, binomial coefficients are a fundamental concept in algebra, specifically when working with binomials and their expansions. These coefficients are represented as \( \binom{n}{k} \), known as "n choose k." They are integral in determining each term of a polynomial expansion of a binomial.

• The formula for a binomial coefficient is given by \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \), where \( n \) is the total number of items, \( k \) is the number of items being chosen, and \( ! \) denotes factorial, which is the product of all positive integers up to that number.
• For example, in the expansion of \( (x + 2y)^4 \), we calculate the coefficients \( \binom{4}{0}, \binom{4}{1}, \ldots, \binom{4}{4} \), which are respectively 1, 4, 6, 4, and 1.
• These numbers help us determine the weights of the terms in the binomial expansion.

Understanding binomial coefficients is crucial for simplifying expressions and solving complex algebraic problems.

Polynomial Expansion

Polynomial expansion is the process of expressing a binomial expression raised to a power as a sum of terms. It involves breaking down the expression into a series of simpler polynomials. The Binomial Theorem offers a straightforward method for achieving this.

• Using the theorem, we can expand \((a + b)^n\) into a sum involving binomial coefficients, powers of \(a\), and powers of \(b\). This expansion is represented as \( \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \).
• For instance, when we apply it to \((x + 2y)^4\), we expand it into a polynomial consisting of terms like \(x^4, 8x^3y,\) and others.
• Each term in this expanded form is derived from multiplying a binomial coefficient with appropriate powers of \(a\) and \(b\).

This process not only simplifies computations but also aids in understanding complex algebraic structures.

Exponentiation

Exponentiation is a mathematical operation involving numbers or expressions being raised to a power. This operation is central to binomial expansion, where the binomial expression is raised to an exponent, often a positive integer.

• In the context of the Binomial Theorem, exponentiation determines the power to which the binomial components \(a\) and \(b\) are raised in each term of the expansion.
• In the example \((x + 2y)^4\), the term \(x^4\) results from raising \(x\) to the power of four, illustrating exponentiation in its simplest form.
• Exponentiation impacts both numbers and variables. For instance, \((2y)^3\) involves raising 2 and \(y\) to the same power simultaneously, resulting in \(8y^3\).

Mastering exponentiation helps in evaluating and simplifying expressions efficiently, which is essential in binomial expansions and beyond.