Problem 25 - Algebra and Trigonometry

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: Understanding the Binomial Theorem

The Binomial Theorem provides us the formula to expand a binomial expression raised to a power, \((a + b)^n\). It states that \( (a + b)^{n} = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k}\) where \(\binom{n}{k}\) is a binomial coefficient and equals \(\frac{n!}{k!(n-k)!}\).

Step 2: Identify the Variables

In the expression \((x + 2y)^4\), identify \(a = x\), \(b = 2y\), and \(n = 4\). This information will be used to apply the Binomial Theorem.

Step 3: Calculate Binomial Coefficients

For \(n = 4\), the coefficients can be calculated as follows: \(\binom{4}{0} = 1\), \(\binom{4}{1} = 4\), \(\binom{4}{2} = 6\), \(\binom{4}{3} = 4\), \(\binom{4}{4} = 1\).

Step 4: Expand the Expression

Use the coefficients and insert them into the formula: \( (x+2y)^4 = \binom{4}{0}x^{4}(2y)^{0} + \binom{4}{1}x^{3}(2y)^{1} + \binom{4}{2}x^{2}(2y)^{2} + \binom{4}{3}x^{1}(2y)^{3} + \binom{4}{4}x^{0}(2y)^{4} \).

Step 5: Simplify Each Term

Calculate and simplify: - \(\binom{4}{0}x^{4}(2y)^{0} = 1 \cdot x^4 \cdot 1 = x^{4}\) - \(\binom{4}{1}x^{3}(2y)^{1} = 4 \cdot x^3 \cdot 2y = 8x^{3}y\) - \(\binom{4}{2}x^{2}(2y)^{2} = 6 \cdot x^2 \cdot 4y^2 = 24x^{2}y^{2}\) - \(\binom{4}{3}x^{1}(2y)^{3} = 4 \cdot x \cdot 8y^3 = 32xy^{3}\) - \(\binom{4}{4}x^{0}(2y)^{4} = 1 \cdot 16y^4 = 16y^{4}\).

Step 6: Combine All Terms

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

Key Concepts

Binomial Expression

A binomial expression is a mathematical expression involving the sum of two terms. It takes the form \( (a + b) \), where both \( a \) and \( b \) can be any expression, like variables or numbers. Binomials are the building blocks for applying the Binomial Theorem.

Understanding binomial expressions is vital because they frequently appear in algebraic manipulations. When raised to a power, binomials allow us to explore more about mathematical expansions and series. For example, the binomial expression \( (x + 2y) \) becomes more intricate when we calculate \( (x + 2y)^4 \). This is where the Binomial Theorem comes into play, providing a structured way to expand the terms systematically.

In summary, recognizing and working with binomial expressions is essential for mastering algebra and calculus. This understanding paves the way to exploring patterns and complex polynomials with ease.

Binomial Coefficients

Binomial coefficients are key components in expanding binomial expressions using the Binomial Theorem. These coefficients are the numbers that quantify how each term in the binomial expression contributes as it's raised to a power.

To calculate a binomial coefficient, use the formula \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \). Here, \( n \) represents the total power of the binomial, while \( k \) indicates the specific term within the expansion. These coefficients are derived from arrangements of ideas or objects, such as determining combinations of items—hence sometimes called combination numbers.

Consider the example \((x + 2y)^4\), where the calculation of coefficients goes as: \( \binom{4}{0} = 1 \), \( \binom{4}{1} = 4 \), \( \binom{4}{2} = 6 \), \( \binom{4}{3} = 4 \), and \( \binom{4}{4} = 1 \).

• These coefficients correspond to the numbers on Pascal's Triangle, a handy tool for finding binomial coefficients quickly.
• As you progress to larger powers or more complex binomials, the concept of binomial coefficients becomes more crucial, providing a foundation for deeper mathematical insight.

Expansion of Powers

The concept of expanding powers of a binomial expression relies on applying the Binomial Theorem. This theorem offers a systematic approach to expand any binomial expression raised to a given power.

Using the Binomial Theorem, you can expand \( (a + b)^n \) by:

• Writing the general form: \( (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k} \).
• Identifying \( a \), \( b \), and \( n \) within your expression.
• Calculating each binomial coefficient for the corresponding terms.
• Substitute the appropriate powers of \( a \) and \( b \), as determined by the binomial coefficients.
• Simplifying the resulting terms to obtain the expanded form.

This method unravels the complexity of expressions raised to high powers into manageable components.

For example, expanding \( (x + 2y)^4 \) involves calculating terms from \( x^4 \) to \( (2y)^4 \) using binomial coefficients, then merging these results. This ends in the expression \( x^4 + 8x^3y + 24x^2y^2 + 32xy^3 + 16y^4 \), showcasing the expanded result of the initial binomial expression.

Thus, mastering expansion of powers equips you with the ability to handle various algebraic calculations with confidence and precision.