Problem 37 - A Graphical Approach to College Algebra

Question

Write the binomial expansion for each expression. $$\left(\sqrt{2} r+\frac{1}{m}\right)^{4}$$

Step-by-Step Solution

Answer

\(2r^4 + 8\sqrt{2} \frac{r^3}{m} + 12\frac{r^2}{m^2} + 4\sqrt{2}\frac{r}{m^3} + \frac{1}{m^4}\)

Step 1: Identify the Formula

The binomial theorem helps us expand expressions of the form \((a + b)^n\). The theorem states that \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\), where \(\binom{n}{k}\) is the binomial coefficient \(\frac{n!}{k!(n-k)!}\).

Step 2: Set the Values

For the expression \(\left(\sqrt{2} r + \frac{1}{m}\right)^4\), set \(a = \sqrt{2} r\), \(b = \frac{1}{m}\), and \(n = 4\).

Step 3: Calculate the Binomial Coefficients

Calculate the binomial coefficients for \(n = 4\):1. \(\binom{4}{0} = 1\)2. \(\binom{4}{1} = 4\)3. \(\binom{4}{2} = 6\)4. \(\binom{4}{3} = 4\)5. \(\binom{4}{4} = 1\)

Step 4: Calculate Each Term of the Expansion

Using the binomial formula, calculate each of the terms:1. First term: \(\binom{4}{0}(\sqrt{2} r)^4\left(\frac{1}{m}\right)^0 = 1 \cdot (2) r^4 = 2r^4\)2. Second term: \(\binom{4}{1}(\sqrt{2} r)^3\left(\frac{1}{m}\right)^1 = 4 \cdot (2\sqrt{2}) r^3 \cdot \frac{1}{m} = 8\sqrt{2} \frac{r^3}{m}\)3. Third term: \(\binom{4}{2}(\sqrt{2} r)^2\left(\frac{1}{m}\right)^2 = 6 \cdot (2) r^2 \cdot \frac{1}{m^2} = 12 \frac{r^2}{m^2}\)4. Fourth term: \(\binom{4}{3}(\sqrt{2} r)^1\left(\frac{1}{m}\right)^3 = 4 \cdot \sqrt{2} r \cdot \frac{1}{m^3} = 4\sqrt{2} \frac{r}{m^3}\)5. Fifth term: \(\binom{4}{4}(\sqrt{2} r)^0\left(\frac{1}{m}\right)^4 = 1 \cdot \frac{1}{m^4} = \frac{1}{m^4}\)

Step 5: Write the Complete Expansion

Combine all the terms to find the expanded form of the original expression: \[2r^4 + 8\sqrt{2} \frac{r^3}{m} + 12\frac{r^2}{m^2} + 4\sqrt{2}\frac{r}{m^3} + \frac{1}{m^4}\].

Key Concepts

Binomial Coefficients

In mathematics, the binomial coefficients play a crucial role when expanding expressions using the Binomial Theorem. These coefficients are numbers that appear in the expansion of a binomial raised to a power. They are represented as \( \binom{n}{k} \), which stands for the number of ways to choose \( k \) elements from a set of \( n \) elements without regard to order.

The formula for calculating binomial coefficients is given by: \[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]where \( n! \) is the factorial of \( n \), calculated as the product of all positive integers up to \( n \).

• For example, in expanding \( (a + b)^4 \), you find the binomial coefficients \( \binom{4}{0}, \binom{4}{1}, \binom{4}{2}, \binom{4}{3}, \) and \( \binom{4}{4} \).
• These coefficients are 1, 4, 6, 4, and 1, respectively.

Understanding these coefficients helps you determine the weights of each term in the polynomial expansion.

Polynomial Expansion

The Binomial Theorem provides a simple method to expand expressions of the form \( (a + b)^n \). This kind of expansion is called a polynomial expansion because the result is a polynomial.

When dealing with \( (\sqrt{2} r + \frac{1}{m})^4 \), we use the theorem to create a series of terms that are added together, each involving different powers of \( a \) and \( b \). This process involves:

• Identifying the values of \( a \) and \( b \) from the expression.
• Calculating each term using binomial coefficients.
• Combining the terms to form the entire polynomial expansion.

Each term in the expansion is derived by the formula \( \binom{n}{k} a^{n-k} b^k \), with \( k \) taking values from 0 to \( n \). This transforms the original binomial expression into an expanded form which consists of terms like \( 2r^4 \) and \( \frac{1}{m^4} \).

This expanded form is helpful for further algebraic manipulation or for evaluating the expression for specific values of \( r \) and \( m \).

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. In the context of the binomial theorem and polynomial expansion, understanding algebraic expressions is key to manipulating and expanding binomials.

When you have an expression like \( (\sqrt{2} r + \frac{1}{m})^4 \), it involves algebraic components:

• \( \sqrt{2} r \) is a product of a constant and a variable.
• \( \frac{1}{m} \) is a fractional term involving a variable in the denominator.

To expand such expressions, clarity on these components is essential. You'll encounter terms raised to powers, and it helps to manage operations like multiplication, power, and division within the expression.

Throughout this process, maintaining consistency with algebraic rules ensures the resulting expansion is both accurate and simple to understand, involving detailed steps like calculating specific powers and combining like terms.