Problem 35 - A Graphical Approach to College Algebra

Question

Write the binomial expansion for each expression. $$\left(\frac{m}{2}-1\right)^{6}$$

Step-by-Step Solution

Answer

The expansion is \( \frac{m^6}{64} - \frac{3m^5}{16} + \frac{15m^4}{16} - \frac{5m^3}{4} + \frac{15m^2}{4} - 3m + 1 \).

Step 1: Identify the Binomial Expression

The expression given is \( \left( \frac{m}{2} - 1 \right)^{6} \). It consists of two terms: the first term \( \frac{m}{2} \) and the second term \( -1 \).

Step 2: Use the Binomial Theorem Formula

The binomial theorem formula is \( (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} \, a^{n-k} \, b^k \). In this expression, \( a = \frac{m}{2} \), \( b = -1 \), and \( n = 6 \).

Step 3: Expand using Binomial Coefficients

Calculate each term of the expansion using the binomial coefficients \( \binom{6}{k} \), for \( k = 0, 1, 2, \ldots, 6 \).

Step 4: Calculate Each Term

Using the formula, calculate the first few terms:- For \( k = 0 \): \( \binom{6}{0} \left(\frac{m}{2}\right)^6 (-1)^0 = \left(\frac{m}{2}\right)^6 \).- For \( k = 1 \): \( \binom{6}{1} \left(\frac{m}{2}\right)^5 (-1)^1 = -6 \left(\frac{m}{2}\right)^5 \).- For \( k = 2 \): \( \binom{6}{2} \left(\frac{m}{2}\right)^4 (-1)^2 = 15 \left(\frac{m}{2}\right)^4 \).Continue this process for \( k = 3, 4, 5, 6 \).

Step 5: Simplify each Term

Simplify each term by calculating powers and coefficients. For example:- \( \left( \frac{m}{2} \right)^6 = \frac{m^6}{64} \).- \( 6 \left( \frac{m}{2} \right)^5 = \frac{6m^5}{32} = \frac{3m^5}{16} \).- Continue simplifying the remaining terms.

Step 6: Write the Complete Expansion

After simplifying all terms, the expansion is:\[ \frac{m^6}{64} - \frac{3m^5}{16} + \frac{15m^4}{16} - \frac{5m^3}{4} + \frac{15m^2}{4} - 3m + 1 \].

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful algebraic tool used for expanding expressions that are raised to a power. When you encounter a binomial, like \( (a + b)^n \), this theorem lets you expand it without having to multiply the expression by itself repeatedly. It saves time and reduces errors.
Here's the formula:

• \( (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} \, a^{n-k} \, b^k \)

In this formula:

• \( \binom{n}{k} \) is a binomial coefficient, calculated as \( \frac{n!}{k! (n-k)!} \), representing the number of ways to choose \( k \) elements from \( n \) elements.
• \( n \) is the power that the binomial is raised to.

You can use this theorem for various algebraic expressions, as long as you have two terms in parentheses raised to any power \( n \).
In our example \( \left(\frac{m}{2}-1\right)^{6} \), we directly apply it to find each term in the expansion.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations (such as addition and multiplication). In mathematics, especially algebra, identifying and manipulating these expressions correctly is essential.
In the expression \( \left(\frac{m}{2} - 1\right)^6 \), we have:

• Two terms: \( \frac{m}{2} \) and \( -1 \).
• The entire expression is raised to the power of 6.

These expressions can be simplified, expanded, or factored depending on what you need to achieve with them.
Identifying terms properly helps in applying the correct mathematical operations, such as what we do with the Binomial Theorem. It’s important to understand and work within the rules of algebra to manipulate these expressions confidently.

Polynomial Expansion

Polynomial expansion refers to the process of expanding expressions that involve powers, resulting in a polynomial. Polynomials are expressions that consist of variables and coefficients combined through addition, subtraction, and multiplication.
When we expand \( (\frac{m}{2} - 1)^6 \), we're turning this into a polynomial by calculating each term:

• Calculate each term using \( \binom{n}{k} \) for different \( k \) values.
• Simplify the results to form terms of the polynomial.

The expansion creates a polynomial which looks like this:

• \( \frac{m^6}{64} \)
• \( - \frac{3m^5}{16} \)
• \( + \frac{15m^4}{16} \)
• \( - \frac{5m^3}{4} \)
• \( + \frac{15m^2}{4} \)
• \( - 3m \)
• \( + 1 \)

This process highlights how we can expand and express compound algebraic expressions as polynomials with various terms. By simplifying each part, we obtain a complete and expanded polynomial, showcasing the depth of mathematical expressions.