Problem 42 - College Algebra with Modeling and Visualization

Question

Use Pascal's triangle to help expand the expression. $$ (2 m-n)^{4} $$

Step-by-Step Solution

Answer

The expanded expression is \( 16m^4 - 32m^3n + 24m^2n^2 - 8mn^3 + n^4 \).

Step 1: Identify the Row in Pascal's Triangle

To expand \( (2m - n)^4 \), identify the 4th row in Pascal's Triangle. The 4th row is \( [1, 4, 6, 4, 1] \), which represents the coefficients for the expanded expression.

Step 2: Apply the Binomial Theorem

According to the Binomial Theorem, \[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]. Here, \( a = 2m \), \( b = -n \), and \( n = 4 \). Substitute these into the theorem using coefficients from Step 1.

Step 3: Calculate Each Term of the Expansion

1. First term: \( 1 \cdot (2m)^4 \cdot (-n)^0 = 16m^4 \).2. Second term: \( 4 \cdot (2m)^3 \cdot (-n)^1 = -32m^3n \).3. Third term: \( 6 \cdot (2m)^2 \cdot (-n)^2 = 24m^2n^2 \).4. Fourth term: \( 4 \cdot (2m)^1 \cdot (-n)^3 = -8mn^3 \).5. Fifth term: \( 1 \cdot (2m)^0 \cdot (-n)^4 = n^4 \).

Step 4: Combine the Terms

Combine all calculated terms to form the expanded expression:\[ 16m^4 - 32m^3n + 24m^2n^2 - 8mn^3 + n^4 \].

Key Concepts

Binomial Theorem

The Binomial Theorem is an important mathematical principle that helps us expand expressions raised to a power. Imagine you have a binomial, like (2m-n), and you need to raise it to the 4th power. Without the Binomial Theorem, this could be complicated.
The theorem tells us that any expression of the form (a+b)^n can be expanded as a sum of terms based on combinations. Each term has a coefficient determined by Pascal's Triangle. Here, a=2m , b=-n , and n=4 .
The formula for expansion is: (2m-n)^4= 1inom{4}{0}(2m)^4(-n)^0 + 4inom{4}{1}(2m)^3(-n)^1 + 6inom{4}{2}(2m)^2(-n)^2 + 4inom{4}{3}(2m)^1(-n)^3 + 1inom{4}{4}(2m)^0(-n)^4. Using coefficients [1,4,6,4,1] from the 4th row of Pascal's Triangle simplifies our work, giving an efficient expansion method.

Polynomial Expansion

When we talk about polynomial expansion, we mean converting a compact expression like (2m-n)^4 into a sum of different terms. Think of it as breaking a bigger task into smaller, manageable pieces.
Each entry in Pascal's Triangle directs us to the coefficients used in the expansion process. In the exercise, we use it to expand (2m-n)^4 into 16m^4 - 32m^3n + 24m^2n^2 - 8mn^3 + n^4.
Here's a simple way to understand the process: - Identify the binomial's components. - Use Pascal's Triangle to find coefficients. - Apply the powers to each part of the binomial, adjusting for positive and negative signs. - Combine the terms: simply adding together the results gives a final expanded polynomial.

Algebraic Expressions

Algebraic expressions are composed of variables and constants brought together by addition, subtraction, multiplication, or division.
In our exercise, (2m-n)^4 is a perfect example. It combines two algebraic terms being raised to the power of four.
Breaking it down:

• Terms: "2m" and "-n" are the building blocks.
• Operators: These show how terms relate (e.g., "-" indicates subtraction).
• Constants and Variables: The number "2" is a constant, while "m" and "n" are variables. Constants remain unchanged but variables can vary.

The art of working with algebraic expressions involves manipulating these parts to simplify or solve them, often making use of specific rules or theorems, like the Binomial Theorem or techniques in polynomial expansion. By mastering these concepts, you can unlock the power of algebra to solve a wide array of problems systematically and efficiently.