Problem 18 - Precalculus: Functions and Graphs

Question

Express as a polynomial. $$\left(4 r^{2}-3 s\right)^{2}$$

Step-by-Step Solution

Answer

The polynomial is \(16r^4 - 24r^2s + 9s^2\).

Step 1: Understand the Problem

We need to expand the expression \((4r^2 - 3s)^2\) by expressing it as a polynomial, which involves simplifying and rewriting the expression without parentheses.

Step 2: Apply the Binomial Expansion Formula

The square of a binomial \((a - b)^2\) is expanded using the formula \((a - b)^2 = a^2 - 2ab + b^2\). Here, we identify \(a = 4r^2\) and \(b = 3s\).

Step 3: Calculate Each Term

Begin by calculating each term in the expansion:1. Find \(a^2\): \( (4r^2)^2 = 16r^4 \).2. Calculate \(-2ab\): \(-2 \cdot 4r^2 \cdot 3s = -24r^2s\).3. Compute \(b^2\): \((3s)^2 = 9s^2\).

Step 4: Combine the Terms

Combine the results from Step 3:\(16r^4 - 24r^2s + 9s^2\).

Step 5: Write the Polynomial

The polynomial expression obtained from expanding \((4r^2 - 3s)^2\) is \(16r^4 - 24r^2s + 9s^2\).

Key Concepts

Binomial Expansion

Binomial expansion is a mathematical method used to expand expressions that are squared or raised to higher powers, typically consisting of two terms. This involves applying the binomial theorem. For our exercise, let's look at the expression \(a-b\)^2. When squared, it can be expanded using the formula: \(a^2 - 2ab + b^2\).
This formula is derived from the general pattern of products when a binomial is multiplied by itself. Understanding how each term is derived in this process is crucial:

• The term \(a^2\) represents the square of the first term.
• The term \(-2ab\) represents twice the product of the two terms.
• The term \(b^2\) represents the square of the second term.

In our original function \(4r^2 - 3s\), recognizing \(a=4r^2\) and \(b=3s\) allows us to apply this expansion formula effectively. By calculating each term, we transform the binomial into a more familiar polynomial format.

Algebraic Expressions

Algebraic expressions include numbers, variables, and operations such as addition, multiplication, and exponents. In our example, we used the expression \(4r^2 - 3s\) which employs both variables and exponents. Understanding these variables and their powers is essential to simplifying expressions.
Each component represents a part of a whole:

• Variables stand in place of unknown values. In \(4r^2\), \(r\) is a variable.
• Coefficients are the numbers multiplying the variables. The 4 in \(4r^2\) is a coefficient.
• Exponents indicate how many times the variable multiplies itself. Here, \(r^2\) means \(r\) is multiplied by itself once, squaring it.

Simplifying algebraic expressions involves identifying like terms and performing operations, ensuring we correctly simplify complex binomials into simpler polynomial forms.

Polynomial Simplification

Simplifying a polynomial refers to expressing it in its simplest form by combining like terms and performing arithmetic operations where possible.
In our initial expression \(16r^4 - 24r^2s + 9s^2\), each term is unique, yet they all originate from the expansion of a binomial. Here's how we simplified it:

• Identify terms that share the same variables and exponents to combine them.
• Perform arithmetic operations like addition or subtraction on these like terms.
• Arrange the terms, starting from the highest to the lowest power, to showcase the polynomial's order.

This makes it easier to understand and apply operations on the polynomial. Mastering polynomial simplification not only aids in solving equations efficiently but also enhances understanding of algebraic structure through a clearer, more concise expression map.