Problem 71 - College Algebra with Modeling and Visualization

Question

Factor the expression completely, if possible. \(4-r^{2} t^{2}\)

Step-by-Step Solution

Answer

The factored expression is \((2 - rt)(2 + rt)\).

Step 1: Recognize the Expression Structure

The given expression is \(4 - r^2 t^2\). Notice that this is a difference of squares because it can be written as \(a^2 - b^2\), where \(a = 2\) and \(b = rt\).

Step 2: Apply the Difference of Squares Formula

Use the difference of squares formula, which states \(a^2 - b^2 = (a - b)(a + b)\). Substituting \(a = 2\) and \(b = rt\), the formula becomes:\[(2 - rt)(2 + rt)\].

Step 3: Verify the Factorization

Multiply the factors \((2 - rt)(2 + rt)\) to ensure that they result in the original expression. Calculating gives:\(2^2 - (rt)^2 = 4 - r^2 t^2\), which matches the original expression.

Key Concepts

Difference of Squares

In algebra, the difference of squares is a concept that involves expressions where two square terms are subtracted from each other. This can be represented by the formula:

• \( a^2 - b^2 = (a-b)(a+b) \).

The expression needs to fit the pattern of one square minus another square. A key aspect of using this formula is identifying the squared terms.
In our example, the expression \( 4 - r^2 t^2 \) is a difference of squares. Here, \( 4 \) is the square of \( 2 \), and \( r^2 t^2 \) is the square of \( rt \). Once you identify this structure, you can factor the expression easily. The difference of squares is a handy shortcut in factorization problems, making it simpler to break down complex polynomial expressions.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and arithmetic operations such as addition and subtraction. Unlike equations, algebraic expressions do not have an equal sign. They can be simple, like \( x + 2 \), or more complex, with multiple terms and factors.
The goal when working with algebraic expressions is often to simplify or factor them. Factorization involves rewriting the expression as a product of its factors. This is useful for solving equations or simplifying calculations.
In our exercise with \( 4 - r^2 t^2 \), we worked towards rewriting it using the difference of squares to find its factors. Understanding each part of the expression helps in identifying what techniques to apply and thus simpler solution pathways.

Polynomials

Polynomials are expressions made up of variables and coefficients, organized in terms of powers with non-negative integer exponents. They are one of the most fundamental structures in algebra. A polynomial can have constants, variables, and exponents but never division by a variable.
In their standard form, polynomials may look like \( ax^n + bx^{n-1} + ... + c \) where \( a, b, \) and \( c \) are coefficients and \( n \) is the degree of the polynomial.

• Constant polynomials: Terms with no variable, like \( 4 \).
• Linear polynomials: First-degree polynomials, like \( x + 1 \).
• Quadratic polynomials: Second-degree, such as \( ax^2 + bx + c \).

In our exercise, the expression \( 4 - r^2 t^2 \) is a specific type of polynomial, evaluated as a simplified quadratic polynomial with a twist: the absence of the middle term, making it a perfect candidate for the difference of squares formula. Understanding the type and degree of a polynomial is essential when discussing factorization and simplification mechanisms.