Problem 52 - Algebra and Trigonometry

Question

Factor the expression completely. $$ 4 t^{2}-9 s^{2} $$

Step-by-Step Solution

Answer

The expression factors to \((2t-3s)(2t+3s)\).

Step 1: Identify the Structure

Recognize that the given expression is of the form \( a^2 - b^2 \), which is the difference of squares. In this case, we have \( a = 2t \) and \( b = 3s \), because \( (2t)^2 = 4t^2 \) and \( (3s)^2 = 9s^2 \).

Step 2: Use the Difference of Squares Formula

Apply the difference of squares formula: \( a^2 - b^2 = (a-b)(a+b) \). Substitute \( a = 2t \) and \( b = 3s \) into the formula to get the factors: \((2t-3s)(2t+3s)\).

Step 3: Verify Your Solution

Ensure correctness by expanding \((2t-3s)(2t+3s)\) to check if it results back in \(4t^2 - 9s^2\). Using the distributive method, \((2t-3s)(2t+3s) = (2t)^2 - (3s)^2 = 4t^2 - 9s^2\). It matches the original expression, confirming the solution is correct.

Key Concepts

Difference of Squares

The difference of squares is a fundamental concept in algebraic expressions. It simplifies certain polynomial equations by breaking them into smaller, more manageable parts. When you have an expression of the form \( a^2 - b^2 \), it is called the difference of squares. This is because you are essentially looking at the subtraction of two perfect square terms.
To factor such an expression, you use the formula:

• \( a^2 - b^2 = (a - b)(a + b) \)

This formula tells us that the expression can be rewritten as the product of two binomials. For example, if you have \( 4t^2 - 9s^2 \), you can identify \( a = 2t \) and \( b = 3s \). Using the formula, the expression becomes \((2t - 3s)(2t + 3s)\).
Using the difference of squares formula simplifies the factoring process and helps you solve polynomial equations more efficiently.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. They form the building blocks for polynomial equations that you often work with in algebra.
An expression like \( 4t^2 - 9s^2 \) can be seen as an algebraic expression because it involves variables \( t \) and \( s \), coefficients \( 4 \) and \( 9 \), and the subtraction operation.
Here are some important points to consider when dealing with algebraic expressions:

• Identify the terms: Each part of the expression separated by a plus or minus sign is a term. In our example, \( 4t^2 \) and \( 9s^2 \) are terms.
• Simplify the expression: Look for patterns such as the difference of squares to simplify.
• Factor completely: Whenever possible, break down the expression further as in rewriting \( 4t^2 - 9s^2 \) to \( (2t-3s)(2t+3s) \).

Understanding algebraic expressions will make it much easier for you to work with even more complex polynomial equations.

Polynomial Equations

Polynomial equations are equations that involve polynomials, which are expressions consisting of variables raised to whole-number exponents and having coefficients. They can be classified based on their degree, which is the highest power of the variable in the equation.
In our example, \( 4t^2 - 9s^2 \) is a polynomial expression, and it can be set equal to zero to form a polynomial equation, \( 4t^2 - 9s^2 = 0 \). Here's how we can approach solving it:

• Use the factored form: From our previous factoring, the expression becomes \((2t-3s)(2t+3s)\).
• Set each factor to zero: This gives the equations \(2t-3s = 0\) and \(2t+3s = 0\).
• Solve for the variable: Solve these equations separately to find the values of \( t \) and \( s \).

Working with polynomial equations often involves such techniques of factoring, making them easier to solve. The key is to identify patterns and apply appropriate algebraic methods.