Problem 21 - Algebra and Trigonometry

Question

17–24 ? Use a Factoring Formula to factor the expression. $$ 8 s^{3}-125 t^{3} $$

Step-by-Step Solution

Answer

The expression factors to \((2s - 5t)(4s^2 + 10st + 25t^2)\).

Step 1: Identify the Type of Expression

The given expression is \(8s^3 - 125t^3\), which is a difference of cubes. Cubes are expressions in the form \(a^3 - b^3\). Here, we need to rewrite \(8s^3\) and \(125t^3\) in terms of cubes.

Step 2: Express Terms as Perfect Cubes

Rewrite \(8s^3\) and \(125t^3\) as cubes: \(8s^3 = (2s)^3\) and \(125t^3 = (5t)^3\). So, the expression becomes \((2s)^3 - (5t)^3\).

Step 3: Apply the Difference of Cubes Formula

The difference of cubes formula is \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\). Identify \(a = 2s\) and \(b = 5t\).

Step 4: Substitute into the Formula

Substitute \(a = 2s\) and \(b = 5t\) into the formula: \((2s - 5t)((2s)^2 + (2s)(5t) + (5t)^2)\).

Step 5: Simplify the Expression

Calculate each part: \((2s)^2 = 4s^2\), \((2s)(5t) = 10st\), and \((5t)^2 = 25t^2\). So, \(a^2 + ab + b^2 = 4s^2 + 10st + 25t^2\).

Step 6: Final Factored Expression

Combine all parts from previous steps: The expression \(8s^3 - 125t^3\) factors to \((2s - 5t)(4s^2 + 10st + 25t^2)\).

Key Concepts

Difference of Cubes

The difference of cubes is an important concept in algebra that refers to the subtraction of two cubed terms. When we talk about cubes, we mean an algebraic expression raised to the third power. For instance, if we have two terms, say \(a\) and \(b\), then their cubes are \(a^3\) and \(b^3\).

To factor a difference of cubes, which looks like \(a^3 - b^3\), we use a specific formula:

• \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)

This formula breaks down the expression into a product of two simpler expressions, making it more manageable.
An example is the expression \(8s^3 - 125t^3\). We recognize it as a difference of cubes because \(8s^3 = (2s)^3\) and \(125t^3 = (5t)^3\). After identifying the cubes, we apply the difference of cubes formula to factor it.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations that form a mathematical phrase. They can represent real-world quantities and are foundational in algebra. Expressions like \(8s^3\) or \(125t^3\) are polynomial expressions that are commonly found in algebra.

An expression like \(8s^3\) is made by multiplying the variable \(s\) by itself two more times and then by 8; similarly, \(125t^3\) involves the variable \(t\) raised to the third power and multiplied by 125. In any algebraic expression, understanding how to manipulate terms using operations like addition, subtraction, multiplication, and division is key.
Algebraic expressions form the basis for higher-level mathematical operations such as factoring, which allows us to simplify and solve equations effectively.

Factoring Formulas

Factoring is a crucial skill in algebra that involves breaking down expressions into their simplest components. Use factoring formulas to simplify or solve equations by finding expressions that multiply together to give the original expression. These formulas include special cases like:

• Difference of cubes: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)
• Sum of cubes: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)

The goal is often to rewrite an expression in a product form that reveals its roots or solutions more clearly.

Using the difference of cubes as an example, it's a tool that lets you break down an expression like \(8s^3 - 125t^3\) into \((2s - 5t)(4s^2 + 10st + 25t^2)\). By applying these formulas, complex expressions become manageable and much easier to work with.