Problem 38 - Elementary and Intermediate Algebra

Question

Factor. $$ s^{3}-64 t^{3} $$

Step-by-Step Solution

Answer

The factorization is \((s - 4t)(s^2 + 4st + 16t^2)\).

Step 1: Recognize the Formula

The expression given is a difference of cubes, which can be factored using the formula \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\). Identify the cube components in each part of the expression: \(s^3\) and \((4t)^3\).

Step 2: Identify \(a\) and \(b\)

From \(s^3 - (4t)^3\), assign \(a = s\) and \(b = 4t\). This will allow us to apply the difference of cubes formula.

Step 3: Apply the Formula

Substitute \(a\) and \(b\) into the formula: \(s^3 - (4t)^3 = (s - 4t)(s^2 + s(4t) + (4t)^2)\).

Step 4: Simplify the Expression

Simplify the expression inside the second parenthesis:1. \(s^2\) remains as is.2. \(s(4t) = 4st\) 3. \((4t)^2 = 16t^2\) Thus, the expression becomes \(s^3 - (4t)^3 = (s - 4t)(s^2 + 4st + 16t^2)\).

Step 5: Verify the Factorization

Multiply the factors to confirm they result in the original expression:1. \((s)(s^2 + 4st + 16t^2) = s^3 + 4s^2t + 16st^2\)2. \((-4t)(s^2 + 4st + 16t^2) = -4ts^2 - 16t^2s - 64t^3 \) Combine like terms: The expression simplifies to \(s^3 - 64t^3\), confirming the factorization is correct.

Key Concepts

Difference of Cubes

When you encounter an expression like \( s^3 - 64t^3 \), you're dealing with a difference of cubes. A difference of cubes refers to any expression of the form \( a^3 - b^3 \). These types of expressions can be factored using a specific formula, which is: \[ a^3 - b^3 = (a-b)(a^2 + ab + b^2) \] In this case, the goal is to recognize and identify the components of the expression that are perfect cubes. Here, \( s^3 \) and \((4t)^3\) are both perfect cubes, making them perfect for using the difference of cubes formula.

• The cube root of \( s^3 \) is \( s \).
• The cube root of \( (4t)^3 \) is \( 4t \).

With this understanding, you can assign \( a = s \) and \( b = 4t \), allowing you to apply the formula effectively.

Algebraic Expressions

Algebraic expressions consist of variables, constants, and operations. They don't generally have an equal sign as an equation would. In the expression \( s^3 - 64t^3 \), we can identify the following:

• \( s^3 \) is a variable term raised to the third power, representing the cube of \( s \).
• \( 64t^3 \) involves both a constant (64) and a variable \( t^3 \), representing the cube of \( 4t \).

Understanding how these elements interact is crucial when factoring complex expressions like this. The task here involves recognizing how the cubes interact in the given subtraction to better understand how to factor them efficiently. By identifying terms as cubes, we simplify the expression by using the difference of cubes formula, making the process systematic and less confusing.

Algebra Steps

The step-by-step process of factoring \( s^3 - 64t^3 \) is methodical and relies on a clear understanding of algebra basics:

Step 1: Recognize the Formula

Quickly identify that \( s^3 - 64t^3 \) is a difference of cubes which utilizes the formula \( a^3 - b^3 = (a-b)(a^2 + ab + b^2) \).

Step 2: Assign Values to \( a \) and \( b \)

Assign \( a = s \) and \( b = 4t \), where \( a^3 = s^3 \) and \( b^3 = (4t)^3 \).

Step 3: Apply the Formula

Plug \( a \) and \( b \) into the standard formula: \( (s - 4t)(s^2 + s t + (4t)^2) \).

Step 4: Simplify the Expression

Perform arithmetic within the expression:

• \( s^2 \) stays the same.
• Calculate \( s \times 4t = 4st \).
• Compute \((4t)^2 = 16t^2 \).

Thus resulting in \( (s - 4t)(s^2 + 4st + 16t^2) \).

Step 5: Verify the Factorization

Multiply the factors back together to ensure they yield the original expression, confirming the correctness of your factorization.