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Magazine Chapter 6: Problem 38
Factor. $$ s^{3}-64 t^{3} $$
Short Answer
Expert verified
The factorization is \((s - 4t)(s^2 + 4st + 16t^2)\).
Step by step solution
01
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 \((4t)^3\).
02
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.
03
Apply the Formula
Substitute \(a\) and \(b\) into the formula: \(s^3 - (4t)^3 = (s - 4t)(s^2 + s(4t) + (4t)^2)\).
04
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 \(s^3 - (4t)^3 = (s - 4t)(s^2 + 4st + 16t^2)\).
05
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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 \( 4t \).
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 \( 4t \).
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 \((4t)^2 = 16t^2 \).
