Problem 63 - Algebra and Trigonometry

Question

Factor the expression completely. $$ 8 x^{3}-125 $$

Step-by-Step Solution

Answer

The expression \( 8x^3 - 125 \) factors to \((2x - 5)(4x^2 + 10x + 25)\).

Step 1: Identify the Expression as a Difference of Cubes

The given expression is \( 8x^3 - 125 \). Notice that this can be written as a difference of cubes, because \( 8x^3 = (2x)^3 \) and \( 125 = 5^3 \). Let's express the expression in the form \( (a^3 - b^3) \), where \( a = 2x \) and \( b = 5 \).

Step 2: Recall the Difference of Cubes Formula

The formula for factoring the difference of cubes is \( a^3 - b^3 = (a-b)(a^2 + ab + b^2) \). In our case, \( a = 2x \) and \( b = 5 \).

Step 3: Apply the Difference of Cubes Formula

Substitute \( a = 2x \) and \( b = 5 \) into the formula: \( a^3 - b^3 = (a-b)(a^2 + ab + b^2) \) becomes \( (2x - 5)((2x)^2 + (2x)(5) + 5^2) \).

Step 4: Simplify the Factored Expression

Calculate each term in the factored expression: \((2x - 5)\) and \((2x)^2 = 4x^2\), \((2x)(5) = 10x\), \(5^2 = 25\). Thus, \((2x - 5)(4x^2 + 10x + 25)\).

Step 5: Verify Solution

Multiply both factors to verify correctness: Expand \((2x - 5)(4x^2 + 10x + 25)\) to ensure it equals the original expression \(8x^3 - 125\). This confirms the factorization is correct.

Key Concepts

Factoring Polynomials

Factoring polynomials means breaking down a complex expression into simpler parts or products. Think of it like unwrapping a present to see what's inside! When a polynomial is 'factored completely,' it means no further factorization is possible, and you've found all the smallest pieces.
Here's a basic approach to factoring:

• Identify if the polynomial is a simple form or complex.
• Check for common patterns, such as differences of squares or cubes.
• Look for any common factors first.
• Apply known formulas like the difference of cubes.

Using these steps makes challenging polynomials easier to handle. The ultimate goal is to express the polynomial as a product of two or more simpler expressions.

Algebra

Algebra is a branch of mathematics that uses symbols and letters to solve equations with various quantities. It's like solving puzzles where letters stand in for numbers until we find their true value.
Key concepts in algebra include:

• Variables: Symbols like \(x\) or \(y\) that represent unknowns or quantities that can change.
• Expressions: Combinations of numbers, variables, and operators like \(+\), \(-\), or \(\times\).
• Equations: Statements that two expressions are equal, often involving variables.
• Formulas: A special equation that shows the relationship between different variables.

Algebra is the foundation for many advanced math topics, helping us model real-world situations and solve practical problems.

Cubed Numbers

Cubed numbers are numbers raised to the power of three, denoted as \(n^3\). Imagine a cube in geometry with all side lengths being \(n\); its volume would be \(n^3\). It's essentially multiplying a number by itself twice more.
For example:

• \(2^3 = 2 \times 2 \times 2 = 8\)
• \(3^3 = 3 \times 3 \times 3 = 27\)
• \(4^3 = 4 \times 4 \times 4 = 64\)

Recognizing cubed numbers helps in factoring expressions like differences of cubes. Just think of each term as a tiny cube stack! Knowing cubes is key when using formulas in algebra and simplifying complex expressions. The difference of cubes formula, for instance, relies on being able to express numbers as cubes, making it easier to solve polynomial equations.