Problem 25 - Precalculus: Functions and Graphs

Question

Factor the polynomial. $$8 x^{3}+64 y^{3}$$

Step-by-Step Solution

Answer

The polynomial factors as \((2x + 4y)(4x^2 - 8xy + 16y^2)\).

Step 1: Identify the Structure

The given polynomial \(8x^3 + 64y^3\) is a sum of cubes. We can write it as \((2x)^3 + (4y)^3\). The sum of cubes formula is \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\).

Step 2: Apply the Formula

In our expression, \(a = 2x\) and \(b = 4y\). Substitute these into the formula: \((2x)^3 + (4y)^3 = (2x + 4y)((2x)^2 - (2x)(4y) + (4y)^2)\).

Step 3: Simplify the Factors

First, calculate \((2x + 4y)\) which remains as is. Then, simplify \((2x)^2 - (2x)(4y) + (4y)^2\) as follows: \((2x)^2 = 4x^2\), \(-(2x)(4y) = -8xy\), and \((4y)^2 = 16y^2\). Thus, \(4x^2 - 8xy + 16y^2\).

Step 4: Final Factored Form

Putting it all together, the factorization of \(8x^3 + 64y^3\) is \((2x + 4y)(4x^2 - 8xy + 16y^2)\).

Key Concepts

Understanding the Sum of Cubes

The concept of the "sum of cubes" is a fundamental factoring technique in algebra. It involves expressions of the form \(a^3 + b^3\).
The sum of cubes formula is given by:

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

This formula helps to break down a complex expression into a product of two factors.
These factors may look simpler and are easier to work with, especially when solving equations or simplifying expressions.

Using this formula requires identifying terms that fit the cube structure. Once identified, you substitute them into the formula for a simplified expression. It's an efficient way to handle cubic terms in polynomial expressions.

Factoring Techniques for Polynomials

Factoring is an essential skill when working with polynomials. It involves breaking down a polynomial into products of simpler polynomials or factors. This technique is not only crucial for solving equations but also for simplifying polynomial expressions, analyzing graphs, and more.

• Identify Special Forms: Look for patterns like the sum of cubes, difference of squares, or perfect square trinomials.
• Use Formulas: Apply formulas such as \(a^2 - b^2 = (a-b)(a+b)\) or the sum of cubes formula.
• Simplify the Expression: After factoring, ensure that the expression is fully simplified for practical use.
• Check Your Work: Verify the factors by expanding them to see if they match the original expression.

Each factorization method helps tackle specific types of polynomials, enhancing your mathematical toolbox for various problems. Mastering these can significantly ease your work with algebraic expressions.

Exploring Polynomial Expressions

Polynomials are fundamental building blocks in mathematics.
They are expressions made up of terms that include numbers and variables raised to whole number exponents. Each term is a product of a constant (known as the coefficient) and a variable raised to an exponent.
For example, in the polynomial \(8x^3 + 64y^3\), there are two terms: \(8x^3\) and \(64y^3\).

Key Characteristics of Polynomials:

• Degree: The highest power of the variable determines a polynomial's degree. In \(8x^3\), the degree is three.
• Standard Form: Polynomials are usually written in descending order of exponent values.
• Operations: You can perform addition, subtraction, multiplication, and division on polynomials.

Understanding these elements helps in recognizing patterns that allow for efficient factoring and manipulation of polynomial expressions, ultimately leading to finding zeros and solving equations.