Problem 22 - College Algebra and Calculus: An Applied Approach

Question

Factor the sum or difference of cubes.\(y^{3}+1000\)

Step-by-Step Solution

Answer

The factored form of \(y^{3} + 1000\) is \((y + 10)(y^2 - 10y + 100)\).

Step 1 Identify the cubes

In the given expression \(y^{3}+1000\), identify the cubes. Here, \(a^3 = y^3\) and \(b^3 = 1000\).

Step 2 Determine the roots

Determine the cubic roots of the identified cubes. In this case, \(a = y\) and \(b = 10\) as the cube root of 1000 is 10.

Step 3 Apply the formula

Plug the appropriate values into the sum of cubes formula \((a^3 + b^3) = (a + b)(a^2 - ab + b^2)\). In the given case, the formula becomes \((y + 10)(y^2 - 10y + 100)\) after substituting the values of a and b.

Key Concepts

Factoring Algebraic Expressions

Factoring algebraic expressions is a fundamental technique in algebra used to simplify complex expressions or solve equations. The process involves rewriting a polynomial as a product of its factors.
Breaking down the expression into simpler components can make it easier to work with and understand.
The goal is to express the original polynomial as a product of two or more simpler polynomials. Here are some general steps you can follow to factor algebraic expressions:

• Look for common factors: Always start by identifying any common factors in the terms.
• Use identities: Familiarize yourself with special identities, like the difference and sum of squares or cubes, to help in factoring.
• Apply techniques: Use different strategies and techniques like grouping, factor by grouping, or using formulas to factor completely.

Understanding how to factor expressions effectively is crucial as it lays the groundwork for solving a variety of algebraic problems, particularly polynomial equations.

Cubic Equations

Cubic equations are polynomial equations where the highest degree of the variable is three. The general form of a cubic equation is:\[ ax^3 + bx^2 + cx + d = 0 \]Cubic equations can have three real roots or a combination of real and complex roots. Solving these equations involves finding values of the variable that make the equation true. Here's how you can approach solving cubic equations:

• Identify the cubes: Determine if the expression is a perfect cube or involves a sum/difference of cubes.
• Factor the equation: Use methods such as the factor theorem, synthetic division, or applying the cubic formulas to factor the equation down to a quadratic.
• Solve the factors: Once factored, solve the resulting simpler equations to find the roots of the original cubic equation.

Solving cubic equations is a bit more involved compared to quadratic equations, but with practice and the right techniques, it becomes a manageable task.

Polynomial Identities

Polynomial identities are equations that are true for every possible value of the variable within the polynomial. These identities are incredibly useful for quickly factoring expressions and solving equations.
The sum and difference of cubes have their specific identities, which can be very helpful:

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

The application of these identities provides a straightforward path to factor otherwise complex polynomial expressions. For example, knowing that \( y^3 + 1000 \) is a sum of cubes allows us to use the identity for the sum of cubes, simplifying the expression into the product of two simpler polynomials.
Learning and applying polynomial identities significantly enhances one's ability to navigate through algebraic problems efficiently.