Problem 52 - A Graphical Approach to College Algebra

Question

Factor each sum or difference of cubes completely. $$r^{3}+27$$

Step-by-Step Solution

Answer

The factored form is \((r + 3)(r^2 - 3r + 9)\).

Step 1: Identify the Sum of Cubes Formula

Recognize that the expression \(r^3 + 27\) is a sum of cubes, where \(a = r\) and \(b = 3\) since \(27 = 3^3\). The sum of cubes formula is \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\).

Step 2: Substitute Variables into Formula

Using the formula from the previous step, substitute \(a = r\) and \(b = 3\) into the formula. Therefore, we have \(a^3 + b^3 = (r + 3)(r^2 - 3r + 9)\).

Step 3: Verify the Factored Form

Ensure that the expression \((r + 3)(r^2 - 3r + 9)\) indeed equals the original \(r^3 + 27\) by expanding the factors: \((r + 3)(r^2 - 3r + 9) = r^3 - 3r^2 + 9r + 3r^2 - 9r + 27 = r^3 + 27\). The factored form is correct.

Key Concepts

Sum of Cubes

The concept of sum of cubes is a specific type of polynomial expression where two terms, each raised to the third power, are added together. A typical example is the expression \(r^3 + 27\). In this scenario, both \(r^3\) and \(27\) (which is \(3^3\)) are perfect cubes.

• The formula for factoring the sum of cubes is: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\).
• This formula provides a straightforward method to transform seemingly complex cubic expressions into simpler binomial and trinomial factors.

Recognizing and applying this formula is essential in algebra since it simplifies polynomial expressions. For example, identifying \(a = r\) and \(b = 3\) allows us to apply the formula to factor the expression \(r^3 + 27\) into \((r + 3)(r^2 - 3r + 9)\). This reduces the complexity and provides a factored form that can be used in further mathematical operations.

Algebra

Algebra is a key branch of mathematics that deals with symbols and the rules for manipulating those symbols. It's often about finding the values of unknowns in equations or breaking down complex problems into easier ones.
The step-by-step solution for \(r^3 + 27\) is a perfect example of algebra in action. It demonstrates the process of:

• Recognizing patterns and structures within mathematical expressions,
• Applying specific formulas like the sum of cubes,
• Simplifying polynomials to solve for easier components.

Algebra provides the tools to factor expressions systematically, making the complex become manageable. Once you understand these concepts, you can solve equations, factor expressions, and efficiently work with formulas, leading to overall better problem-solving skills.

Polynomial Expressions

Polynomial expressions are algebraic expressions that consist of variables, coefficients, and non-negative integer exponents. They can take various forms, such as linear, quadratic, or cubic, like the one in the original problem.
Understanding polynomials involves knowing how to manipulate them to factor or expand them correctly. For example, the original expression \(r^3 + 27\) is a cubic polynomial. It combines a term involving a variable raised to the third power and a constant term.
Decomposition into simpler polynomials through methods like the sum of cubes formula results in easier manipulation. By recognizing polynomials as flexible mathematical expressions, we can:

• Evaluate them with substitutions,
• Find roots or solutions,
• Graph their behavior.

Ultimately, mastering polynomial expressions and their operations makes tackling higher-degree polynomials and complex algebra problems much more approachable.