Problem 75 - Algebra A Combined Function

Question

Factor. $$ x^{3}+125 $$

Step-by-Step Solution

Answer

The factored form is \((x + 5)(x^2 - 5x + 25)\).

Step 1: Identify the Sum of Cubes

The expression given is \(x^3 + 125\). Here, \(x^3\) is already a cube. Notice that 125 is \(5^3\). This expression can be represented as a sum of cubes of the form \(a^3 + b^3\), where \(a = x\) and \(b = 5\).

Step 2: Recall the Sum of Cubes Formula

The sum of cubes formula is \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\). For our expression, substitute \(a = x\) and \(b = 5\) into the formula.

Step 3: Apply the Sum of Cubes Formula

Plug \(a = x\) and \(b = 5\) into the formula, resulting in: \[(x + 5)(x^2 - 5x + 25)\].

Step 4: Verify the Factored Form

To ensure correctness, expand the factored form \((x + 5)(x^2 - 5x + 25)\) and check if it simplifies back to \(x^3 + 125\). Upon expanding, you will see it results in the original expression, confirming the factorization is correct.

Key Concepts

Factoring

Factoring is a fundamental algebraic process involving rewriting expressions as a product of simpler expressions. In the context of cubic expressions, this often involves identifying special forms like the sum of cubes. To factor the sum of cubes, you'll need to recognize if the expression matches the standard form. For example, in the expression \(x^3 + 125\), each term is a cube: \(x^3\) and \(125 = 5^3\).

Once identified, use the sum of cubes formula: \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\). This formula breaks the expression into two factors. Applying it to \(x^3 + 125\), where \(a = x\) and \(b = 5\), gives the factored form \((x + 5)(x^2 - 5x + 25)\).

Factoring is about breaking down, simplifying, and understanding relationships within expressions. It requires knowledge of standard forms and formulas. In algebra, knowing these tools allows for more complex problem solving and helps verify solutions.

Algebra

Algebra serves as the foundation of math involving symbols and the rules for manipulating those symbols. It provides the language and tools we use to express and solve problems. When dealing with algebraic expressions, such as cubic equations, understanding how to manipulate their forms is key.

The problem \(x^3 + 125\) is tackled using algebraic methods. Recognizing and using formulas like the sum of cubes formula is essential in algebra. These formulas are powerful shortcuts that simplify complex expressions, making them easier to analyze and solve.

Beyond simple computations, algebra also involves logical thinking and the ability to justify each step in problem-solving. This means showing that a factorization is correct by expansion. Expanding \((x+5)(x^2-5x+25)\) back to \(x^3+125\) confirms the solution is valid and highlights the consistency of algebraic principles.

Cubic Expressions

Cubic expressions are polynomial equations where the highest power is three. These equations often appear in real-world contexts, modeling phenomena from physics to finance. Understanding cubic expressions and how to manipulate them is crucial to tackling higher-level math problems.

In the example \(x^3 + 125\), the expression can be identified as a cubic expression because it includes the cube of a variable. The solution process involves recognizing these types of patterns and applying formulas specifically designed for them, like the sum of cubes.

Learning to identify different kinds of cubic expressions and the corresponding identities to simplify them enhances problem-solving skills. By recognizing a cubic expression and converting it through algebraic manipulation, students gain insights into the structure and behavior of polynomials. Cubic expressions provide a unique challenge, blending creativity with logical processes to master their handling.