Problem 58 - College Algebra Essentials

Question

Factor using the formula for the sum or difference of two cubes. $$x^{3}+64$$

Step-by-Step Solution

Answer

The factored form of the expression \(x^3 + 64\) is \((x + 4)(x^2 - 4x + 16)\)

Step 1: Identifying a and b

We compare the given expression \(x^3 + 64\) to the formula for the sum of two cubes where \(a^3 + b^3\), to determine the values of \(a\) and \(b\). Here, \(a^3 = x^3\) implies \(a = x\) and \(b^3 = 64\) implies \(b = 4\).

Step 2: Using Formula

Plug these values into the formula for the sum of cubes, so we have:\((x + 4)(x^2 - 4x + 16)\)

Step 3: Verification

Multiply out the factored form to ensure that it simplifies back to the original expression, \(x^3 + 64\).

Key Concepts

Polynomial Factorization

Factoring polynomials is an essential skill in algebra that allows us to simplify complex algebraic expressions and solve polynomial equations more efficiently. For the sum of cubes, the factorization process involves breaking down an expression like x^3 + 64 into a product of simpler polynomials.

The formula for the factoring of a sum of cubes is always of the form a^3 + b^3 = (a + b)(a^2 - ab + b^2). It's a special factoring technique used when identifying perfect cubes within a polynomial expression.

In the exercise at hand, identifying the cubic terms is the key to using this technique. The term 64 is recognized as 4^3, allowing us to set up the equation with a = x and b = 4. Then, by applying the formula, we can factor the expression as \(x + 4)(x^2 - 4x + 16)\). The crucial takeaway is that recognizing and applying these patterns in polynomial factorization can turn a daunting algebraic task into a more manageable one.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. When it comes to a cubic expression like x^3 + 64, the goal is to manipulate this expression into a more useful form, typically for the purpose of graphing the function, finding its roots, or simplifying the expression for further calculation.

Cubic expressions, one type of algebraic expression, are especially interesting since they represent three-dimensional geometries in certain applications. They also have intriguing properties, such as having at least one real root.

Understanding the structure of these expressions is crucial for tasks like factoring. For instance, noting that 64 is a cube and can be written as 4^3 means it's easier to recognize the sum of cubes pattern. Further, being comfortable converting between different forms of algebraic expressions enhances one's mathematical fluency and problem-solving flexibility.

Cubic Equations

A cubic equation is a type of polynomial equation that involves a variable raised to the third power. The general form of a cubic equation is \(ax^3 + bx^2 + cx + d = 0\), where \(a \eq 0\).

These equations are known for having complex behaviors and can have up to three real roots, which are the solutions to the equation. In the provided exercise x^3 + 64, to turn the expression into an equation, we would set it equal to zero, leading to \(x^3 + 64 = 0\). Solving this would require factoring the sum of cubes first, then using the Zero Product Property to find the roots.

Understanding how to factor cubic equations is valuable as it can reveal insights into the nature of their roots without necessarily using complex numerical methods or graphing. For students and mathematicians alike, studying cubic equations is an excellent practice in gaining deeper insight into polynomial behaviors.