Problem 50 - College Algebra

Question

In Exercises \(49-56,\) factor using the formula for the sum or difference of two cubes. $$x^{3}+64$$

Step-by-Step Solution

Answer

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

Step 1: Identify a and b

The formula for the sum of cubes is \(a^{3} + b^{3} = (a + b)(a^{2} - ab + b^{2})\). In this problem, a and b must satisfy this equation: \(x^{3} + 64 = a^{3} + b^{3}\). It can be observed that \(x^{3} = a^{3}\) suggests a should be x. Likewise, \( 64 = b^{3}\) implies b should be 4 because \(4^{3} = 64\).

Step 2: Use the Formula for Factoring the Sum of Cubes

Substitute the values of a and b into the formula: \((x + 4)(x^{2} - 4x + 16)\)

Step 3: Solution: Factor Expression

The fully factored expression of \(x^{3} + 64\) is \( (x + 4)(x^{2} - 4x + 16) \). The second factor cannot be further factored, as it cannot be written as the difference of squares and does not have rational roots.

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols such as addition, subtraction, multiplication, and division. In the context of the exercise provided, the algebraic expression is the sum of two 'cubed' terms, which signifies raising a number or variable to the third power. In our example, the expression is x^{3} + 64, where x is a variable and 64 is a constant.

To solve such an expression by factoring, it's crucial to recognize that the cube of a variable and the cube of a constant can sometimes be re-written through a specific factoring method, turning the original expression into a product of simpler expressions. This approach not only simplifies the problem but also unveils its underlying structure and can lead to further simplifications or solutions in the context of an equation.

The Art of Polynomial Factoring

Polynomial factoring is a process of expressing a polynomial as the product of its factors, which are also polynomials. This technique is akin to finding the 'building blocks' of algebraic constructions. The polynomial in our exercise, x^{3} + 64, is more easily handled and understood once we identify its constituent parts.

To do this effectively, one typically looks for common factors, applies known factoring formulas like the difference of squares, or leverages special patterns such as the sum and difference of cubes. Factoring polynomials is not only a fundamental skill in algebra but also serves as a crucial step when solving equations, simplifying expressions, and analyzing function properties. A systematic approach, which includes identifying possible factors and applying relevant formulas, simplifies the original polynomial, turning a potentially intimidating problem into a manageable one.

Applying the Sum of Cubes Formula

The sum of cubes formula is a special algebraic identity that allows factoring expressions of the form a^{3} + b^{3}. This formula is expressed as a^{3} + b^{3} = (a + b)(a^{2} - ab + b^{2}), effectively splitting the sum of two cubes into a product of a binomial and a trinomial.

In the exercise, the expression x^{3} + 64 is a classic example of a sum of cubes. By applying the formula, we first determine a and b: x for a (since x^{3} is the cube of x) and 4 for b (since 64 = 4^{3}). Subsequently, substituting a and b into the sum of cubes formula, we factor the given expression into (x + 4)(x^{2} - 4x + 16). This transformation is not just a mechanical process but reveals a deeper insight into the structure of the expression, illustrating the elegance of algebraic identities in simplifying complex algebraic forms.