Problem 48 - Algebra for College Students

Question

Use the sum-of-two-cubes or the difference-of-two-cubes pattern to factor each of the following. $$x^{3}+8$$

Step-by-Step Solution

Answer

Factor as \((x+2)(x^2-2x+4)\).

Step 1: Identify the Type of Cubes

The expression given is \(x^3 + 8\). Notice that it fits the sum of two cubes formula because \(8 = 2^3\). Thus, \(x^3+8=(x)^3+(2)^3\).

Step 2: Apply the Sum of Two Cubes Formula

The sum of cubes formula is \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\). Here, \(a = x\) and \(b = 2\). Thus, substituting these into the formula gives \((x+2)(x^2-2x+4)\).

Step 3: Verify Factoring

To ensure the expression is factored correctly, expand \((x+2)(x^2-2x+4)\). Expanding gives \(x^3 -2x^2 + 4x + 2x^2 - 4x + 8 = x^3 + 8\), confirming the factorization is correct.

Key Concepts

Sum of Cubes

The sum of cubes is a special pattern used in algebra to factor expressions of the form \(a^3 + b^3\). This method is particularly useful, as it allows simplification of complex expressions into manageable factors. The general formula for the sum of cubes is:

• \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)

Consider an example from our exercise, where we factor \(x^3 + 8\). Here, \(8\) is rewritten as \(2^3\). Thus, the expression becomes \(x^3 + 2^3\). By applying the sum of cubes formula:

• Set \(a = x\) and \(b = 2\)
• Substitute into the formula: \((x + 2)(x^2 - 2x + 4)\)

This breaks down the expression into two simpler polynomial factors. Understanding this pattern is heavily used in algebra to deconstruct and reassemble polynomial expressions.

Difference of Cubes

While the exercise specifically demonstrated the sum of cubes, it's crucial to also understand the difference of cubes, which complements it. The difference of cubes aids in factoring expressions of the form \(a^3 - b^3\). Its formula is slightly different:

• \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)

Despite them seeming similar, the signs differ between the two identities, reflecting the plus-minus nature of addition and subtraction. This formula finds use when faced with subtraction-type cube problems. Recognizing when to use either formula is key to mastering factorization of polynomials.

Polynomials

Polynomials are expressions made up of variables and coefficients, combined using operations like addition, subtraction, and multiplication. In our exercise, \(x^3 + 8\) is a polynomial with two terms, termed a binomial.

• Each term consists of a coefficient (here, the implied coefficient is 1 for \(x^3\)) and a variable raised to a power.
• The degree of the polynomial refers to the largest exponent, which in this case is 3, hence it's known as a cubic polynomial.

Polynomials are foundational elements in algebra and serve multiple mathematical purposes, including modeling real-world phenomena, solving equations, and the basis for further algebraic concepts like polynomial functions and higher degree systems. Factorization like our example helps simplify these expressions for easier manipulation or solving.

Algebraic Expressions

Algebraic expressions are a broader category that includes polynomials like \(x^3 + 8\). They consist of numbers, variables, and operation signs combined in a meaningful manner. Expressions can range from simple, containing just one term, to complex involving many terms.

• The term "expression" implies there isn’t necessarily an equality (like in equations) — it’s a "phrase" rather than a "sentence".
• Algebraic expressions are critical in allowing us to generalize mathematical ideas, express patterns, or translate real-world problems into mathematical form.

In algebra, the goal often is to simplify these expressions, which can be done by factoring techniques such as sums or differences of cubes. Understanding their structure assists students in transitioning from simple arithmetic to complex problem solving.