Problem 37 - College Algebra with Corequisite Support

Question

For the following exercises, factor the polynomials. \(x^{3}+216\)

Step-by-Step Solution

Answer

The factorization of \(x^3 + 216\) is \((x + 6)(x^2 - 6x + 36)\).

Step 1: Identify the Type of Polynomial

The given polynomial is a sum of cubes: \(x^3 + 216\). This can be rewritten as \(x^3 + 6^3\), which is a sum of two perfect cubes.

Step 2: Apply the Sum of Cubes Formula

The formula for factoring the sum of cubes is \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\). Identify \(a = x\) and \(b = 6\), then substitute into the formula.

Step 3: Substitute and Simplify

Substitute \(a = x\) and \(b = 6\) into the sum of cubes formula: \((x + 6)(x^2 - x \cdot 6 + 6^2)\). Simplify the expression to \((x + 6)(x^2 - 6x + 36)\).

Step 4: Verify the Factorization

Multiply the factors back to check: \((x + 6)(x^2 - 6x + 36) = x^3 + 216\). The factorization is correct.

Key Concepts

Sum of Cubes

A sum of cubes is a type of algebraic expression where two cube numbers are added together. In the expression \(x^3 + 216\), both \(x^3\) and \(216\) are perfect cubes.
The number \(216\) is actually \(6^3\) because multiplying 6 by itself twice equals 216. Therefore, our expression can be rewritten using these cubes: \(x^3 + 6^3\).

• The sum of cubes formula is given by: \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\).
• Identifying \(a\) and \(b\) is crucial for effectively using this formula. Here, \(a = x\) and \(b = 6\).
• When using the formula, replace \(a\) and \(b\) with these values to break down the expression into factors.

This process makes it relatively simple to factor the sum of cubes, allowing us to rewrite it as a multiplication of binomial and trinomial expressions.

Factoring Techniques

Factoring techniques are methods used to rewrite polynomials as the product of simpler expressions. There are various techniques, but for a sum of cubes like \(x^3 + 216\), a special formula is used.
This formula is different from factoring a difference of squares or a simple quadratic equation.

• Recognizing the type of polynomial is key. For example, perfect cubes suggest the sum or difference of cubes formula might apply.
• Once identified, apply the relevant formula step-by-step to simplify the expression.
• In our example, using the sum of cubes formula makes the factoring straightforward and manageable.

While other methods might apply to different types of polynomials, using the correct technique for the problem type ensures an accurate factorization.

Algebraic Expressions

Algebraic expressions like \(x^3 + 216\) are combinations of numbers, variables, and operations. Factoring these expressions simplifies them and can make solving equations involving them more efficient.
Understanding the nature of this expression is vital for effective manipulation.

• Expressions can be categorized by their degree, the highest power of the variable involved. Here, it’s a cubic expression due to the \(x^3\) term.
• Simplifying expressions can involve distributing numbers, combining like terms, or factoring.
• Factoring transforms a complex expression into a product of simpler terms, making it easier to understand and work with further.

Starting with the factorization of basic algebraic expressions is a foundation for tackling more complex problems in algebra.