Problem 7 - Elementary and Intermediate Algebra

Question

For each of the following polynomials, which factoring method would you use first? $$ x^{3}+27 $$

Step-by-Step Solution

Answer

Use the sum of cubes formula to factor: \(x^3 + 27 = (x + 3)(x^2 - 3x + 9)\).

Step 1: Identify the Type of Polynomial

The given polynomial is in the form of \(x^3 + 27\). We recognize this as a sum of cubes since it can be expressed in the form \(a^3 + b^3\), where \(a = x\) and \(b = 3\) because \(3^3 = 27\).

Step 2: Recall the Formula for Factoring a Sum of Cubes

The sum of cubes formula is \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\). This formula will allow us to factor \(x^3 + 27\).

Step 3: Apply the Sum of Cubes Formula

In our polynomial, substitute \(a = x\) and \(b = 3\) into the sum of cubes formula: \(x^3 + 27 = (x + 3)(x^2 - 3x + 9)\).

Step 4: Verify the Factored Form

To ensure accuracy, expand \((x + 3)(x^2 - 3x + 9)\) to check if it yields \(x^3 + 27\). This confirms that the polynomial is factored correctly.

Key Concepts

Understanding the Sum of Cubes

The sum of cubes is an important concept when dealing with certain types of polynomial expressions. Let's take a closer look at what it means to factor a sum of cubes. A polynomial can be identified as a sum of cubes if it fits the mold of the format \(a^3 + b^3\). In our exercise, the polynomial \(x^3 + 27\) fits perfectly as \(x^3\) is a cube of \(x\) and \(27\) is a cube of \(3\). This allows us to apply the sum of cubes formula for factoring:

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

By substituting \(a = x\) and \(b = 3\), we can effortlessly factor the polynomial into two simpler expressions for further calculations. Understanding and using the sum of cubes formula effectively allows for the simplification and solving of polynomial expressions that would otherwise be more complex.

Overview of Factoring Methods

Factoring is a fundamental method used to simplify polynomial expressions. There are several ways to factor polynomials, each useful for specific types of expressions. Here’s a rundown of common methods:

• **Greatest Common Factor (GCF):** This involves factoring out the highest number that divides all the terms of the polynomial.
• **Difference of Squares:** Used for expressions like \(a^2 - b^2 = (a - b)(a + b)\).
• **Trinomials:** Polynomials of the form \(ax^2 + bx + c\) can be factored into the product of two binomials.
• **Sum of Cubes:** As we see in our example, \(x^3 + 27\) factors using this method.
• **Difference of Cubes:** Similar to the sum of cubes but involves subtraction: \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\).

Choosing the right method involves recognizing the form of the polynomial and applying the appropriate formula to break it down. This makes complex computations manageable and allows for easier mathematical manipulations.

Exploring Polynomial Expressions

Polynomial expressions comprise variables and coefficients, involving operations of addition, subtraction, multiplication, and non-negative integer exponents. They form the building blocks of algebra and appear in various mathematical problems.

• **Standard Form:** Typically, polynomials are expressed in descending order of their powers. For example, the expression \(x^3 + 27\) is a cubic polynomial.
• **Degree:** The degree of a polynomial is the highest power of the variable present. In \(x^3 + 27\), the degree is three.
• **Terms:** These are the individual parts of the expression separated by plus or minus signs. In our case, \(x^3\) and \(27\) are terms.

Understanding the structure of polynomials is essential in determining the appropriate operations or transformations, such as factoring, to simplify or solve them. Recognizing these elements provides a firm foundation for tackling a variety of algebraic challenges.