Problem 4 - Algebra 2

Question

Factor each polynomial. \(x^{3}-27\)

Step-by-Step Solution

Answer

The polynomial \(x^3 - 27\) factors to \((x - 3)(x^2 + 3x + 9)\).

Step 1: Identify the Form of the Polynomial

The polynomial given is in the form of a difference of cubes, which is expressed as \(a^3 - b^3\). In this problem, \(x^3\) is the cube term and \(27\) can be rewritten as \(3^3\). Hence, the expression \(x^3 - 27\) is a difference of cubes.

Step 2: Use the Difference of Cubes Formula

The formula for factoring a difference of cubes is: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\), where \(a = x\) and \(b = 3\) in this case.

Step 3: Substitute Values into the Formula

Substitute \(a = x\) and \(b = 3\) into the difference of cubes formula:\((x - 3)(x^2 + 3x + 9)\). This completes the factorization of the polynomial.

Key Concepts

Difference of Cubes

The difference of cubes is a special type of polynomial that can be factored using a specific formula. This formula applies when we have an expression like \(a^3 - b^3\), where both \(a\) and \(b\) are perfect cubes. In our problem, the expression \(x^3 - 27\) is recognized as a difference of cubes. Here, \(x^3\) is the cube of \(x\), and \(27\) is the cube of \(3\).
To factor such expressions, we use the formula:

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

This formula allows us to break down and simplify polynomials that might first appear complex. By recognizing the difference of cubes, we make solving such problems much easier.

Polynomial Expression

A polynomial expression is a mathematical expression that consists of variables and coefficients. These expressions involve operations like addition, subtraction, multiplication, and non-negative integer exponents of variables. In the case of our example, \(x^3 - 27\) is a polynomial expression consisting of two terms.
Each term in a polynomial is composed of a coefficient (a constant value), a variable (like \(x\)), and an exponent (which shows the power to which the variable is raised). For instance:

• In \(x^3\), the term has a coefficient of 1, a variable \(x\), and an exponent 3.
• In \(27\), the term is a constant, which can also be seen as \(3^3\), a perfect cube.

Polynomial expressions are foundational in algebra and can represent a wide range of mathematical relationships. Understanding these helps lay the groundwork for higher-level math and applications.

Algebraic Formula

An algebraic formula is a mathematical equation involving variables and constants. It provides a rule or pattern that helps solve problems or understand relationships between different mathematical elements. For instance, the difference of cubes formula \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\) is an algebraic formula that's invaluable in factoring polynomials.
Using algebraic formulas involves identifying the formula that applies to your specific problem, substituting the appropriate values into the formula, and simplifying the expression as needed. In our exercise, once we identify \(a = x\) and \(b = 3\), we can easily substitute these values into the difference of cubes formula to get

• \((x - 3)(x^2 + 3x + 9)\)

This process highlights the power of algebraic formulas in simplifying and solving complex expressions in a systematic way.