Problem 38 - College Algebra

Question

For the following exercises, factor the polynomials. $$ 27 y^{3}-8 $$

Step-by-Step Solution

Answer

The factored form of \( 27y^3 - 8 \) is \((3y - 2)(9y^2 + 6y + 4)\).

Step 1: Identify the Form

The polynomial given is of the form \( a^3 - b^3 \), which represents a difference of cubes. A difference of cubes can be factored using the formula \( a^3 - b^3 = (a-b)(a^2 + ab + b^2) \).

Step 2: Determine Values of a and b

We need to express \( 27y^3 - 8 \) as \( a^3 - b^3 \). Notice that \( 27y^3 = (3y)^3 \) and \( 8 = 2^3 \). Therefore, let \( a = 3y \) and \( b = 2 \).

Step 3: Apply the Difference of Cubes Formula

Use the values of \( a \) and \( b \) to apply the formula: \( (3y)^3 - 2^3 = (3y - 2)((3y)^2 + (3y)(2) + 2^2) \).

Step 4: Simplify Each Part of the Formula

Calculate each term separately: \((3y)^2 = 9y^2\), \((3y)(2) = 6y\), and \(2^2 = 4\). Substitute these values back into the equation: \((3y - 2)(9y^2 + 6y + 4)\).

Step 5: Final Factored Form

Thus, the polynomial \( 27y^3 - 8 \) is factored as \((3y - 2)(9y^2 + 6y + 4)\).

Key Concepts

Difference of Cubes

Understanding the difference of cubes is fundamental in factoring certain types of polynomials. A polynomial expression in the form of \( a^3 - b^3 \) represents what mathematicians call a "difference of cubes." Here, both terms in the expression are perfect cubes. The method to factor such expressions relies on the specific algebraic formula: \( a^3 - b^3 = (a-b)(a^2 + ab + b^2) \), where:

• \( a \) is the cube root of the first term.
• \( b \) is the cube root of the second term.

This formula breaks down the difference into a multiplication of a binomial \((a - b)\) and a trinomial \((a^2 + ab + b^2)\), greatly simplifying the expression.

In the exercise given, \( 27y^3 - 8 \), we recognize this as \((3y)^3 - 2^3\). Thus, \( a = 3y \) and \( b = 2 \). Applying the formula enables the expression's simplification into a product of easier terms.

Polynomial Expressions

Polynomial expressions are algebraic expressions that include variables raised to powers and coefficients. These expressions can be combinations of constants, variables, and exponents. For instance, \( 27y^3 - 8 \) is a simple form of a polynomial with one variable, "\( y \)," raised to the power of three.

• "Degree" indicates the highest power in the polynomial.
• "Coefficients" are numerical factors multiplied by the variable terms.

This particular polynomial has a degree of 3, making it a cubic polynomial. Recognizing this helps in identifying potential factoring formulas like the "difference of cubes."

To tackle factoring, one first examines the structure of the polynomial to apply suitable factoring methods. Understanding these basics, such as identifying terms and their relationships, forms the first step in mastering polynomial manipulation.

Algebraic Formulas

Algebraic formulas are powerful tools for solving and simplifying algebraic expressions, especially in factoring processes. In the context of a "difference of cubes," specific formulas allow us to simplify expressions that are not easily factorable otherwise.

• The formula \( a^3 - b^3 = (a-b)(a^2 + ab + b^2) \) is used for the difference of cubes.
• These formulas reduce complex problems into simpler parts.
• They provide clear, structured methods for obtaining factored forms.

By recognizing the type of algebraic expression, such as a difference of cubes, and applying the correct formula, one can break down and solve seemingly complex problems efficiently. This is crucial in many areas of algebra and higher mathematics, reinforcing problem-solving skills and ensuring the mastery of factoring techniques.