Problem 40 - College Algebra with Corequisite Support

Question

For the following exercises, factor the polynomials. \(b^{3}-8 d^{3}\)

Step-by-Step Solution

Answer

\((b - 2d)(b^2 + 2bd + 4d^2)\)

Step 1: Recognize the Pattern

The given polynomial is \(b^3 - 8d^3\). This is a difference of cubes, which follows the general form \(a^3 - b^3\). The formula for factoring a difference of cubes is \((a - b)(a^2 + ab + b^2)\).

Step 2: Identify \(a\) and \(b\)

In our polynomial, \(b^3\) is the cube of \(b\), and \(8d^3\) is the cube of \((2d)\). Therefore, \(a = b\) and \(b = 2d\).

Step 3: Apply the Difference of Cubes Formula

Using the formula \((a - b)(a^2 + ab + b^2)\), substitute \(a = b\) and \(b = 2d\): \((b - 2d)(((b)^2) + (b)(2d) + (2d)^2)\).

Step 4: Simplify the Expression

Calculate each part: \((b)^2 = b^2\), \((b)(2d) = 2bd\), and \((2d)^2 = 4d^2\). Substitute back into the expression: \((b - 2d)(b^2 + 2bd + 4d^2)\).

Key Concepts

Difference of Cubes

When you have a polynomial of the form \( b^3 - 8d^3 \), you're dealing with a difference of cubes. This is because each term is a perfect cube: \( b^3 \) and \( (2d)^3 \). Identifying this form is crucial because it allows you to apply a specific formula. The formula for factoring a difference of cubes is:

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

In our example, you determine what is being cubed in each term. Here, \( b^3 \) tells us \( a = b \), and since \( 8d^3 = (2d)^3 \), \( b = 2d \).
This pattern recognition is essential whenever you factor polynomials, particularly when dealing with cubes. Once you've identified the terms \( a \) and \( b \), you can apply the difference of cubes formula to factor the expression.

Polynomial Expressions

Polynomial expressions are combinations of variables and coefficients using operations of addition, subtraction, multiplication, and non-negative integer exponents. They represent a wide range of mathematical expressions. Understanding the structure of these expressions is vital for factoring, simplifying, and performing other algebraic operations.
In the problem \( b^3 - 8d^3 \), each term represents a power of a variable:

• \( b^3 \) is a cubic (degree 3) term involving the variable \( b \).
• \( -8d^3 \) is also a cubic term, expressed as \( (2d)^3 \), involving the variable \( d \).

A useful tip when working with polynomials is to recognize common forms such as simple binomial identities or special products like cubes. This recognition can simplify what otherwise seem like complex expressions into manageable factors.

Algebraic Identities

Algebraic identities are pre-established truths in algebra that simplify complex problems. They provide elegant, powerful methods to manipulate and simplify expressions. In the context of our polynomial, the difference of cubes identity is applied:

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

This identity helps break down the polynomial into more approachable expressions. Utilizing identities like these, you can transform intimidating polynomial expressions into products of simpler terms. In fact, these identities are the same that students often memorize and utilize in exams to quickly and accurately factor or expand expressions.
In practice, substituting the values in the identity reveals a structured way to factor polynomials, exhibiting the practical application of theoretical algebraic identities. This is why learning and understanding these identities is a crucial aspect of mastering algebra.