Problem 40 - College Algebra

Question

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

Step-by-Step Solution

Answer

The polynomial factors as \((b - 2d)(b^2 + 2bd + 4d^2)\).

Step 1: Identify the Form

Notice that the given polynomial \( b^3 - 8d^3 \) is a difference of cubes. It can be expressed in the form \( a^3 - b^3 \) where \( a = b \) and \( b = 2d \).

Step 2: Recall the Formula for Difference of Cubes

The formula for factoring a difference of cubes \( a^3 - b^3 \) is \( (a-b)(a^2+ab+b^2) \).

Step 3: Apply the Difference of Cubes Formula

Using the values found in Step 1, substitute \( a = b \) and \( b = 2d \) into the difference of cubes formula: \( (b - 2d)((b)^2 + (b)(2d) + (2d)^2) \).

Step 4: Simplify the Expression

Simplify the expression from Step 3: - The first term is \( b - 2d \). - The square of \( b \) is \( b^2 \). - The product \( b \times 2d \) is \( 2bd \). - The square of \( 2d \) is \( 4d^2 \). So the expression is \( (b - 2d)(b^2 + 2bd + 4d^2) \).

Step 5: Write the Final Factored Form

Combine the simplified terms to write the final factored polynomial: \( (b - 2d)(b^2 + 2bd + 4d^2) \).

Key Concepts

Difference of Cubes

Understanding the difference of cubes is key to factoring certain types of polynomials. A polynomial in the form of \( a^3 - b^3 \) is known as a "difference of cubes". This is because it involves subtracting one cube from another.

The formula for factoring a difference of cubes is:
\[a^3-b^3 = (a-b)(a^2+ab+b^2)\]This formula is based on rewriting the expression so it can be broken down into two smaller polynomial factors. The first factor \( (a - b) \) represents the linear difference between the cubic roots. The second factor \( (a^2 + ab + b^2) \) is a quadratic polynomial that includes the squares and cross products of the roots.

When recognizing the pattern of the difference of cubes, it helps to identify what each term represents. Like in the example \( b^3 - 8d^3 \), you identify \( a = b \) and \( b = 2d \) to match the pattern. Once you see this structure, applying the formula becomes straightforward.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operations. Being comfortable with these expressions is crucial when factoring polynomials, especially differences of cubes.

Variables like \( b \) and \( d \) represent quantities that can change. In the expression \( b^3 - 8d^3 \), these variables are raised to powers and multiplied, forming complex terms. The goal in factoring is to simplify these expressions by finding common parts through certain identities or formulas, such as the difference of cubes formula.

When working with algebraic expressions, it's helpful to:

• Identify common factors or similar terms.
• Use known formulas like factoring techniques.
• Visually reorganize terms if necessary to see underlying patterns.

Understanding these components allows for proper manipulation and simplification, bringing out factors that make equations solvable or expressions more understandable.

Polynomial Identities

Polynomial identities are standardized equations or formulas used to simplify or manipulate polynomials. Recognizing these identities can save you time and effort in solving algebraic problems.

The identity for the difference of cubes is one such example. It is a powerful tool that can transform the specific cubic polynomial form into a product of two easier-to-manage factors. This adds to polynomial identities like the sum/difference of squares or the sum of cubes.

Here’s how polynomial identities help:

• They allow us to rewrite complicated expressions in simpler forms.
• They provide a clear pathway to factor or expand polynomials.
• They connect different expressions, making it easier to solve equations systematically.

In our example, applying the polynomial identity for the difference of cubes directly shows the relationship between the initially perceived complex term \( b^3 - 8d^3 \) and its factors \( (b - 2d)(b^2 + 2bd + 4d^2) \). This connection underscores the usefulness of knowing and remembering these identities.