Problem 52 - Algebra for College Students

Question

Use the sum-of-two-cubes or the difference-of-two-cubes pattern to factor each of the following. $$1-8 x^{3}$$

Step-by-Step Solution

Answer

The factored form of \(1 - 8x^3\) is \((1 - 2x)(1 + 2x + 4x^2)\)."

Step 1: Identify the difference of two cubes

Recognize that the expression can be rewritten as a difference of cubes. The expression given is \(1 - 8x^3\). Notice that \(1 = 1^3\) and \(8x^3 = (2x)^3\). So, it is of the form \(a^3 - b^3\) with \(a = 1\) and \(b = 2x\).

Step 2: Recall the difference of cubes formula

For any two terms \(a\) and \(b\), the difference of cubes formula is \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\). We'll use this formula to factor the expression.

Step 3: Apply the formula

Using the formula \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\), substitute \(a = 1\) and \(b = 2x\) into the expression. This gives: \(1^3 - (2x)^3 = (1 - 2x)(1^2 + 1 \cdot 2x + (2x)^2)\).

Step 4: Simplify the factored expression

Simplify the part \((1^2 + 1 \cdot 2x + (2x)^2)\): \(1^2 = 1\), \(1 \cdot 2x = 2x\), and \((2x)^2 = 4x^2\). Thus, \((1 - 2x)(1 + 2x + 4x^2)\).

Step 5: Write the final factored form

The expression \(1 - 8x^3\) is fully factored as \((1 - 2x)(1 + 2x + 4x^2)\).

Key Concepts

Difference of Cubes

When dealing with certain polynomial expressions, recognizing specific patterns can help in their simplification. One such pattern is the **Difference of Cubes**. This occurs when you have an expression in the form of \( a^3 - b^3 \). To factor a difference of cubes effectively, you use the formula:

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

The formula breaks down the difference into two binomial expressions. The first binomial, \( (a - b) \), is straightforward—it's the inherent difference between the two cube roots: \( a \) and \( b \). The second is a trinomial \( (a^2 + ab + b^2) \), which combines square and multiplication terms of \( a \) and \( b \) in a balanced way. This trinomial acts as a stabilizing element that maintains equality in the initial expression within the factorized form.
Identifying cube roots and accurately applying this formula are crucial. For the problem \( 1 - 8x^3 \), we see that \( a = 1 \) and \( b = 2x \), translating this into the formula yields \( (1 - 2x)(1^2 + 1 \cdot 2x + (2x)^2) \). Simplifying, we achieve the factored form: \( (1 - 2x)(1 + 2x + 4x^2) \). This method allows for a structured path to untangling complex polynomials.

Algebraic Expressions

The term **Algebraic Expressions** encompasses a wide range of mathematical phrases that can include numbers, variables, and sometimes operators like addition or multiplication. They are foundational constructs in algebra, allowing us to generalize mathematical concepts.
Components of an algebraic expression are often termed as **terms**, which might look like simple numbers (constants), single variables, or products of numbers and variables. In terms of expression complexity, some might involve higher power terms like the cubes involved in our original example. For example, in the expression \( 1 - 8x^3 \), you have:

• "1" as the constant term, representing \(1^3\).
• "-8x^3" as the variable term, which can be rewritten as \((2x)^3\) to highlight the cube.

An efficient manipulation of algebraic expressions often involves identifying these terms and their interactions, like cubic terms. Knowing how to recognize specific forms like cubes, squares, or generic monomials is critical for successful factoring or expansion of expressions in broader algebraic contexts.

Polynomial Identities

In algebra, **Polynomial Identities** serve as reliable truths that help simplify expressions. These identities, akin to algebraic rules, provide a framework for transforming and solving polynomial equations. One famous category strongly tied to our example is the cube-related identities, particularly the sum and difference of cubes.
Commonly used polynomial identities include:

• \( a^2 - b^2 = (a - b)(a + b) \) for difference of squares
• \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \) for sum of cubes
• \( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \) for difference of cubes

In these identities, the beauty lies in their ability to break complex polynomials into simplified products of binomials and trinomials, allowing further applications like factoring, equation solving, or evaluations. This structured approach not only aids in computing but also deepens our comprehension of polynomial behaviors. Using these polynomial identities effectively requires a firm grasp of the inherent patterns in expressions, and applying them correctly can bridge gaps between seemingly unrelated algebraic forms.