Factor and Solve Sum/Difference of Cubes

Algebra-2

1. Fundamental Concepts

Definition: The sum and difference of cubes are special polynomial forms that can be factored using specific formulas.
Sum of Cubes: $$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$$
Difference of Cubes: $$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$$

2. Key Concepts

Factorization: Use the sum or difference of cubes formula to factor the polynomial.

Solving Equations: Set each factor equal to zero and solve for the variable.

Application: Used in various fields such as physics, engineering, and geometry to simplify and solve complex equations.

3. Examples

Example 1 (Basic)

Problem: Factor and solve the equation $$x^3 + 8 = 0$$

Step-by-Step Solution:

Identify the sum of cubes: $$x^3 + 8 = x^3 + 2^3$$
Apply the sum of cubes formula: $$x^3 + 2^3 = (x + 2)(x^2 - 2x + 4)$$
Set each factor equal to zero:
- $$x + 2 = 0 \Rightarrow x = -2$$
- $$x^2 - 2x + 4 = 0$$ (This quadratic has no real roots, as the discriminant is negative: $$b^2 - 4ac = (-2)^2 - 4(1)(4) = 4 - 16 = -12$$)

Validation: Substitute $$x = -2$$ into the original equation: $$(-2)^3 + 8 = -8 + 8 = 0$$ ✓

Example 2 (Intermediate)

Problem: Factor and solve the equation $$27y^3 - 1 = 0$$

Step-by-Step Solution:

Identify the difference of cubes: $$27y^3 - 1 = (3y)^3 - 1^3$$
Apply the difference of cubes formula: $$(3y)^3 - 1^3 = (3y - 1)((3y)^2 + (3y)(1) + 1^2) = (3y - 1)(9y^2 + 3y + 1)$$
Set each factor equal to zero:
- $$3y - 1 = 0 \Rightarrow y = \frac{1}{3}$$
- $$9y^2 + 3y + 1 = 0$$ (This quadratic has no real roots, as the discriminant is negative: $$b^2 - 4ac = 3^2 - 4(9)(1) = 9 - 36 = -27$$)

Validation: Substitute $$y = \frac{1}{3}$$ into the original equation: $$27\left(\frac{1}{3}\right)^3 - 1 = 27 \cdot \frac{1}{27} - 1 = 1 - 1 = 0$$ ✓

4. Problem-Solving Techniques

Identify the Form: Recognize if the polynomial is a sum or difference of cubes.
Apply the Formula: Use the appropriate sum or difference of cubes formula to factor the polynomial.
Solve Each Factor: Set each factor equal to zero and solve for the variable. Check for extraneous solutions.
Check Solutions: Substitute the solutions back into the original equation to verify correctness.