(x^3-27)

Jasonbradley

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Jun 17, 2024

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Factoring the Difference of Cubes: (x^3 - 27)

The expression (x^3 - 27) represents the difference of cubes. This is a special type of binomial that can be factored using a specific formula.

The Formula

The formula for factoring the difference of cubes is:

a³ - b³ = (a - b)(a² + ab + b²)

Applying the Formula to (x³ - 27)

1. Identify a and b:

   • a = x (the cube root of x³)
   • b = 3 (the cube root of 27)

2. Substitute a and b into the formula:

   • (x³ - 27) = (x - 3)(x² + 3x + 9)

The Factored Form

Therefore, the factored form of (x³ - 27) is (x - 3)(x² + 3x + 9).

Important Note

The quadratic factor (x² + 3x + 9) cannot be factored further using real numbers. This is because it has no real roots.

Example: Solving an Equation

Let's say we need to solve the equation x³ - 27 = 0.

1. Factor the equation: (x - 3)(x² + 3x + 9) = 0

2. Set each factor to zero:

   • x - 3 = 0
   • x² + 3x + 9 = 0

3. Solve for x:

   • x = 3
   • The quadratic equation has no real roots.

Therefore, the only real solution to the equation x³ - 27 = 0 is x = 3.