(x-1)(x+3)(x-4)

3 min read Jun 17, 2024

Factoring and Solving the Cubic Equation: (x-1)(x+3)(x-4)

This article will explore the properties and solutions of the cubic equation represented by the factored form: (x-1)(x+3)(x-4).

Understanding the Factored Form

The given expression is already in factored form, which makes it easy to understand its roots (also known as solutions or zeros). Each factor represents a linear expression that becomes zero for a specific value of x.

(x-1): This factor equals zero when x = 1.
(x+3): This factor equals zero when x = -3.
(x-4): This factor equals zero when x = 4.

Therefore, the equation (x-1)(x+3)(x-4) = 0 has three distinct roots: x = 1, x = -3, and x = 4.

Expanding the Factored Form

To obtain the standard form of the cubic equation, we can expand the factored expression:

(x-1)(x+3)(x-4) = (x² + 2x - 3)(x-4)

= x³ - 2x² - 11x + 12

This gives us the cubic equation in standard form: x³ - 2x² - 11x + 12 = 0.

Graphing the Cubic Equation

The graph of this cubic equation will intersect the x-axis at the points where x = 1, x = -3, and x = 4, confirming our earlier findings about the roots. The graph will have a general "S" shape, typical of cubic functions.

Summary

The factored form (x-1)(x+3)(x-4) represents a cubic equation with three distinct roots: x = 1, x = -3, and x = 4. These roots can be easily identified from the factored form and visually confirmed by graphing the equation. The expanded form of the equation is x³ - 2x² - 11x + 12 = 0. This information can be valuable for analyzing the behavior and solutions of the cubic equation in various mathematical applications.