Understanding the Roots of the Cubic Equation (x^3 2x^2 - x - 2 0)

This article will guide you through the process of finding the roots of the cubic equation x^3 2x^2 - x - 2 0. We will explore various methods including the rational root theorem, synthetic division, and factorization techniques.

Introduction to the Rational Root Theorem

The Rational Root Theorem states that any possible rational root of a polynomial equation with integer coefficients is a factor of the constant term divided by a factor of the leading coefficient. For the equation x^3 2x^2 - x - 2 0, the constant term is -2 and the leading coefficient is 1.

The factors of -2 are ±1, ±2, and the factors of 1 are ±1. Thus, the possible rational roots are ±1, ±2.

Testing the Possible Rational Roots

Let's test the possible rational roots:
x 1

1^3 2(1)^2 - 1 - 2 1 2 - 1 - 2 0

Since this evaluates to 0, x 1 is a root of the equation.

Synthetic Division to Factor the Polynomial

Now that we have identified one root, we can use synthetic division to factor the polynomial.

Divide x^3 2x^2 - x - 2 by x - 1.
The quotient is x^2 3x 2.

The original polynomial can be written as:

x^3 2x^2 - x - 2 (x - 1)(x^2 3x 2)

Next, we factor x^2 3x 2. This can be factored as:

x^2 3x 2 (x 1)(x 2).

Therefore, the complete factorization of the original polynomial is:

x^3 2x^2 - x - 2 (x - 1)(x 1)(x 2)

Identifying the Roots of the Polynomial

From the factorization, we can see that the roots of the equation are:

x 1 x -1 x -2

Verification Using Vieta's Formulas

According to Vieta's formulas, for the equation x^3 2x^2 - x - 2 0:

1 r s -2 (where r and s are the other two roots).

1 * r * s -2.

r * s -3.

r * s -2.

Solving these equations, we find that r -2 and s -1, which confirms our previous factorization.

Using Desmos for Polynomial Graphing

Desmos, the beautiful free math graphing calculator, can be very helpful in visualizing the roots of the polynomial. By plotted the cubic equation, the graph will intercept at x -2, x -1, and x 1.

To plot the equation in Desmos, simply enter the following into the expression bar:

y x^3 2x^2 - x - 2

Observe the x-intercepts to see the roots of the polynomial.

Conclusion

The roots of the cubic equation x^3 2x^2 - x - 2 0 are x 1, x -1, and x -2. We have shown how to find these roots using the rational root theorem, synthetic division, and factorization techniques.

Understanding these methods is crucial for solving polynomial equations and can be applied to a variety of problems in mathematics and engineering.

Enjoy exploring polynomial equations with the help of Desmos and other powerful mathematical tools.