Question

Find all the roots of the equation.  Show your step-by-step work. x3−x2−x−2=0

Answer 1

To find the roots of the equation x^3 - x^2 - x - 2 = 0, you can use the Rational Root Theorem. This theorem allows you to test possible rational roots and find the actual roots of the equation. You can then write the equation in factored form using the roots as factors.

Step-by-step explanation:

To find the roots of the equation x^3 - x^2 - x - 2 = 0, you can use a method called the Rational Root Theorem. This theorem allows you to test possible rational roots and find the actual roots of the equation. Here are the steps:

1. Find the possible rational roots by listing all the factors of the constant term (in this case, 2) and dividing them by the factors of the leading coefficient (in this case, 1). The possible rational roots are: ±1, ±2.
2. Use synthetic division to check the possible roots. Start with the first possible root, 1, and divide the equation by (x - 1) using synthetic division. The result should give you a quotient without a remainder.
3. If the remainder is 0, then 1 is a root of the equation. The quotient obtained from synthetic division represents the new quadratic equation. Solve the quadratic equation using any method you prefer (factoring, completing the square, quadratic formula) to find the remaining roots.
4. If the remainder is not 0, move on to the next possible root and repeat the process until you find a root.
5. Once you have found all the roots, write the equation in factored form by using the roots as factors. In this case, the factored form would be: (x - 1)(x + 2)(x + 1) = 0.

Answer 2

The roots of the cubic equation x^3 - x^2 - x - 2 = 0 are x = 1, x = i√2, and x = -i√2. This is determined through the Rational Roots Theorem and synthetic division, followed by the square root method for solving the resulting quadratic equation.

Step-by-step explanation:

The equation provided is a cubic equation: x3 − x2 − x − 2 = 0. Finding the roots of a cubic equation can be done by various methods, such as factorization, synthetic division, or the Rational Roots Theorem.

1. First, we can try to find a root by guess and check or by using the Rational Roots Theorem, which suggests we try factors of the constant term (±2, ±1).
2. We find that x = 1 is a root because (1)3 − (1)2 − 1 − 2 = 0.
3. Once we have one root, we can perform synthetic division or factor out (x - 1) to reduce the cubic equation to a quadratic equation.
4. After factoring, we obtain (x - 1)(x2 + 2) = 0. The quadratic equation x2 + 2 = 0 can't be factored further, so we use the square root method to find the complex roots.
5. Solving x2 + 2 = 0 we get two complex roots x = ±i√2.

Therefore, the roots of the equation x3 − x2 − x − 2 = 0 are x = 1, x = i√2, and x = -i√2.