Which of the following is a root of the equation x3 + 4x2 + x – 6 = 01–102

To find the root of the equation x^3 + 4x^2 + x - 6 = 0, we can use various methods such as factoring, synthetic division, or numerical methods like Newton's method.

Let's start by trying to factor the equation. We can try to factor out any common factors or use the rational root theorem to find possible rational roots.

The rational root theorem states that if a rational number p/q is a root of the equation, then p must be a factor of the constant term (-6 in this case) and q must be a factor of the leading coefficient (1 in this case).

The factors of -6 are ±1, ±2, ±3, and ±6. The factors of 1 are ±1. So, the possible rational roots are ±1, ±2, ±3, and ±6.

We can substitute each of these values into the equation to check if they satisfy it. By trying different values, we find that x = 1 is a root of the equation.

To confirm this, we can use synthetic division to divide the equation by (x - 1).

Performing synthetic division, we get:

1 | 1 4 1 -6 | 1 5 6 -------------- 1 5 6 0

The result of the synthetic division is 1x^2 + 5x + 6.

Now, we have a quadratic equation 1x^2 + 5x + 6 = 0. We can solve this equation by factoring or using the quadratic formula.

Factoring the quadratic equation, we get (x + 2)(x + 3) = 0.

Setting each factor equal to zero, we have x + 2 = 0 and x + 3 = 0.

Solving these equations, we find x = -2 and x = -3.

Therefore, the roots of the equation x^3 + 4x^2 + x - 6 = 0 are x = 1, x = -2, and x = -3.