Question

The equation x⁴-x³-x²-3x-6=0has roots -1 and 2. Find the remaining roots. (Leave your answers in the simplest radical form.)

A) 1+√2 and 1-√2
B) -1+√2 and -1-√2
C) 2+√3 and 2-√3
D) -2+ √3and -2-√3

Answer 1

To find the remaining roots of the polynomial equation x⁴-x³-x²-3x-6=0 with known roots -1 and 2, we must factor out these roots and solve the resulting quadratic equation using the quadratic formula.

Step-by-step explanation:

The equation given is x⁴-x³-x²-3x-6=0 and we're told it has roots -1 and 2. Since these are roots, the equation can be factored by (x + 1) and (x - 2). When we perform the polynomial division or use synthetic division with these factors, we can find the other factor, which should also be a quadratic equation. The quotient will then be of the form ax²+bx+c=0. Once we obtain this quadratic equation, we can solve for the remaining roots by applying the quadratic formula, -b ± √(b² - 4ac) / (2a).

The quadratic formula will provide the solutions for x, which represent the remaining two roots of the original equation. We'll then simplify these solutions to get the answer in the simplest radical form.