Question

Identify all of the real roots of x^4 − x^3 − 9x^2 + 11x + 6 = 0.

Answer 1

Answer:

4 roots: 2 , -3, 1+√2, 1 - √2

Step-by-step explanation:

x⁴ − x³ − 9x² + 11x + 6 = 0    leading: 1 (x⁴)    Trailing: 6 (1,2,3,6)

possible roots: trailing/leading (±1 , ±2, ±3, ±6)

Test: x = 2     (2)⁴ - (2)³ - 9*(2)² + 11*(2) + 6 = 0

        x = -3    (-3)⁴ - (-3)³ - 9*(-3)² + 11*(-3) + 6 = 0

By rational roots test: possible roots (x-2)(x+3) are factors

Long division:  (x⁴ − x³ − 9x² + 11x + 6) / (x-2)(x+3) = x²-2x-1

roots of x²-2x-1: x= (-b±√b²-4ac) / 2a

x = (2 ± √4+4) / 2 = (2 ± 2√2) / 2 = 1 ± √2