Final answer:
To find the constants a and b such that 4x⁴+ax²+11x+b is divisible by x²-x+2, we set the remainder of the division to zero and solve the resulting system of equations.
Step-by-step explanation:
If the polynomial 4x⁴+ax²+11x+b is divisible by x²-x+2, we can use the remainder theorem to find the values of a and b. According to the theorem, if a polynomial f(x) is divisible by x-k, then f(k) = 0. The roots of x²-x+2 = 0 are complex and cannot be used directly for simple substitution; however, we can perform polynomial division to divide p(x) by x²-x+2, setting the remainder equal to zero to solve for a and b. Performing polynomial long division or using synthetic division, we set up:
(4x⁴ + ax² + 11x + b) ÷ (x² - x + 2)
The remainder of this division must equal zero; hence the coefficients of x and the constant term in the remainder must both be zero. By equating these to zero and solving the resulting system of equations, we can find the exact values of a and b. This will involve finding values for a and b such that the terms x and the constant in the remainder disappear.