Question

ii) Given that x^4-2x^3+ax^2-2 =x(x-1)(x+1)Q(x)+4x^2+bx+c is an identity, state the degree of the polynomial Q(x).
ii) Find the value of a, of b and of XC. Hence, state the remainder when x^4-2x^3+ax^2-2 is divided by x^3-x

Answer 1

Q is linear and monic, since it is multiplied by a cubic to produce a 4th-degree polynomial.
(x^3-x)(x+k)+(4x^2+bx+c) = x^4 + kx^3 + 3x^2 + (b-k)x + c
so now we have
b = -2
c = -2
a = 3
check:
x^4-2x^3+3x^2-2 = (x^3-x)(x-2) + 4x^2-2x-2