Question: A polynomial P(x) is the product of (x^3+ax^2-x-2) and x-b.
Q1) The coefficients of x^3 and x in P(x) are zero. Find the values of a and b.
Q2) Hence factorise P(x) completely and find the roots of the equation P(x)=0
P = (x^3+a*x^2-x-2)*(x-b)
Expanding P,
P = x^4 - x^3*b + a*x^3 - a*x^2*b - x^2 + x*b - 2*x + 2*b
P = x^4 + (-b + a)*x^3 + (-1 - a*b)*x^2 + (b - 2)*x + 2*b
The coefficients of x^3 and x are zero, hence
-b + a = 0
b - 2 - 0
These two simultaneous eqns give us: a = 2, b = 2.
Substituting for a and b into the equation for P(x),
P = x^4 - 5*x^2 + 4
P = u^2 - 5u + 4, u = x^2
P = (u - 1)(u - 4)
i.e. u = 1, u = 4
or x^2 = 1, x^2 = 4
i.e. x = 1, x = 2