A polynomial P(x) is the product of (x^3+ax^2-x-2) and x-b.

Question

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

this is for a level cambridge. Thanks for any help!

Answer 1

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