Advanced function division

Question

The polynomial px^3 – x^2 + qx – 2 has no remainder when divided by (x – 1) and a remainder of –18 when divided by
(x + 2). What are the values of p and q? 

Answer 1

The Remainder Theorem states that if a polynomial function f(x) is divided by x-c, then the remainder is f(c). 

 

Since the polynomial has a remainder of zero when x = 1 and a remainder of -18 when x = -2, we have:

 

    p(1)³ - (1)² + q(1) - 2 = 0  and p(-2)³ - (-2)² + q(-2) - 2 = -18

 

Simplifying, we get the system of equations:   p + q = 3

                                                                -8p - 2q = -12

 

Multiply the first equation by 2 :   2p + 2q = 6

                                                -8p - 2q = -12

 

Adding the equations, we get -6p = -6   

 

So, p = 1

 

Since p = 1 and p+q = 3, q = 2.