Question

Given that x-1 and x+2 are factors of f(x)=x³+px+q where p and q are integers, find p and q. Hence, find the remainder when x³+px+q is divided by (x + 1).​

Answer 1

Answer:

• p = -3
• q = 2
• remainder = 4

Step-by-step explanation:

We want to find the constants 'a', 'p', and 'q' such that ...

  (x -a)(x -1)(x +2) ≡ x³ +px +q

  (x -a)(x² +x -2) ≡ x³ +px +q

  x³ +x² -2x -ax² -ax +2a ≡ x³ +px +q

This gives rise to three equations:

  1 -a = 0 . . . . coefficient of x²

  -2 -a = p . . . coefficient of x

  2a = q . . . . . constant

The first of these tells us a=1. Then, p=-3 and q=2.

The values of p and q are -3 and 2, respectively.

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The remainder from division by (x+1) is the value of the cubic for x=-1. That value is ...

  (-1)³ -3(-1) +2 = -1 +3 +2 = 4

The remainder from division by (x+1) is 4.