Using the remainder theorem to evaluate a polynomial

Question

Use the remainder theorem to find P(2) for P(x) = x^4 - 3x^3 - 6x^2 + 9.

Specifically, give the quotient and the remainder for the associated division and the value of P(2).

QUOTIENT = ?

REMAINDER = ?

P(2) = ?

Answer 1

P(2) = 16-24 -24+9 = -23 = the remainder when x-2 divides the 4th degree polynomial

quotient is x^3 -x^2 - 8x -16

use synthetic or long division

remainder = -23

P(2) = -23

for synthetic division, copy the coefficients of the 4th degree polynomial

2 | 1 -3 -6 0 9

__2__-2_-16 -32

1 -1 -8 -16 -23

1,-1 & -8 are coefficients of x^3, x^2 and x, -16 is the constant term, -23 is the remainder

x^3 -x^2 -8x-16 with remainder -23