Question

Given that (x + 3) is a factor of the polynomial (p(x) = 2x³ + ax² - 24x + b), and the remainder when (p(x)) is divided by (x - 2) is -15. Find the remainder when (p(x)) is divided by (x + 1).

a) -18
b) -24
c) -12
d) -21

Answer 1

To find the remainder when p(x) = 2x³ + ax² - 24x + b is divided by (x + 1), we use the given information that (x + 3) is a factor and the remainder when divided by (x - 2) is -15 to solve for 'a' and 'b', and then evaluate p(-1).

Step-by-step explanation:

The problem is about finding the remainder when a polynomial p(x) = 2x³ + ax² - 24x + b is divided by (x + 1). We are given that (x + 3) is a factor, which tells us that p(-3) = 0. To find the value of 'a' and 'b', we use the remainder theorem, which states that if a polynomial f(x) is divided by (x - c), the remainder is f(c). First, we use the fact that p(-3) = 0 to set up an equation: 2(-3)³ + a(-3)² - 24(-3) + b = 0. Then, we use the information that p(2) = -15, giving us another equation: 2(2)³ + a(2)² - 24(2) + b = -15. These two equations will help us solve for 'a' and 'b'. After that, we find the remainder when p(x) is divided by (x + 1) by evaluating p(-1).