Question

The remainders when f(x) = x³ + ax² + bx + c is divided by (x - 1), (x + 2) and (x - 2) are respectively 2, -1 and 15. Find the quotient and remainder when f(x) is divided by (x + 1).

Answer 1

To find the remainder of the polynomial f(x) = x³ + ax² + bx + c when divided by (x + 1), apply the Remainder Theorem and solve for a, b, and c using the given remainders for division by (x - 1), (x + 2), and (x - 2). Then, calculate f(-1).

Step-by-step explanation:

We are given that when the polynomial f(x) = x³ + ax² + bx + c is divided by (x - 1), (x + 2), and (x - 2), the remainders are 2, -1, and 15, respectively. To find the remainder when f(x) is divided by (x + 1), we can use the Remainder Theorem, which states that if a polynomial f(x) is divided by (x - r), the remainder is f(r).

Let's evaluate f(x) at x = -1: f(-1) = (-1)³ + a(-1)² + b(-1) + c. Now using the information that f(1) = 2, f(-2) = -1, and f(2) = 15, we can create a system of equations to solve for a, b, and c.

Substituting these into the polynomial we get:

• f(1) = 2: 1 + a + b + c = 2
• f(-2) = -1: -8 + 4a - 2b + c = -1
• f(2) = 15: 8 + 4a + 2b + c = 15

By solving this system of equations, we can find the coefficients a, b, and c, and then substitute x = -1 into the polynomial to get the corresponding remainder.