Please help with Polynomial function with remainder

Question

If f(x) = x⁴ + 5x³ + ax + b, where a and b are constants, and the remainder when f(x) is divided by (x-2) is equal to the remainder when f(x) is divided by (x+1)

a) How can I find the value of a.

b) Given that (x+3) is a factor of f(x), how do I find the value of b?

Answer 1

f(x) = x⁴ + 5x³ + ax + b

By the Remainder Theorem, the remainder when f(x) is divided by x - c is f(c).

So, f(2) = f(-1).

2a + b + 56 = -a + b - 4

3a = -60

a = -20

By the Factor Theorem, if x - c is a factor of f(x), then f(c) = 0.

So, f(-3) = 0

f(x) = x⁴ + 5x³ - 20x + b

f(-3) = 81 - 135 + 60 + b = 0

b = -6

Answer 2

See the videa.