Given that x + 2 and x - 3 are factors of x³ + ax + b. Calculate the values of a and b. Also, find the remaining factor.

By factor theorem,

If (x - a) is a factor of polynomial f(x), then remainder f(a) = 0.

⇒ x + 2 = 0

⇒ x = -2.

Since, x + 2 is a factor of x³ + ax + b

∴ Remainder = 0.

⇒ (-2)³ + (-2)a + b = 0

⇒ -8 - 2a + b = 0

⇒ b = 2a + 8 …………….(1)

x - 3 = 0

⇒ x = 3.

Since, x - 3 is a factor of x³ + ax + b

∴ Remainder = 0.

⇒ (3)³ + (3)a + b = 0

⇒ 27 + 3a + b = 0

⇒ b = -27 - 3a …………….(2)

From (1) and (2), we get:

⇒ 2a + 8 = -27 - 3a

⇒ 2a + 3a = -27 - 8

⇒ 5a = -35

⇒ a = $\dfrac{-35}{5}$

⇒ a = -7.

Substituting value of a in equation (1), we get :

⇒ b = 2a + 8 = 2 × -7 + 8 = -14 + 8 = -6.

f(x) = x³ + ax + b = x³ - 7x - 6.

Substituting x = -1 in f(x), we get :

⇒ (-1)³ - 7(-1) - 6

⇒ -1 + 7 - 6

⇒ -1 + 1

⇒ 0.

∴ (x + 1) is a factor of x³ - 7x - 6.

Hence, a = -7, b = -6 and (x + 1) is the remaining factor.