If x² - 3x + 2 is a factor of x³ - ax² + b, find the values of a and b.

Factorising, x² - 3x + 2.

⇒ x² - 2x - x + 2

⇒ x(x - 2) - 1(x - 2)

⇒ (x - 1)(x - 2).

So, we can say (x - 1) and (x - 2) are the factors of x³ - ax² + b.

By factor theorem,

A polynomial f(x) has a factor (x - a), if and only if, f(a) = 0.

Given, 1st factor

⇒ x - 1 = 0

⇒ x = 1.

Substituting x = 1 in x³ - ax² + b, will give remainder = 0.

⇒ 1³ - a(1)² + b = 0

⇒ 1 - a + b = 0

⇒ a = b + 1 ………(Eq. 1)

Given, 2nd factor

⇒ x - 2 = 0

⇒ x = 2.

Substituting x = 2 in x³ - ax² + b, will give remainder = 0.

⇒ 2³ - a(2)² + b = 0

⇒ 8 - 4a + b = 0

⇒ 4a = b + 8

Substituting value of a from Eq 1 in above equation :

⇒ 4(b + 1) = b + 8

⇒ 4b + 4 = b + 8

⇒ 4b - b = 8 - 4

⇒ 3b = 4

⇒ b = $\dfrac{4}{3}$

⇒ a = b + 1 = $\dfrac{4}{3} + 1 = \dfrac{4 + 3}{3} = \dfrac{7}{3}$.

Hence, a = $\dfrac{4}{3}$ and b = $\dfrac{7}{3}$.