If a polynomial f(x) = x⁴ - 2x³ + 3x² - ax - b leaves remainders 5 and 19 when divided by (x - 1) and (x + 1) respectively, find the values of a and b. Hence, determine the remainder when f(x) is divided by (x - 2).

When a polynomial f(x) is divided by (x - 1), the remainder is 5 and when it is, divided by (x - 2), the remainder is 7. Find the remainder when it is divided by (x - 1)(x - 2).

The polynomial 3x³ + 8x² - 15x + k has (x - 1) as a factor. Find the value of k. Hence factorize the resulting polynomial completely.

While factorizing a given polynomial, using remainder and factor theorem, a student finds that (2x + 1) is a factor of 2x³ + 7x² + 2x - 3.

(a) Is the student's solution correct stating that (2x + 1) is a factor of the given polynomial ? Given a valid reason for your answer.

(b) Factorize the given polynomial completely.