(3x + 5) is a factor of the polynomial (a - 1)x³ + (a + 1)x² - (2a + 1)x - 15. Find the value of 'a'. For this value of 'a', factorise the given polynomial completely.

Using remainder theorem, find the value of k if on dividing 2x³ + 3x² - kx + 5 by (x - 2) leaves a remainder 7.

Find the value of 'a' if x - a is a factor of the polynomial 3x³ + x² - ax - 81.

While factorizing a given polynomial, using remainder and factor theorem, a student finds that x + 3 is a factor of 2x³ - x² - 5x - 2.

(a) Is the student's, solution correct stating that (x + 3) is a factor of the given polynomial?

(b) Give a valid reason for your answer.

(c) Factorize the given polynomial completely.