What number should be added to x³ - 9x² - 2x + 3 so that the remainder may be 5 when divided by (x - 2) ?

Let a be added to x³ - 9x² - 2x + 3 so that the remainder may be 5 when divided by (x - 2).

By remainder theorem,

If f(x), a polynomial in x, is divided by (x - a), the remainder = f(a).

∴ On dividing x³ - 9x² - 2x + 3 + a by (x - 2)

Remainder = (2)³ - 9(2)² - 2(2) + 3 + a

Since, remainder = 5.

∴ 8 - 9(4) - 4 + 3 + a = 5

⇒ 8 - 36 - 1 + a = 5

⇒ a - 29 = 5

⇒ a = 5 + 29 = 34.

Hence, 34 must be added to x³ - 9x² - 2x + 3 so that the remainder may be 5 when divided by (x - 2).