What must be added to the polynomial 2x³ – 3x² – 8x, so that it leaves a remainder 10 when divided by 2x + 1?

By the remainder theorem, the remainder when dividing by 2x + 1 is the value of the polynomial at x = −½.

P(−½) = 2(−⅛) − 3(¼) − 8(−½) = −¼ − ¾ + 4 = 3.

The remainder is 3 but we want 10. Adding a number c raises the remainder by c, so c = 10 − 3 = 7.