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

Let k be the number to be added.

New polynomial:

p(x) = 2x³ − 3x² − 8x + k

Divisor:

2x + 1

Its zero is:

x = −1/2

By the Remainder Theorem:

p(−1/2) = 10

2(−1/2)³ − 3(−1/2)² − 8(−1/2) + k = 10

−1/4 − 3/4 + 4 + k = 10

3 + k = 10

k = 7

Answer:

7 must be added.

For a divisor 2x + 1, substitute x = −1/2 in the polynomial. The resulting value must equal the required remainder 10.