What must be added to x³ + 7x² + 3x + 2 so that the result is completely divisible by (x + 2)?

1. −40

2. −16

3. 16

4. 40

Using remainder theorem,

If polynomial P(x) is divided by (x - a), the remainder is P(a).

Given,

x + 2 = 0

x = -2.

Substituting x = -2 into the polynomial x³ + 7x² + 3x + 2, we get :

⇒ P(-2) = (-2)³ + 7(-2)² + 3(-2) + 2

= -8 + 28 - 6 + 2

= 20 - 6 + 2

= 16.

For a polynomial to be completely divisible by x + 2, the remainder must be 0.

We have a remainder of 16, we need to add a number k such that the new remainder becomes zero i.e.,:

k + 16 = 0

k = -16.

Hence, option 2 is the correct option.