A polynomial in x is x³ + 5x² - kx - 24. Which of the following is a factor of the given polynomial so that the value of k is 2?

1. (x + 2)

2. (x - 3)

3. (x + 4)

4. (x - 4)

By factor theorem,

(x - a) is a factor of f(x), if f(a) = 0.

Given,

Polynomial = x³ + 5x² - kx - 24

If k = 2, then :

Polynomial = x³ + 5x² - 2x - 24.

Dividing polynomial by (x + 4) or substituting -4 in polynomial, we get :

⇒ (-4)³ + 5(-4)² - 2(-4) - 24

⇒ -64 + 5(16) + 8 - 24

⇒ -64 + 80 + 8 - 24

⇒ 88 - 88

⇒ 0.

Since, on substituting -4 in polynomial, we get remainder = 0.

∴ (x + 4) is the factor of x³ + 5x² - kx - 24, when k = 2.

Hence, Option 3 is the correct option.