What is the value of p if the polynomial 2x³ + px² - 16x - 105 is exactly divisible by (x + 5)?

To find the value of p, we can use the factor theorem.

If the polynomial 2x³ + px² - 16x - 105 is exactly divisible by (x + 5), then (x + 5) must be a factor of the polynomial.

By the factor theorem, if (x + 5) is a factor of the polynomial, then the polynomial evaluated at x = -5 should be equal to 0.

Substitute x = -5 into the polynomial:

2(-5)³ + p(-5)² - 16(-5) - 105 = 0
-250 + 25p + 80 - 105 = 0
-270 + 25p = 0
25p = 270
p = 270/25
p = 10.8

Therefore, the value of p is 10.8.

Given Polynomial: 2x³ + px² - 16x - 105

Divisor: (x - 5)

By Factor Theorem: If a polynomial f(x) is exactly divisible by (x - a), then f(a) = 0.

Substitute x = 5 into the given polynomial:

2(5)³ + p(5)² - 16(5) - 105 = 0
500 + 25p - 80 - 105 = 0
25p = 315
p = 12.6

Therefore, the value of p for which the given polynomial is exactly divisible by (x - 5) is 12.6.