When $$x^4 - px^3+ 2x^2 -5x + 8$$​ is divided by x -1, the remainder is 2p. The value of p is:

Correct option is D

Given:

​$$f(x) = x^4 - px^3 + 2x^2 - 5x + 8$$ 

and the condition that when this polynomial is divided by x−1, the remainder is 2p.

Concept Used:

The Remainder Theorem states that the remainder when a polynomial f(x) is divided by x − c is f(c).

Solution:

We are given the polynomial

$$f(x) = x^4 - px^3 + 2x^2 - 5x + 8$$ ​

As, the divisor is (x−1)

Substituting x = 1 into f(x):

​$$f(1) = (1)^4 - p(1)^3 + 2(1)^2 - 5(1) + 8$$​​

​$$f(1) = 1 − p + 2 − 5 + 8$$

$$f(1)=6−p$$ 

Remainder when dividing by (x−1) is 2p, so:

$$f(1)=2p$$ 

$$6−p=2p$$

3p = 6 

p = 2

Thus, the value of p is 2.