Polynomial Remainder Theorem Explanation

The Remainder Theorem states that when a polynomial $f(x)$ is divided by a linear divisor $(x - a)$, the remainder is equal to $f(a)$.

Applying the Remainder Theorem

The given polynomial is $f(x) = x^{4} - px^{3} + 2x^{2} - 5x + 8$. The divisor is $(x - 1)$, which means $a = 1$. According to the Remainder Theorem, the remainder is $f(1)$.

Substitute $x=1$ into the polynomial:

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

Simplify the expression:

$f(1) = 1 - p(1) + 2(1) - 5 + 8$

$f(1) = 1 - p + 2 - 5 + 8$

Combine the constant terms:

$f(1) = (1 + 2 + 8) - 5 - p$

$f(1) = 11 - 5 - p$

$f(1) = 6 - p$

Solving for the Value of p

We are given that the remainder is $2p$. Equating the calculated remainder $f(1)$ to the given remainder:

$6 - p = 2p$

To solve for $p$, add $p$ to both sides of the equation:

$6 = 2p + p$

Combine the terms involving $p$:

$6 = 3p$

Finally, divide by 3 to find the value of $p$:

$p = \frac{6}{3}$

$p = 2$

Thus, the value of $p$ is 2.