Answer: You're Welcome!
Explanation:
The Polynomial Remainder Theorem states that if you divide a polynomial P(x) by a linear divisor of the form x - a, the remainder is equal to P(a). In other words, if you substitute x = a into the polynomial, the result is the remainder.
In your case, you have the polynomial P(x) = x^4 + 13x^3 - 12x^2 - 18x - 51, and you want to find the value of P(x) when x = 4. According to the Polynomial Remainder Theorem, this is equivalent to finding the remainder when P(x) is divided by x - 4.
So, let's evaluate P(4):
P(4) = 4^4 + 13 * 4^3 - 12 * 4^2 - 18 * 4 - 51
Calculating this expression will give you the value of the polynomial when x = 4:
P(4) = 256 + 832 - 192 - 72 - 51
P(4) = 773
Therefore, the value of the polynomial P(x) when x = 4 is 773.