The polynomial 2x^3+px^2-x+6 has a remainder of zero when divided by x-2. Calculate p.

P(x)=2x^3+px^2-x+6 = (x-2)Q(x)=0

Since (x-2) is a factor of P(x), evalulate P(2) and equate the result to zero:

2(2)^3+p(2)^2-(2)+6 = 0
16+4p-2+6=0
4p=-20
p=-5

Thank you!!!!

To calculate the value of p, we can use the Remainder Theorem. According to this theorem, if a polynomial f(x) is divided by x - c and the remainder is zero, then f(c) = 0.

In this case, the polynomial 2x^3 + px^2 - x + 6 is divided by x - 2, and the remainder is zero. Therefore, we have:

f(2) = 0

Substituting x = 2 into the polynomial:

2(2)^3 + p(2)^2 - 2 + 6 = 0

Simplifying the equation:

16 + 4p - 2 + 6 = 0
20 + 4p = 0

Now, we can solve for p:

4p = -20
p = -20/4
p = -5

Therefore, the value of p is -5.