Questions: Use the Remainder Theorem to evaluate the polynomial function. P(z)=2z^3-4z^2+3z-3 ; P(-2) P(-2)=

Solution Steps

To evaluate the polynomial function $P(z) = 2z^3 - 4z^2 + 3z - 3$ at $z = -2$ using the Remainder Theorem, substitute $z = -2$ into the polynomial and compute the result. The Remainder Theorem states that the remainder of the division of a polynomial $P(z)$ by $z - a$ is $P(a)$.

To evaluate the polynomial $P(z) = 2z^3 - 4z^2 + 3z - 3$ at $z = -2$, we substitute $-2$ into the polynomial:

$$P(-2) = 2(-2)^3 - 4(-2)^2 + 3(-2) - 3$$

Calculating each term separately:

1. $2(-2)^3 = 2 \cdot (-8) = -16$
2. $-4(-2)^2 = -4 \cdot 4 = -16$
3. $3(-2) = -6$
4. The constant term is $-3$

Now, we combine all the calculated terms:

$$P(-2) = -16 - 16 - 6 - 3$$

Calculating this gives:

$$P(-2) = -41$$

Final Answer