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

Use the Remainder Theorem to evaluate the polynomial function.

P(z)=2z^3-4z^2+3z-3 ; P(-2)

P(-2)=
Transcript text: Use the Remainder Theorem to evaluate the polynomial function. \[ \begin{array}{l} \quad P(z)=2 z^{3}-4 z^{2}+3 z-3 ; P(-2) \\ P(-2)= \end{array} \] Need Help? Read It Submit Answer
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Solution

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Solution Steps

To evaluate the polynomial function P(z)=2z34z2+3z3 P(z) = 2z^3 - 4z^2 + 3z - 3 at z=2 z = -2 using the Remainder Theorem, substitute z=2 z = -2 into the polynomial and compute the result. The Remainder Theorem states that the remainder of the division of a polynomial P(z) P(z) by za z - a is P(a) P(a) .

Step 1: Evaluate the Polynomial

To evaluate the polynomial P(z)=2z34z2+3z3 P(z) = 2z^3 - 4z^2 + 3z - 3 at z=2 z = -2 , we substitute 2 -2 into the polynomial:

P(2)=2(2)34(2)2+3(2)3 P(-2) = 2(-2)^3 - 4(-2)^2 + 3(-2) - 3

Step 2: Calculate Each Term

Calculating each term separately:

  1. 2(2)3=2(8)=16 2(-2)^3 = 2 \cdot (-8) = -16
  2. 4(2)2=44=16 -4(-2)^2 = -4 \cdot 4 = -16
  3. 3(2)=6 3(-2) = -6
  4. The constant term is 3 -3
Step 3: Combine the Results

Now, we combine all the calculated terms:

P(2)=161663 P(-2) = -16 - 16 - 6 - 3

Calculating this gives:

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

Final Answer

Thus, the value of P(2) P(-2) is

41 \boxed{-41}

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