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We are given a polynomial function \( P(x) = x^4 + ax^3 + bx^2 + cx + d \), with the conditions:

\[
P(1) = 2, \quad P(2) = 4, \quad P(3) = 6, \quad P(4) = 8
\]
We need to determine the value of \( P(5) \).

### Step 1: Express the polynomial using the given points

The general form of the polynomial is:
\[
P(x) = x^4 + ax^3 + bx^2 + cx + d
\]
We can substitute the values for \( P(1), P(2), P(3), P(4) \) into the equation to get a system of equations for \( a, b, c, d \).

#### For \( P(1) = 2 \):
\[
1^4 + a \cdot 1^3 + b \cdot 1^2 + c \cdot 1 + d = 2
\]
This simplifies to:
\[
1 + a + b + c + d = 2 \quad \text{(Equation 1)}
\]

#### For \( P(2) = 4 \):
\[
2^4 + a \cdot 2^3 + b \cdot 2^2 + c \cdot 2 + d = 4
\]
This simplifies to:
\[
16 + 8a + 4b + 2c + d = 4 \quad \text{(Equation 2)}
\]

#### For \( P(3) = 6 \):
\[
3^4 + a \cdot 3^3 + b \cdot 3^2 + c \cdot 3 + d = 6
\]
This simplifies to:
\[
81 + 27a + 9b + 3c + d = 6 \quad \text{(Equation 3)}
\]

#### For \( P(4) = 8 \):
\[
4^4 + a \cdot 4^3 + b \cdot 4^2 + c \cdot 4 + d = 8
\]
This simplifies to:
\[
256 + 64a + 16b + 4c + d = 8 \quad \text{(Equation 4)}
\]

### Step 2: Solve the system of equations

We now have the following system of four equations:
1. \( 1 + a + b + c + d = 2 \)
2. \( 16 + 8a + 4b + 2c + d = 4 \)
3. \( 81 + 27a + 9b + 3c + d = 6 \)
4. \( 256 + 64a + 16b + 4c + d = 8 \)

By solving this system of equations, we can find the values of \( a, b, c, d \).

Let me calculate this for you.The solution to the system of equations is:

\[
a = -10, \quad b = 35, \quad c = -48, \quad d = 24
\]

Thus, the polynomial becomes:
\[
P(x) = x^4 - 10x^3 + 35x^2 - 48x + 24
\]

### Step 3: Calculate \( P(5) \)

Now, let's substitute \( x = 5 \) into the polynomial to find \( P(5) \):

\[
P(5) = 5^4 - 10 \cdot 5^3 + 35 \cdot 5^2 - 48 \cdot 5 + 24
\]

Let me calculate that for you.The value of \( P(5) \) is \( 34 \).

Would you like further details or have any questions?

Here are 5 related questions to consider:
1. How do you solve a system of equations involving polynomials?
2. What are other methods for finding the coefficients of a polynomial given multiple points?
3. How can you use synthetic division for evaluating polynomials?
4. What is the significance of solving for multiple points on a polynomial curve?
5. How do the values of \( a, b, c, d \) affect the shape of a polynomial graph?

**Tip:** When solving polynomial problems with multiple conditions, organizing them into a system of equations is a key step towards finding the solution efficiently.

Diberikan polinom P(x) = x^4 + ax^3 + bx^2 + cx + d. Jika P(x) memenuhi P(1) = 2, P(2) = 4, P(3) = 6, dan P(4) = 8, nilai dari P(5) adalah...

Learn how to solve a polynomial problem where P(1) = 2, P(2) = 4, P(3) = 6, and P(4) = 8. We determine the coefficients of a fourth-degree polynomial and find P(5) using algebraic techniques. This problem demonstrates key concepts in polynomial algebra and systems of equations.

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