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.