Let f(x) be a fourth degree polynomial with co-coefficient of x power 4 is 1 such that f(-1) = -1, f(2) = -4, f(-3) = -9, and f(4) = -16. Then find the value of f(1).

We are given that \( f(x) \) is a fourth-degree polynomial of the form:

\[
f(x) = x^4 + ax^3 + bx^2 + cx + d
\]
where the leading coefficient of \( x^4 \) is 1. We are also provided with the following conditions:

\[
f(-1) = -1, \quad f(2) = -4, \quad f(-3) = -9, \quad f(4) = -16
\]

Our goal is to find the value of \( f(1) \).

### Step 1: Set up the equations

Using the conditions \( f(-1) = -1 \), \( f(2) = -4 \), \( f(-3) = -9 \), and \( f(4) = -16 \), we can substitute these values into the polynomial to create a system of equations.

#### 1. For \( f(-1) = -1 \):
\[
(-1)^4 + a(-1)^3 + b(-1)^2 + c(-1) + d = -1
\]
\[
1 - a + b - c + d = -1 \quad \Rightarrow \quad -a + b - c + d = -2 \quad \text{(Equation 1)}
\]

#### 2. For \( f(2) = -4 \):
\[
(2)^4 + a(2)^3 + b(2)^2 + c(2) + d = -4
\]
\[
16 + 8a + 4b + 2c + d = -4 \quad \Rightarrow \quad 8a + 4b + 2c + d = -20 \quad \text{(Equation 2)}
\]

#### 3. For \( f(-3) = -9 \):
\[
(-3)^4 + a(-3)^3 + b(-3)^2 + c(-3) + d = -9
\]
\[
81 - 27a + 9b - 3c + d = -9 \quad \Rightarrow \quad -27a + 9b - 3c + d = -90 \quad \text{(Equation 3)}
\]

#### 4. For \( f(4) = -16 \):
\[
(4)^4 + a(4)^3 + b(4)^2 + c(4) + d = -16
\]
\[
256 + 64a + 16b + 4c + d = -16 \quad \Rightarrow \quad 64a + 16b + 4c + d = -272 \quad \text{(Equation 4)}
\]

### Step 2: Solve the system of equations

We now solve the system of four equations:

1. \( -a + b - c + d = -2 \)
2. \( 8a + 4b + 2c + d = -20 \)
3. \( -27a + 9b - 3c + d = -90 \)
4. \( 64a + 16b + 4c + d = -272 \)

Let me solve this system.The solution to the system of equations is:

\[
a = -2, \quad b = -14, \quad c = 14, \quad d = 24
\]

Thus, the polynomial \( f(x) \) is:

\[
f(x) = x^4 - 2x^3 - 14x^2 + 14x + 24
\]

### Step 3: Find \( f(1) \)

Now, we substitute \( x = 1 \) into the polynomial:

\[
f(1) = (1)^4 - 2(1)^3 - 14(1)^2 + 14(1) + 24
\]
\[
f(1) = 1 - 2 - 14 + 14 + 24 = 23
\]

Thus, the value of \( f(1) \) is:

\[
f(1) = 23
\]

Let me know if you would like further details on the process.

Here are 5 related questions to expand your understanding:

1. How do you find the roots of a polynomial equation?
2. Can you explain the process of solving a system of nonlinear equations?
3. How do the degrees of polynomials affect their graph shapes?
4. What are the methods to factor a fourth-degree polynomial?
5. How can you determine if a polynomial has any symmetry properties?

**Tip:** Always check for special cases like symmetry or known root values when solving polynomial problems. It may simplify the process.

Let f(x) be a fourth degree polynomial with the coefficient of x^4 as 1 such that f(-1) = -1, f(2) = -4, f(-3) = -9, and f(4) = -16. Then find the value of f(1).

Discover how to solve a fourth-degree polynomial problem where f(x) is defined with specific values at f(-1), f(2), f(-3), and f(4). This example walks through setting up a system of equations and solving for the polynomial coefficients. Ideal for Grades 10-12 students learning about polynomials and algebra.

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