Consider the following.
P(x) = 4x^4 − x^3 − 8x^2 + 18x − 4
Find all the zeros of the polynomial function. (Hint: First determine the rational zeros. Enter your answers as a comma-separated list. Enter all answers including repetitions.)

To find all the zeros of the polynomial \( P(x) = 4x^4 - x^3 - 8x^2 + 18x - 4 \), we will follow these steps:

### Step 1: Use the Rational Root Theorem
The Rational Root Theorem helps identify possible rational roots by looking at the factors of the constant term and the leading coefficient.

- The constant term is \( -4 \), so the factors of \( -4 \) are \( \pm 1, \pm 2, \pm 4 \).
- The leading coefficient is \( 4 \), so the factors of \( 4 \) are \( \pm 1, \pm 2, \pm 4 \).

The possible rational roots are the combinations of these factors:  
\[
\pm 1, \pm \frac{1}{2}, \pm 2, \pm 4
\]

### Step 2: Test the possible rational roots
We will use synthetic division or direct substitution to check which of these values are roots.

1. Test \( x = 1 \):
\[
P(1) = 4(1)^4 - (1)^3 - 8(1)^2 + 18(1) - 4 = 4 - 1 - 8 + 18 - 4 = 9 \quad (\text{Not a root})
\]

2. Test \( x = -1 \):
\[
P(-1) = 4(-1)^4 - (-1)^3 - 8(-1)^2 + 18(-1) - 4 = 4 + 1 - 8 - 18 - 4 = -25 \quad (\text{Not a root})
\]

3. Test \( x = 2 \):
\[
P(2) = 4(2)^4 - (2)^3 - 8(2)^2 + 18(2) - 4 = 4(16) - 8 - 32 + 36 - 4 = 64 - 8 - 32 + 36 - 4 = 56 \quad (\text{Not a root})
\]

4. Test \( x = -2 \):
\[
P(-2) = 4(-2)^4 - (-2)^3 - 8(-2)^2 + 18(-2) - 4 = 4(16) + 8 - 32 - 36 - 4 = 64 + 8 - 32 - 36 - 4 = 0 \quad (\text{Root found!})
\]

Thus, \( x = -2 \) is a root.

### Step 3: Perform synthetic division
Now that we know \( x = -2 \) is a root, we can divide \( P(x) \) by \( (x + 2) \) using synthetic division to find the quotient polynomial.

After dividing, the quotient is:

\[
4x^3 - 9x^2 + 10x - 2
\]

### Step 4: Find the remaining roots
We now solve the cubic equation \( 4x^3 - 9x^2 + 10x - 2 = 0 \).

Testing possible rational roots again on this cubic, we find that \( x = \frac{1}{2} \) is a root.

After performing synthetic division again, the quotient becomes:

\[
4x^2 - 7x + 4
\]

### Step 5: Solve the quadratic equation
The quadratic equation \( 4x^2 - 7x + 4 = 0 \) can be solved using the quadratic formula:

\[
x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(4)(4)}}{2(4)} = \frac{7 \pm \sqrt{49 - 64}}{8} = \frac{7 \pm \sqrt{-15}}{8}
\]

Since the discriminant is negative, the solutions are complex:

\[
x = \frac{7 \pm i\sqrt{15}}{8}
\]

### Step 6: List all the roots
The zeros of \( P(x) \) are:
\[
x = -2, \frac{1}{2}, \frac{7 + i\sqrt{15}}{8}, \frac{7 - i\sqrt{15}}{8}
\]

These include one real double root \( x = -2 \), a rational root \( x = \frac{1}{2} \), and two complex conjugate roots.

Would you like more details on any step? Let me know!

Here are some related questions:

1. How do you perform synthetic division step by step?
2. What is the Rational Root Theorem and how does it work?
3. How do you solve a quadratic equation with complex solutions?
4. What are the steps to factor a cubic polynomial?
5. How does the quadratic formula handle negative discriminants?

**Tip:** Complex conjugate roots always appear in pairs when dealing with polynomials with real coefficients.

Learn how to find all the zeros of the polynomial P(x) = 4x^4 − x^3 − 8x^2 + 18x − 4 using the Rational Root Theorem, synthetic division, and the quadratic formula. This comprehensive solution includes real and complex roots and is suitable for advanced high school students and early college learners.

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