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.