To find the polynomial $P(x)$ with the given properties, we will use the following information:

1. Degree 3: The polynomial has a degree of 3, so it can be expressed in the form $P(x) = a(x - r_1)(x - r_2)(x - r_3)$, where $r_1, r_2,$ and $r_3$ are its roots.
2. Real coefficients: The coefficients of the polynomial are real. If there is a complex root, its complex conjugate must also be a root.
3. Zeros at $x = 1 + 4i$ and $x = 3$: The conjugate of $1 + 4i$, which is $1 - 4i$, must also be a root due to the real coefficients.
4. $y$-intercept at $(0, 204)$: This means $P(0) = 204$.

Given this, the roots of the polynomial are $x = 1 + 4i$, $x = 1 - 4i$, and $x = 3$. The polynomial can be written as:

$P(x) = a(x - (1 + 4i))(x - (1 - 4i))(x - 3)$

Step 1: Expand the Factors

First, we multiply the factors involving the complex roots:

$(x - (1 + 4i))(x - (1 - 4i))$

$= (x - 1 - 4i)(x - 1 + 4i)$

$= ((x - 1) - 4i)((x - 1) + 4i)$

$= (x - 1)^2 - (4i)^2$

$= (x - 1)^2 + 16$

$= x^2 - 2x + 1 + 16$

$= x^2 - 2x + 17$

Now, the polynomial becomes:

$P(x) = a(x^2 - 2x + 17)(x - 3)$

Step 2: Expand the Remaining Terms

$P(x) = a(x^3 - 3x^2 - 2x^2 + 6x + 17x - 51)$

$= a(x^3 - 5x^2 + 23x - 51)$

Step 3: Determine $a$ Using the y-Intercept

Since the polynomial has a y-intercept of $204$, we substitute $x = 0$ and set $P(0) = 204$:

$P(0) = a(0^3 - 5(0)^2 + 23(0) - 51) = 204$

$-51a = 204$

$a = -\frac{204}{51} = -4$

Final Polynomial

$P(x) = -4(x^3 - 5x^2 + 23x - 51)$

$P(x) = -4x^3 + 20x^2 - 92x + 204$

Thus, the polynomial is:

$\boxed{P(x) = -4x^3 + 20x^2 - 92x + 204}$

Would you like a more detailed explanation or have any questions?

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Questions

1. What are complex conjugate roots, and why must they appear in polynomials with real coefficients?
2. How do we determine the constant $a$ in a polynomial given a specific point?
3. Why is it necessary to expand the polynomial when finding the final expression?
4. How can we verify that the polynomial has the correct y-intercept?
5. What is the significance of the degree of a polynomial?
6. How does the Fundamental Theorem of Algebra relate to this problem?
7. Can a polynomial with real coefficients have an odd number of non-real roots?
8. How can you determine the end behavior of a polynomial function?

Tip

When dealing with polynomials, always check for complex conjugate pairs to ensure the coefficients remain real.