Find a formula for the polynomial
𝑃(𝑥)
with
degree 3
real coefficients
zeros at 𝑥=1+4𝑖 and 𝑥=3
𝑦-intercept at (0,204)
𝑃(𝑥)=

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.

Learn how to construct a polynomial P(x) of degree 3 with real coefficients and specific roots at x = 1+4i, x = 1-4i, and x = 3, along with a y-intercept at (0, 204). This problem involves polynomial expansion, complex conjugate roots, and application of the Fundamental Theorem of Algebra. Suitable for Grades 10-12 students studying advanced algebra and polynomial functions.

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