Let’s solve the problem step-by-step:

Problem:

We need to find a degree 3 polynomial $P(x)$ with:

1. Real coefficients.
2. Zeros at $1$ and $3i$.
3. Lead coefficient equal to $1$.

Step-by-Step Solution:

1. Zeros and Conjugates:

   • For polynomials with real coefficients, if $3i$ is a root, then its complex conjugate, $-3i$, must also be a root. Thus, the roots of the polynomial are $1$, $3i$, and $-3i$.

2. Forming Factors:

   • If $a$ is a root of a polynomial, then $(x - a)$ is a factor.
   • The factors corresponding to the roots are: $(x - 1), (x - 3i), \text{ and } (x + 3i).$

3. Combine the Complex Conjugates:

   • Multiply $(x - 3i)$ and $(x + 3i)$ to get rid of the imaginary part: $(x - 3i)(x + 3i) = x^2 - (3i)^2 = x^2 - (-9) = x^2 + 9.$

4. Form the Polynomial:

   • Multiply all factors together to create the polynomial: $P(x) = (x - 1)(x^2 + 9).$

5. Expand:

   • Expand $P(x)$: $P(x) = (x - 1)(x^2 + 9) = x(x^2 + 9) - 1(x^2 + 9) = x^3 + 9x - x^2 - 9.$
   • Combine like terms: $P(x) = x^3 - x^2 + 9x - 9.$

6. Final Polynomial:

   • The polynomial is: $P(x) = x^3 - x^2 + 9x - 9.$

Let me know if you'd like further clarifications or a deeper explanation!

5 Related Questions:

1. How do we ensure a polynomial with complex roots has real coefficients?
2. Why do conjugates of imaginary roots always appear in polynomials with real coefficients?
3. Can the leading coefficient be changed, and how would that affect the polynomial?
4. What is the general process for expanding polynomials with higher-degree roots?
5. How would the polynomial change if we used a different lead coefficient?

Tip:

Always confirm that your polynomial has the required degree and satisfies the given constraints (e.g., lead coefficient and real coefficients).