To form a polynomial $f(x)$ with real coefficients given its degree and zeros, we need to follow these steps:

Steps:

1. Identify the degree and zeros: From the problem, the degree of the polynomial is determined by the number of zeros. If the zeros are complex, remember that non-real zeros of polynomials with real coefficients must come in conjugate pairs.

2. Write factors based on zeros:

   • If $a$ is a real zero, include a factor $(x - a)$.
   • If $a + bi$ is a complex zero, include a factor $(x - (a + bi))(x - (a - bi))$, which simplifies to $(x^2 - 2ax + a^2 + b^2)$.

3. Multiply the factors: Once all the factors are identified, multiply them together to form the polynomial.

4. Simplify: Combine like terms to write the polynomial in standard form.

Example Problem:

Suppose the polynomial has degree 4 and the given zeros are:

• $2$ (real zero),
• $-3$ (real zero),
• $1 + 2i$ (complex zero, which implies that $1 - 2i$ is also a zero).

Forming the Polynomial:

1. Write the factors:

   • For $2$, the factor is $(x - 2)$.
   • For $-3$, the factor is $(x + 3)$.
   • For $1 + 2i$ and $1 - 2i$, use $(x - (1 + 2i))(x - (1 - 2i))$, which simplifies to $(x^2 - 2x + 5)$ (from the identity $(x - (a + bi))(x - (a - bi)) = x^2 - 2ax + a^2 + b^2$).

2. Multiply the factors: $f(x) = (x - 2)(x + 3)(x^2 - 2x + 5)$

3. Expand: First, multiply $(x - 2)$ and $(x + 3)$: $(x - 2)(x + 3) = x^2 + 3x - 2x - 6 = x^2 + x - 6$

   Now, multiply $(x^2 + x - 6)$ by $(x^2 - 2x + 5)$: $f(x) = (x^2 + x - 6)(x^2 - 2x + 5)$ Expand this: $f(x) = x^4 - 2x^3 + 5x^2 + x^3 - 2x^2 + 5x - 6x^2 + 12x - 30$ Combine like terms: $f(x) = x^4 - x^3 - 3x^2 + 17x - 30$

So, the polynomial is: $f(x) = x^4 - x^3 - 3x^2 + 17x - 30$

Let me know if you'd like a breakdown of any steps!

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Related Questions:

1. What happens if a zero has multiplicity greater than 1?
2. How do we handle polynomials with more than two complex roots?
3. Can you find a polynomial where all roots are complex numbers?
4. How does the Fundamental Theorem of Algebra help in finding polynomial roots?
5. What is the relationship between the degree of a polynomial and the number of turning points?

Tip: When multiplying complex conjugates, always simplify using $(x^2 - 2ax + a^2 + b^2)$ to avoid handling imaginary terms directly.