Form a polynomial f(x) with real coefficients having the given degree and zeros  

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.

Form a polynomial f(x) with real coefficients having the given degree and zeros

Learn how to form a polynomial f(x) with real coefficients from given zeros. This step-by-step guide covers polynomials of degree 4, including both real and complex zeros, and uses the Fundamental Theorem of Algebra. Suitable for high school students in Grades 10-12.

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