Write a polynomial function of least degree with integral coefficients that has the given zeros -5, 2+i

To construct a polynomial with integral coefficients from the given zeros, we need to remember that complex zeros always come in conjugate pairs if the polynomial has real (and integral) coefficients. 

You are given the zeros:
- \( -5 \)
- \( 2 + i \)

Since \( 2 + i \) is a zero, its conjugate \( 2 - i \) must also be a zero.

Thus, the polynomial will have the following zeros:
- \( x = -5 \)
- \( x = 2 + i \)
- \( x = 2 - i \)

### Steps to write the polynomial:

1. **Write the factors corresponding to the zeros:**
   - For \( x = -5 \), the factor is \( (x + 5) \).
   - For \( x = 2 + i \) and \( x = 2 - i \), the factors are \( (x - (2 + i)) \) and \( (x - (2 - i)) \), respectively.

2. **Multiply the complex conjugate factors:**

   Let's first multiply \( (x - (2 + i)) \) and \( (x - (2 - i)) \):
   \[
   (x - (2 + i))(x - (2 - i)) = [(x - 2) - i][(x - 2) + i]
   \]
   This is a difference of squares:
   \[
   = (x - 2)^2 - i^2 = (x - 2)^2 - (-1) = (x - 2)^2 + 1
   \]
   Expanding \( (x - 2)^2 \):
   \[
   (x - 2)^2 = x^2 - 4x + 4
   \]
   So,
   \[
   (x - (2 + i))(x - (2 - i)) = x^2 - 4x + 4 + 1 = x^2 - 4x + 5
   \]

3. **Multiply the result by \( (x + 5) \):**

   Now, multiply \( (x + 5) \) by \( x^2 - 4x + 5 \):
   \[
   (x + 5)(x^2 - 4x + 5)
   \]
   Use the distributive property:
   \[
   = x(x^2 - 4x + 5) + 5(x^2 - 4x + 5)
   \]
   \[
   = x^3 - 4x^2 + 5x + 5x^2 - 20x + 25
   \]
   Combine like terms:
   \[
   = x^3 + x^2 - 15x + 25
   \]

### Final Polynomial:
The polynomial function of least degree with integral coefficients that has the given zeros is:
\[
f(x) = x^3 + x^2 - 15x + 25
\]

Would you like further details or have any questions?

Here are 5 related questions:
1. What are complex conjugates, and why must they appear together in polynomials with real coefficients?
2. How do we simplify expressions using the difference of squares?
3. Can a polynomial have non-integer coefficients if its zeros are complex?
4. How do we verify that a given polynomial has specific zeros?
5. What is the degree of a polynomial, and how is it determined from its zeros?

**Tip:** Always remember to check for complex conjugates when dealing with polynomials with real coefficients.

Learn how to construct a polynomial of least degree with integral coefficients given the zeros -5 and 2 + i. This problem involves key algebraic concepts such as complex conjugates, factoring, and the difference of squares. Suitable for high school students in Grades 10-12.

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