Question

write a polynomial function of least degree with integral coefficients that has the given zeros
18. -2,3+i
20.2,5,1+i

Answer 1

To write a polynomial function with integral coefficients that has the given zeros, we can use the fact that complex roots come in conjugate pairs. This means that if \( a + bi \) is a root, then \( a - bi \) must also be a root.

### Problem 18: Zeros -2, 3+i

**Given Zeros:**
- \(-2\)
- \(3+i\) (its conjugate \(3-i\) is also a root)

**The roots of the polynomial are:**
- \(-2\)
- \(3+i\)
- \(3-i\)

**Finding the polynomial:**

1. **Form factors from the roots:**
- From the root \(-2\), the factor is \(x + 2\).
- From the roots \(3+i\) and \(3-i\), the associated quadratic factor can be formed as follows:
\[
(x - (3+i))(x - (3-i)) = (x - 3 - i)(x - 3 + i)
\]
Using the difference of squares:
\[
= (x - 3)^2 - (i^2) = (x - 3)^2 + 1
\]
Calculate \((x - 3)^2\):
\[
(x - 3)^2 = x^2 - 6x + 9
\]
Therefore:
\[
(x - 3)^2 + 1 = x^2 - 6x + 9 + 1 = x^2 - 6x + 10
\]

2. **Combine the factors:**
\[
f(x) = (x + 2)(x^2 - 6x + 10)
\]

3. **Expand:**
\[
f(x) = x(x^2 - 6x + 10) + 2(x^2 - 6x + 10) = x^3 - 6x^2 + 10x + 2x^2 - 12x + 20
\]
\[
= x^3 - 4x^2 - 2x + 20
\]

Thus, the polynomial function is:
\[
f(x) = x^3 - 4x^2 - 2x + 20
\]

---

### Problem 20: Zeros 2, 5, 1+i

**Given Zeros:**
- \(2\)
- \(5\)
- \(1+i\) (its conjugate \(1-i\) is also a root)

**The roots of the polynomial are:**
- \(2\)
- \(5\)
- \(1+i\)
- \(1-i\)

**Finding the polynomial:**

1. **Form factors from the roots:**
- From \(2\), the factor is \(x - 2\).
- From \(5\), the factor is \(x - 5\).
- From the roots \(1+i\) and \(1-i\):
\[
(x - (1+i))(x - (1-i)) = (x - 1 - i)(x - 1 + i)
\]
Using the difference of squares:
\[
= (x - 1)^2 - (i^2) = (x - 1)^2 + 1
\]
Calculate \((x - 1)^2\):
\[
(x - 1)^2 = x^2 - 2x + 1
\]
Therefore:
\[
(x - 1)^2 + 1 = x^2 - 2x + 1 + 1 = x^2 - 2x + 2
\]

2. **Combine the factors:**
\[
f(x) = (x - 2)(x - 5)(x^2 - 2x + 2)
\]

3. **Expand:**
- First, expand \((x - 2)(x - 5)\):
\[
= x^2 - 7x + 10
\]
- Now expand:
\[
f(x) = (x^2 - 7x + 10)(x^2 - 2x + 2)
\]
- Using distributive property:
\[
= x^4 - 2x^3 + 2x^2 - 7x^3 + 14x^2 - 14x + 10x^2 - 20x + 20
\]
Combine like terms:
\[
= x^4 - 9x^3 + 26x^2 - 34x + 20
\]

Thus, the polynomial function is:
\[
f(x) = x^4 - 9x^3 + 26x^2 - 34x + 20
\]

### Summary:
- For zeros \(-2\) and \(3+i\): \(f(x) = x^3 - 4x^2 - 2x + 20\)
- For zeros \(2\), \(5\), and \(1+i\): \(f(x) = x^4 - 9x^3 + 26x^2 - 34x + 20\)