Three of the roots of the equation z^5 + bz^4 + cz^3+ dz^2 + ez + f=0, where h, c. d. e. f belong to the set of real numbers, are -2, 2i and 1 + i. Find the values of b. c. d. e and f.

The given equation is:

\[
z^5 + bz^4 + cz^3 + dz^2 + ez + f = 0
\]

We are told that three of the roots are \( z = -2 \), \( z = 2i \), and \( z = 1 + i \). Since the coefficients of the equation are real, the complex roots must occur in conjugate pairs. Therefore, the additional roots are \( z = -2i \) (the conjugate of \( 2i \)) and \( z = 1 - i \) (the conjugate of \( 1 + i \)).

Thus, the five roots are:
\[
z = -2, \quad z = 2i, \quad z = -2i, \quad z = 1 + i, \quad z = 1 - i
\]

### Step 1: Forming the Polynomial from Roots
The polynomial with these roots can be written as:
\[
(z + 2)(z - 2i)(z + 2i)(z - (1+i))(z - (1-i)) = 0
\]

### Step 2: Simplifying Each Factor
1. \( (z - 2i)(z + 2i) = z^2 + 4 \) (Difference of squares)
2. \( (z - (1+i))(z - (1-i)) = (z - 1 - i)(z - 1 + i) = (z - 1)^2 + 1 = z^2 - 2z + 2 \)

Thus, the equation becomes:
\[
(z + 2)(z^2 + 4)(z^2 - 2z + 2) = 0
\]

### Step 3: Expanding the Polynomial
First, expand \( (z^2 + 4)(z^2 - 2z + 2) \):
\[
(z^2 + 4)(z^2 - 2z + 2) = z^4 - 2z^3 + 2z^2 + 4z^2 - 8z + 8
\]
\[
= z^4 - 2z^3 + 6z^2 - 8z + 8
\]

Now multiply this by \( (z + 2) \):
\[
(z + 2)(z^4 - 2z^3 + 6z^2 - 8z + 8) = z^5 - 2z^4 + 6z^3 - 8z^2 + 8z + 2z^4 - 4z^3 + 12z^2 - 16z + 16
\]
\[
= z^5 + 0z^4 + 2z^3 + 4z^2 - 8z + 16
\]

### Step 4: Comparing with the Given Polynomial
The expanded polynomial is:
\[
z^5 + 0z^4 + 2z^3 + 4z^2 - 8z + 16 = 0
\]

By comparing this with the original equation \( z^5 + bz^4 + cz^3 + dz^2 + ez + f = 0 \), we can find the values of \( b \), \( c \), \( d \), \( e \), and \( f \):
- \( b = 0 \)
- \( c = 2 \)
- \( d = 4 \)
- \( e = -8 \)
- \( f = 16 \)

Thus, the values are:
\[
b = 0, \quad c = 2, \quad d = 4, \quad e = -8, \quad f = 16
\]

Would you like a detailed explanation of any part of this solution?

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Here are 5 related questions for further practice:
1. What happens if one of the roots is a real multiple of another?
2. How would the equation change if all coefficients were complex?
3. How do you find the polynomial for any set of real and complex roots?
4. What would the solution look like if the roots included irrational numbers?
5. How can we generalize this method to higher-degree polynomials?

**Tip:** When roots are complex, remember to always account for their conjugates if the polynomial has real coefficients.

Three of the roots of the equation z^5 + bz^4 + cz^3 + dz^2 + ez + f = 0, where b, c, d, e, f belong to the set of real numbers, are -2, 2i, and 1 + i. Find the values of b, c, d, e, and f.

This problem explores how to find the coefficients of a 5th-degree polynomial with both real and complex roots. Using algebraic concepts such as polynomial factorization and the conjugate root theorem, we provide a step-by-step solution. Suitable for high school students in Grades 10-12.

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