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

Degree 5;
zeros: -8; -i; -8+i

To form a polynomial with the given degree and zeros, we need to use the concept of complex conjugate zeros.

The zeros given are:
1. -8 (real zero)
2. -i (complex zero)
3. -8+i (complex zero)
Since -i is a zero, its conjugate, i, is also a zero. Similarly, -8+i has a conjugate -8-i.

To find the polynomial, we can start by writing the factors using the zeros:

(x - (-8))(x - (-i))(x - (-8+i))(x - (-8-i))(x - i)

Simplifying the expressions inside brackets:

(x + 8)(x + i)(x - 8 - i)(x - 8 + i)(x - i)

Expanding these brackets:

(x + 8)(x + i)(x^2 - 8x + 8i - ix - 8i + 64 + 8x - 64 - 8i + i^2)(x - i)

Simplifying further:

(x + 8)(x + i)(x^2 - 8x - ix - 8i - 8i + 64 - 8i + 1)(x - i)

(x + 8)(x + i)(x^2 - 8x - 16i + 65)(x - i)

(x + 8)(x + i)(x^2 - 8x + 65 - 16i)(x - i)

To have the polynomial with real coefficients, we combine the complex parts:

(x + 8)(x + i)(x^2 - 8x + 65 - 16i)(x - i)

Expanding:

(x + 8)(x + i)(x^3 - ix^2 - 8ix + 8x^2 - 65x + 8xi + 65i - 16i^2)(x - i)

(x + 8)(x^4 - ix^3 - 8ix^2 + 8x^3 - 65x^2 + 8ix - 65ix + 16 - x + 65i - 8ix^2 + 8x^3 - 8x^2i + 65xi - 16i^2)(x - i)

(x^5 - ix^4 - 8ix^3 + 8x^4 - 65x^3 + 8ix^2 - 65ix^2 + 16x - -x^2 + 65ix - 8ix^2 + 8x^3 - 8x^2i + 65xi - 16i^2x - xi^4 - 8ix^3i + 8xi^4 - 65ix^3 + 8ix^2i^2 - 65ix^2i - 16xi + xi^2 - 65xi^2 + 16xi^3 + xi^4 - 65xi^3 - 16xi^2 - xi^4 + 65xi^3 + 16xi^2 + xi^6)(x - i)

Simplifying the expression:

x^5 + 15x^4 - 158x^3 - 194x^2 + 9x + 49

must have complex conjugates of complex zeros given:

(x+8)(x+i)(x-i)(x+8-i)(x+8+i)

Multiply out.
Have fun.