Form a polynomial (f) with real coefficients having the given degree and zeros.
Degree 5; zeros: - 7 ; −i; 5+i
Let a represent the leading coefficient. The polynomial is f(x)=a
(x + 7)(x + i)(x - i)(x - 5 - i)(x - 5 + i)
Expanding the factors, we get:
f(x) = a(x^2 + 7x + ix + 7i)(x - ix - i^2)(x - 5 - ix - i)(x - 5 + ix + i)
f(x) = a(x^2 + 7x + ix + 7i)(x + ix + 1)(x - 5 - ix - i)(x - 5 + ix + i)
f(x) = a(x^2 + 8x + 7i)(x + ix + 1)(x - 5 - ix - i)(x - 5 + ix + i)
f(x) = a(x^2 + 8x + 7i)(x + ix + 1)(x - 5 - ix - i)(x - 5 + ix + i)
f(x) = a(x^2 + 8x + 7i)(x + ix + 1)(x - 5 - ix - i)(x - 5 + ix + i)
Expanding the remaining factors, we get:
f(x) = a(x^2 + 8x + 7i)(x + ix + 1)(x - 5 - ix - i)(x - 5 + ix + i)
f(x) = a(x^2 + 8x + 7i)(x^2 - 5x - ix - 5ix + 25 + 5i - i - 5i - i^2)(x^2 - 5x + ix - 5ix + 25 - 5i + i + 5i - i^2)
f(x) = a(x^2 + 8x + 7i)(x^2 - 5x - 6ix + 25 + 4i)(x^2 - 5x - 6ix + 25 + 4i)
f(x) = a(x^4 + 8x^3 + 7ix^2 - 5x^3 - 40x^2 - 35ix + 6ix^2 + 48ix + 42i - 25x^2 - 200x - 175i + 4ix^3 + 32ix^2 + 28i^2)(x^2 - 10x + 49)
f(x) = a(x^4 + 3x^3 + 11ix^2 - 225x - 91i)(x^2 - 10x + 49)
Therefore, the polynomial with real coefficients is:
f(x) = a(x^6 - 10x^5 + 49x^4 + 3x^5 - 30x^4 + 147x^3 + 11ix^4 - 110ix^3 + 539ix^2 - 225x^2 + 2250x - 11025 + 3x^3 - 30x^2 + 147x + 11ix^2 - 110ix + 539i - 225x + 2250 - 11025i)
f(x) = a(x^6 - 7x^5 + 19x^4 - 181x^3 + 832x^2 - 4244x + 19979)