Answer :
The polynomial with complex coefficients of the smallest possible degree is found as: P(x) = x⁴ + 17x² + 16.
Explain the term complex coefficients?
- A complex coefficient is indeed a complex number that is a factor of some variable, as opposed to a complex number, which is a standalone entity.
The given zeroes for the polynomial is-
4i and 1i.
From the conjugate zeroes theorem, for the polynomial having eal coefficients.
Then, -4i and -i is also the zeroes of the polynomial.
Thus, forming the polynomial P(x).
P(x) = (x + i)(x - i).(x + 4i)(x - 4i)
Using the property.
P(x) = (x² - i²).(x² - (4i)²)
We know that, i² = -1
P(x) = (x² + 1).(x² + 16)
Simplifying the equation;
P(x) = x⁴ + 17x² + 16
In which the coefficient of the highest power (x⁴) is 1.
Thus, the polynomial with complex coefficients of the smallest possible degree is found as: P(x) = x⁴ + 17x² + 16.
To know more about the complex coefficients, here
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