Question

How do you find the polynomial function P of lowest degree, having rational coefficients, with the given zero: 3i?

Answer 1

Hint: The term with the highest degree is called the leading term because it is usually written first. The coefficient of the leading term is called the leading coefficient. A polynomial having its highest degree zero is called a constant polynomial, it has no variables, only constants. If given zero is 3i, then -3i would also be a zero of the function, because complex zeros always occur in conjugate pairs. Thus, we need to find the factors of the polynomial and evaluate it.

Complete step-by-step answer:
If given zero is 3i, then -3i would also be a zero of the function, because complex zeros always occur in conjugate pairs. Thus, the factors of the polynomial would be:
 \[\left( {x + 3i} \right)\left( {x - 3i} \right)\]
Hence, multiplying the terms, we get:
 \[ = {x^2} - 3xi + 3xi - 9{i^2}\]
We, know that \[ - 3xi + 3xi = 0\] and \[\left( {{i^2} = - 1} \right)\] , hence substituting it we get:
 \[ = {x^2} - 9\left( { - 1} \right)\]
Evaluating we get:
 \[ = {x^2} + 9\]
Therefore, we have:
 \[\left( {x + 3i} \right)\left( {x - 3i} \right) = {x^2} + 9\]
So, the correct answer is “ \[{x^2} + 9\] ”.

Note: We must note that, if the degree of the polynomial function is even, the function behaves the same way at both ends (as x increases, and as x decreases). If the leading coefficient is positive, the function increases as x increases and decreases. If the leading coefficient is negative, the function decreases as x increases and decreases.