How do you find a polynomial function of lowest degree with rational coefficients that the given number of some of its zeroes as $ -5i $ , 3?

Hint: We start solving the problem by recalling the fact that the polynomial function with rational coefficients will have conjugate irrational and complex zeros together. We then find the conjugate of the complex root and then make use of the fact that the equation of the polynomial with zeroes a, b, c,… is defined as $ \left( x-a \right)\left( x-b \right)\left( x-c \right).. $ to proceed further. We then make the necessary calculations to get the required equation of the lowest degree polynomial.

Complete step by step answer:
According to the problem, we are asked to find the polynomial function of lowest degree with rational coefficients that the given number of some of its zeroes as $ -5i $, 3.
We can see that $ -5i $ is a complex zero.
We know that the polynomial function with rational coefficients will conjugate irrational and complex zeros together.
So, the conjugate of the complex zero $ -5i $ is also the zero of the given polynomial. We know that the conjugate of the complex number $ a\pm ib $ is $ a\mp ib $.
So, the given polynomial must have the zeros $ -5i $, $ 5i $, and 3. This makes the lowest degree of the polynomial as 3.
We know that the equation of the polynomial with zeroes a, b, c,… is defined as $ \left( x-a \right)\left( x-b \right)\left( x-c \right).. $ .
So, the equation of the lowest degree polynomial is $ \left( x+5i \right)\left( x-5i \right)\left( x-3 \right) $ .
 $ \Rightarrow \left( x+5i \right)\left( x-5i \right)\left( x-3 \right)=\left( {{x}^{2}}-25{{i}^{2}} \right)\left( x-3 \right) $ .
 $ \Rightarrow \left( x+5i \right)\left( x-5i \right)\left( x-3 \right)=\left( {{x}^{2}}-25\left( -1 \right) \right)\left( x-3 \right) $ .
 $ \Rightarrow \left( x+5i \right)\left( x-5i \right)\left( x-3 \right)=\left( {{x}^{2}}+25 \right)\left( x-3 \right) $ .
 $ \Rightarrow \left( x+5i \right)\left( x-5i \right)\left( x-3 \right)={{x}^{3}}-3{{x}^{2}}+25x-75 $.
So, we have found the lowest degree polynomial with zeroes $ -5i $ , 3 as $ {{x}^{3}}-3{{x}^{2}}+25x-75 $ .
$ \therefore $ The required lowest degree polynomial with zeroes $ -5i $ , 3 is $ {{x}^{3}}-3{{x}^{2}}+25x-75 $ .

Note:
 We should perform each step carefully in order to avoid confusion and calculation mistakes while solving this problem. We should keep in mind that the least degree of the polynomial will be equal to the total number of zeroes (inclusive of conjugate irrational and complex zeroes). Whenever we get this type of problem, we first need to add the conjugate roots (if required) to get the required answer. Similarly, we can expect problems to find the lowest degree of the polynomial with some of its zeroes are $ -3i $, $ \sqrt{3} $, –2.