How do you write a polynomial in standard form given zeros -1 and 3 + 2i?
Since we are given the zeroes of the polynomial function, we can write the solution in terms of factors.
Whenever a complex number exists as one of the zeros, there is at least one more, which is the complex conjugate of the first. A complex conjugate is a number where the real parts are identical and the imaginary parts are of equal magnitude but opposite sign. Thus, the problem stated should have 3 zeros:
In general, given 3 zeros of a polynomial function, a, b, and c, we can write the function as the multiplication of the factors
In this case, we can show that each of a, b, and c are zeroes of the function:
Since the value of the function at x=a, b and c is equal to 0, then the function
With the generalized form, we can substitute for the given zeroes,
From here, we can put it in standard polynomial form by multiplying all the terms:
Collecting terms, and substituting
Multiplying terms again:
Which yields a final answer: