If –3 + i is a root of the polynomial function f(x), which of the following must also be a root of f(x)?

Step-by-step explanation:

REcall that f(x) is a polynomial whose one of its roots is -3+i. The fundamental algebra theorem states that any polynomial of degree n has n complex roots. In the real case, it can be also interpreted as any polynomial can be factored in factors of degree at most 2.

Consider that given a polynomial of degree 2 of the form the solutions are given by

In this case, the fact that x is real or complex depends on the number which is called the discriminant. When this number is negative, we have that x is a complex root. Let say that and that , so the roots are given by

this means that, whenever we have a complex root, the other root is the complex conjugate. Recall that the complex conjugate of a complex number of the form a+bi is obtained by changing the sign of the imaginary part, that is a-bi.

So, in our case since -3+i is a root, then -3-i necessarily is another root.

If -3 + i is a root then -3 - i is too.

Therefore, the answer is -3 - i

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step-by-step explanation: