Question

If a polynomial function (f(x)) has roots (3+√5) and -6, what must be a factor of (x(x))?

a) ((x + (3-√5)))
b) ((x-(3-√5)))
c) ((x + (5+√3)))
d) ((x - (5-√3)))

Answer 1

The polynomial function with roots (3+√5) and -6 must have (x - (3-√5)) as a factor per the Conjugate Root Theorem, making the correct answer option b.

Step-by-step explanation:

When it comes to identifying the factors of a polynomial function, understanding the nature of roots is essential. Given the roots of a polynomial function f(x), we can form factors of the function. If a function has a root of (3+√5), then by the Conjugate Root Theorem (applicable if the coefficients of the polynomial are real), it must also have the conjugate root, which is (3-√5), as a root. Thus, the factor corresponding to this root would be (x - (3-√5)). Similarly, if -6 is a root, then (x + 6) would be another factor of the polynomial function. Therefore, the factor corresponding to the given root (3+√5) is (x - (3-√5)), which makes option b the correct answer.