Final answer:
A polynomial function with rational coefficients that has the roots √3, 2, and -i is P(x) = x^5 - 2x^4 - 3x^3 + 6x^2 - 2x - 6. This includes the conjugate root i to ensure rational coefficients.
Step-by-step explanation:
To write a polynomial function P(x) with rational coefficients that has the given roots √3, 2, and -i, we need to remember the Fundamental Theorem of Algebra and the fact that non-real roots of polynomials with rational coefficients come in conjugate pairs.
First, write the factors corresponding to each root. The given roots are:
- x = √3 => x2 - 3 (since x2 = 3)
- x = 2 => x - 2
- x = -i => x + i (since if -i is a root, its conjugate i must also be a root.)
For the root -i, we include its conjugate, i, even though it is not given in the problem, to ensure the coefficients are rational.
The conjugate root pairs x + i and x - i can be combined into a quadratic: (x + i)(x - i) = x2 + 1.
Multiplying out these factors gives us P(x):
P(x) = (x2 - 3)(x - 2)(x2 + 1) = x5 - 2x4 - 3x3 + 6x2 - 2x - 6.
Therefore, P(x) = x5 - 2x4 - 3x3 + 6x2 - 2x - 6 is the polynomial function with rational coefficients and the given roots.