Question

advanced algebra: concepts and connections ic sem b mcclendon writing polynomial functions from complex roots if a polynomial function, f(x), with rational coefficients has roots 3 and \\(\sqrt{7}\\), what must also be a root of f(x)? options: 3i, -3, i\\(\sqrt{7}\\), -\\(\sqrt{7}\\)

Explanation:

Step1: Apply conjugate root theorem

For polynomials with rational coefficients, irrational roots of the form $\sqrt{a}$ come with their conjugate $-\sqrt{a}$, and complex roots of the form $bi$ come with their conjugate $-bi$.

Step2: Identify required conjugate roots

Given roots $3$ (rational, no conjugate needed) and $\sqrt{7}$ (irrational, needs conjugate $-\sqrt{7}$).

Answer:

$-\sqrt{7}$