Question

writing polynomial functions from complex roots
a polynomial function, ( f(x) ), with rational coefficients has roots ( 0 ), ( 4 ), and ( 3 + sqrt{11} ). what must also be a root of ( f(x) )?
options:
( -3 + isqrt{11} )
( 3 + isqrt{11} )
( 3 - sqrt{11} )
( -3 - sqrt{11} )

Explanation:

Step1: Recall conjugate root theorem

For polynomials with rational coefficients, irrational roots of the form $a+\sqrt{b}$ (where $a$ is rational, $\sqrt{b}$ is irrational) have their conjugate $a-\sqrt{b}$ as a root.

Step2: Identify the given irrational root

The given root is $3+\sqrt{11}$. Its conjugate is $3-\sqrt{11}$.

Answer:

$\boldsymbol{3 - \sqrt{11}}$