If a polynomial function f(x) has roots 0, 4, and 3+ sqrt11 what must also be a root of f(x)?

Step-by-step explanation:

If you want rational coefficients then you would want the conjugate of any irrational zero given.

The question is equivalent to what is the conjugate of .

The conjugate of is .

In general, the conjugate of a+b is a-b

or the conjugate of a-b is a+b.

Or maybe you like this explanation more:

Let

Subtract 3 on both sides:

Square both sides:

Subtract 11 on both sides:

Use difference of squares to factor. I apply .

So you have either

or

Solve both for x-3 and then x.

Add sqrt(11) on both sides for first equation and subtract sqrt(11) on both sides for second equation:

or

Add 3 on both sides:

 or