can there be a radical in a polynomial?

Answer 1

Absolutely. A polynomial is just a type of equation, and you can have any values (radicals, integers etc) as the coefficients for each term (x^2, x etc) in the polynomial.

Note that some combinations of constants will give "real" solutions when using the quadratic equation, and some will give "imaginary" values which occur when you are taking the square root of a negative value.

Answer 2

Yes and no. You can certainly have radicals for any of your coefficients. What you can't have, is one of your variables under a radical. So (√3)x + 2y is still a polynomial, but 3(√x) + 2y is not.

Answer 3

No. Polynomials, despite meaning "many names", can only have positive integer exponents on variables. That means a polynomial can look like this:

1 - x + x^2 - x^3

But not like this:

x^(-1) + sqrt(x)