Question

Which expression is a prime polynomial?

Answer 1

Answer:

A polynomial is prime if it can not be factored into polynomials of lower degree also with integer coefficients.

For example, the first option:

x^3 + b*x^2 can be rewritten as:

(x - 0)*(x^2 + b*x)

So it is not prime.

The second option:

x^2 -4x - 12

Because here we can factor this into:

(x + 2)*(x - 6) = x^2 - 6x + 2*x - 12 = x^2  - 4x - 12

Now, the third option is a two variable polynomial, here the degree is equal to the sum of the degrees of both variables.

x^4 + 8*x*y^3

(x - 0)*(x^3 + 8*y^3)

So each side has a lower degree than the original polynomial, then it is not prime.

4th option:

x^2 - b^3

This can be written as:

(x + b^(3/2))*(x - b^(3/2))

Now, here we have a problem.

If for example, b = 1, this would not be a prime.

because 1^(3/2) = 1.

But if b^(3/2) is not an integer, then we can not factorize the initial polynomial into lower degree polynomials with only integer coefficients, then we can not be 100% sure that this is not a prime polynomial, then this is the correct option.