PLEASE HELP!!!!!!!!!!!

Question

Write an example of a cubic polynomial that is prime and then explain how you know it is prime. Your polynomial cannot be a monomial. Write your answer as a sentence.

Answer 1

A very simple example would be f(x) = x³+2. From the rational root theorem, we know that any rational roots of f(x) must be in the set {1,-1,2,-2}. None of these values are zeros of f. Therefore, f has no rational roots and thus it cannot be factored as (x-a)(x-b)(x-c), nor as (x-a)(x^2+bx+c).

Make sense?

Answer 2

A cubic polynomial can be represented like ax3+bx2+cx+d

We need to find a prime polynomial meaning we need to find values for a,b,c, and d where we cannot factor out the above polynomial anymore.

A easy prime polynomial that is not a monomial is setting b and c equal to zero and we can set a =2 and d=13 given us a cubic binomial:

2x³+13.

You can modify this above cubic binomial by setting b and c to prime numbers like b=5 and c=7 giving us:

2x³+5x2+7x+13

which is also an prime polynomial.

I hope this was helpful, here is a polynomial factoring calculator if you want to check if you polynomial is factorable or not: https://www.mathportal.org/calculators/polynomials-solvers/polynomial-factoring-calculator.php