Prove that the polynomial x² + x + 1 cannot be written as a product of first degree polynomials.

Given, a second degree polynomial x² + x + 1

Need to prove the given equation cannot be written as product of first degree polynomial

⇒ we know that a polynomial equation of degree 2, x² + (a + b)x + ab can be written as (x + a)(x + b)

⇒ Here, x² + x + 1 is written as x² + (a + b)x + ab

⇒ coefficient on either side are equal, we get

⇒ a + b = 1 and ab = 1

⇒ We need to find the values of a, b such that it satisfies the given equation to get the factors of first degree polynomial

⇒ Since, a + b = 1 and ab = 1 it is not possible to find out the values of a, b which satisfy the equation x² + x + 1

Hence, x² + x + 1 cannot be splited into factors of first degree polynomial