Question

Prove that none of the polynomials listed below has first degree factors:

(i) x² + x + 1

(ii) x⁴ + 1

(iii) x² − x + 1

(iv) x⁴ + x² + 1

Answer 1

According to the Factor theorem, to find the first degree factors of a polynomial, we need to find those numbers x, which make the given polynomial zero.

(i)

The first degree factors of the polynomial x² + x + 1 can be obtained by finding out the solutions of the equation x² + x + 1 = 0.

_{On comparing with the general quadratic equation, ax}² _{+ bx + c = 0, we have:}

_{a = 1, b = 1 and c = 1}

⇒ x =

As we have obtained a negative value for the discriminant, the polynomial x² + x + 1 has no first degree factors.

(ii)

The first degree factors of the polynomial x⁴ + 1 can be obtained by finding out the solutions of the equation x⁴ + 1 = 0.

Let x² = p

So, the equation becomes p² + 1 = 0.

⇒ p² = −1

⇒ p =

_{As the value of p is not real, the value of x is also not real.}

Thus, the polynomial x⁴ +1 has no first degree factors.

(iii)

The first degree factors of the polynomial x² − x + 1 can be obtained by finding out the solutions of the equation x² − x + 1 = 0.

_{On comparing with the general quadratic equation, ax}² _{+ bx + c = 0, we have:}

_{a = 1, b = −1 and c = 1}

⇒ x =

As we have obtained a negative value for the discriminant, the polynomial x² − x + 1 has no first degree factors.

(iv)

The first degree factors of the polynomial x⁴ + x² + 1 can be obtained by finding out the solutions of the equation x⁴ + x² + 1 = 0.

Let x² = p

⇒ p² + p + 1 = 0

_{On comparing with the general quadratic equation, ax}² _{+ bx + c = 0, we have:}

_{a = 1, b = 1 and c = 1}

⇒ p =

As we have obtained a negative value for the discriminant, the polynomial x⁴ + x² + 1 has no first degree factors.