one factor of x4 +x2 +1 is

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List Possible Rational Roots: List the possible rational roots of the polynomial $ x^4 + x^2 + 1 $ using the Rational Root Theorem.Rational Root Theorem Not Applicable: The Rational Root Theorem is not applicable here because the polynomial $ x^4 + x^2 + 1 $ has no term with $ x^3 $ or $ x $, and it is not a polynomial of the form $ ax^n + bx + c $. We need to look for other methods to factor this polynomial.Recognize Quadratic Form: Recognize that $ x^4 + x^2 + 1 $ is a quadratic in form with respect to $ x^2 $. Let $ y = x^2 $, then the polynomial becomes $ y^2 + y + 1 $.Factor Quadratic Using Formula: Attempt to factor the quadratic $ y^2 + y + 1 $ using the quadratic formula to find its roots.Calculate Discriminant: The quadratic formula is $ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $, where $ a = 1 $, $ b = 1 $, and $ c = 1 $ for the quadratic $ y^2 + y + 1 $.No Real Roots: Calculate the discriminant $ \Delta = b^2 - 4ac = 1^2 - 4(1)(1) = 1 - 4 = -3 $.Polynomial Cannot Be Factored: Since the discriminant $ \Delta $ is negative, there are no real roots for the quadratic $ y^2 + y + 1 $, and thus it cannot be factored over the real numbers.Since the polynomial $ x^4 + x^2 + 1 $ cannot be factored over the real numbers, we can say that one of its factors is itself, $ x^4 + x^2 + 1 $.

one factor of $x^4 +x^2 +1$ is

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one factor of $x^4 +x^2 +1$ is