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

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Greatest common factor

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Q. one factor of

x^4 +x^2 +1

is

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.
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, $x^4 + x^2 + 1$ .

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