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Polynomial Factoring

A factored polynomial reveals its roots, a key concept in understanding the behavior of these expressions.

Descartes' Rule of Signs

         

Determine the number of sign changes in

\[M(x) = x^6 + x^5 + x - 2.\]

\[ T(x) = x^4 + x^3 - x^2 - 1 \]

By determining the types sign changes in \(T(x)\) and \(T(-x)\), determine the number of non-real complex roots of \(T(x) \).

Given that the equation \[ x^9 - x^7 + 4 x^4 + x^2+100 = 0 \] has either 1, 5, or 9 real roots, determine the number of its positive roots.

Let \(J(x) = x^5 - 4x^4 + x^3 + 2x^2 + 3 \). By comparing the values of \(J(1) \) and \(J(2) \), find the number of real roots of \( J(x) = 0 \).

Is there any negative root to the polynomial

\[U(x) = x^7 + x- 7 ?\]

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