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Problem : How many variations are in the function f (x) = x6 -6x4 +2x2 + 2x - 7?

3.

Problem : How many positive and negative real roots might the function f (x) = 2x3 -3x2 - 4x + 1 have?

f (x) has two variations, so it may have two or zero positive roots. f (- x) has one variation, so it has one negative root.

Problem : What are the possible rational roots of f (x) = x3 -4x2 + 3x - 6? Use Descartes' Rule of Signs and the Rational Root Theorem.

Descartes' Rule of Signs indicates that there are either three or one positive real roots, and no negative real roots. The Rational Root Theorem indicates that the possible rational roots are ±1, ±2, ±3, and ±6. Knowing that there are no negative roots, the possible rational roots are 1, 2, 3, and 6.

Problem : Find all real roots of x3 + x2 - 4x - 2.

x = { -2, - , 2}.

Problem : Find all real roots of x4 +5x3 - 15x + 9.

x = { -3, 1, -3.7912…, 0.7912…}
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