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1. Determine how the number of real roots of the equation

g(x) = 4x3 - 17x2 + 10x + k = 0

depends upon k. Are there any cases for which the equation has exactly two distinct real roots?

1.3 Continue the analysis of the polynomial equation

f(x) = x7 + 5x6 + x4 - x3 + x2 - 2 = 0,

investigated in subsection 1.1.1, as follows.

(a) By writing the ?fth-degree polynomial appearing in the expression for ft(x) in the form 7x5 + 30x4 + a(x - b)2 + c, show that there is in fact only one positive root of f(x) = 0.

(b) By evaluating f(1), f(0) and f(-1), and by inspecting the form of f(x) for negative values of x, determine what you can about the positions of the real roots of f(x) = 0.

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