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Q.

Consider the polynomial equation $x^{4} - 2 x^{3} + 3 x^{2} - 4 x + 1 = 0$ which one of the following statements describes correctly the solution set of the equation.

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a

Two real and two non real complex zeroes

b

Four non real complex zeroes

c

Two positive and two negative zeroes

d

Four positive zeroes

answer is D.

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Detailed Solution

Let $f (x) = x^{4} - 2 x^{3} + 3 x^{2} - 4 x + 1 = 0$
$\begin{aligned} f^{'} (x) = 4 x^{3} - 6 x^{2} + 6 x - 4 ∴ f^{'} (x) = 0 \Rightarrow x = 1 \\ f^{''} (x) = 12 x^{2} - 12 x + 6 = 6 (2 x^{2} - 2 x + 1) > 0 \end{aligned}$
$∴ x = 1$ is point of min imum
$f (0) = 1 > 0, f (1) < 0 f (2) > 0$

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