Consider \text{P} \left(\right. \text{x} \left.\right) = \text{x} + \text{3} + \frac{13 x + 7}{x^{2} - 3 x - 1} . If x1, x2 and x3 are the roots of the equation P(x) = 0 then the value of \prod_{i = 1}^{3} \left(\right. 1 + x

Engineering

Mathematics

Introduction to Complex Number

Question

Consider $P (x) = x + 3 + \frac{13 x + 7}{x^{2} - 3 x - 1} .$ If x₁, x₂ and x₃ are the roots of the equation P(x) = 0 then the value of $\prod_{i = 1}^{3} (1 + x_{i}^{2})$ is

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P(x) = 0

$\Rightarrow x + 3 + \frac{13 x + 7}{x^{2} - 3 x - 1} = 0$

x³ + 3x + 4 = (x – x_i)(x – x₂)(x – x₃)

putting x = i and –i

4 + 2i = (i – x₁)(i – x₂)(i – x₃) …(i)

4 – 2i = (–i – x₁)(–i – x₂)(–i – x₃) …(ii)

(i) × (ii)

$(1 + x_{1}^{2}) (1 + x_{2}^{2}) (1 + x_{3}^{2}) = 20 .$