First slide

Theory of equations

Question

If $α, β, γ, σ$ are the roots of the equation $x^{4} + 4 x^{3} - 6 x^{2} + 7 x - 9 = 0$ then the value of $(1 + α^{2}) (1 + β^{2}) (1 + γ^{2}) (1 + σ^{2})$ is

Moderate

A

5
B

9
C

11
D

13

Solution

Since $α, β, γ, σ$ are the roots of the given equation $∴ x^{4} + 4 x^{3} - 6 x^{2} + 7 x - 9 = (x - α) (x - β) (x - γ) (x - σ)$
Putting $x = i$ and then $x = = - i,$ we get
and $\begin{array}{l} 1 - 4 i + 6 + 7 i - 9 = (i - α) (i - β) (i - γ) (i - σ) \\ 1 + 4 i + 6 - 7 i - 9 = (- i - α) (- i - β) (- i - γ) (- i - σ) \end{array}$
Multiplying these two, we get
$\begin{array}{l} (- 2 + 3 i) (- 2 - 3 i) = (1 + α^{2}) (1 + β^{2}) (1 + γ^{2}) (1 + σ^{2}) \\ \Rightarrow 13 = (1 + α^{2}) (1 + β^{2}) (1 + γ^{2}) (1 + σ^{2}) \end{array}$

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