If $\alpha = 1$ and $\beta = 1 + i\sqrt{2}$, where $i = \sqrt{- 1}$ are two roots of the equation$x^

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If $\alpha = 1$ and $\beta = 1 + i\sqrt{2}$, where $i = \sqrt{- 1}$ are two roots of the equation

$x^{3} + {ax}^{2} + bx + c = 0,a,b,c \in \mathbb{R}$, then $\int_{- 1}^{1}\mspace{2mu}\left( x^{3} + ax^{2} + bx + c \right)dx$ is equal to :

Accepted Answer

Sol. [Key Concept: Sum/product of the roots $1,\ 1 \pm \ i\sqrt{2}$ give $a,b,c$; only even powers survive the symmetric integral $\int_{- 1}^{1}$.]Roots are $1 \pm i\sqrt{2}$ & 1$- a =$ sum of roots$${a = - 3 }{- c = 1(1 + \sqrt{2}i)(1 - i\sqrt{2}) }{c = - 3 }{I = \int_{- 1}^{1}\mspace{2mu}\left( x^{3} - 3x^{2} + bx - 3 \right)dx }{= 2\int_{0}^{1}\mspace{2mu}\left( - 3x^{2} - 3 \right)dx }{= 2\left( - x^{3} - 3x \right)_{0}^{1} = - 8 }$$

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