Question

6.
The roots of x3 + x2 - x - 1 = 0 are
a) (-1,-1,1)
b) (1, 1, - 1)
d) (1, 1, 1)
c) (- 1,-1,-1)
please give a right option with step​

Answer 1

SOLUTION

TO CHOOSE THE CORRECT OPTION

$\sf{ The \: roots \: of \: \: {x}^{3 } + {x}^{2} - x - 1 = 0 \: \: are \: }$

a) (-1,-1 , 1 )

b) (1, 1, - 1)

d) (1, 1, 1 )

c) (- 1,-1,-1 )

EVALUATION

Here the given equation is

$\sf{ {x}^{3 } + {x}^{2} - x - 1 = 0 \: \: }$

$\implies \sf{ {x}^{2 } (x + 1)-1( x + 1) = 0 \: \: }$

$\implies \sf{ ( {x}^{2} - 1)( x + 1) = 0 \: \: }$

$\implies \sf{ (x + 1) ( x - 1)( x + 1) = 0 \: \: }$

$\implies \sf{ {( x + 1)}^{2}(x - 1) = 0 \: \: }$

$\sf{ implies \: \: {( x + 1)}^{2} = 0 \: \: or \: \: (x - 1) = 0 \: \: }$

Now

$\sf{ {( x + 1)}^{2} = 0 \: \: gives \: \: x = - 1 \: , \: - 1 \: \: }$

Again

$\sf{ {( x - 1)} = 0 \: \: gives \: \: x = 1 \: \: \: }$

FINAL ANSWER

$\sf{ The \: roots \: of \: \: {x}^{3 } + {x}^{2} - x - 1 = 0 \: \: are \: }$

a) ( -1 , - 1 , 1 )

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