Find the remainder when $x^{100}$ is divided by $x^{2} - 3 x + 2$

(2^{100} - 1) x + (2^{100} - 1)

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(2^{100} - 1) x + (1^{100} - 2)

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(2^{100} - 1) x + (2 - 2^{100})

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(2^{100} - 2) x + (2 - 2^{100})

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Solution

The correct option is C $(2^{100} - 1) x + (2 - 2^{100})$
$x^{100} = p (x) (x - z) (x - 1) + (a x + b)$
$x = 1 1^{100} = a + b \Rightarrow a + b = 1$
$x = 2 2^{100} = 0 + 2 a + b \Rightarrow a = 2^{100} - 1 b = 2 - 2^{100}$
$R (x) = (2^{100} - 1) x + (2 - 2^{100})$

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