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GCD of Polynomials

H.C.F of x3...

Question

H.C.F of $(x^{3} - 3 x + 2)$ and $(x^{2} - 4 x + 3)$ is:

(x - 1)

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{(x - 1)}^{2}

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(x - 1) (x + 2)

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(x - 1) (x - 3)

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Solution

The correct option is A $(x - 1)$
Given $p (x) = x^{3} - 3 x + 2$
Let $x = 1$
$p (1) = 1 - 3 + 2 = 0$ . So, $(x - 1)$ is a factor of $p (x)$
$\Rightarrow$ $x^{2} (x - 1) + x (x - 1) - 2 (x - 1)$
$\Rightarrow$ $p (x) = (x^{2} + x - 2) (x - 1)$
$\Rightarrow$ $p (x) = (x^{2} + 2 x - x - 2) (x - 1)$
$\Rightarrow$ $p (x) = (x - 1) (x - 1) (x + 2)$
And, $q (x) = x^{2} - 4 x + 3$
$\Rightarrow$ $q (x) = x^{2} - 3 x - x + 3$
$\Rightarrow$ $q (x) = x (x - 3) - 1 (x - 3)$
$\Rightarrow$ $q (x) = (x - 3) (x - 1)$
$\Rightarrow$ $q (x) = (x - 3) (x - 1)$
Hence, HCF $= (x - 1)$

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