Find the HCF and LCM of $3 x^{2} - 6 x$ , $x^{2} - 4 x + 4$ and $x^{2} + x - 6$

HCF =

x - 2

and LCM =

3 x (x - 2)^{2} (x + 3)

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HCF =

x + 3

and LCM =

3 x (x - 3)^{2} (x + 1)

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HCF =

x - 1

and LCM =

3 x (x - 3)^{2} (x + 2)

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HCF =

x

and LCM =

3 x (x - 3)^{2} (x + 3)

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Solution

The correct option is A HCF = $x - 2$ and LCM = $3 x (x - 2)^{2} (x + 3)$
Factorise all the polynomials
${3 x}^{2} - 6 x = 3 x (x - 2)$
$x^{2} - 4 x + 4 = (x - 2) (x - 2)$
$x^{2} + x - 6 = (x - 2) (x + 3)$
The least common multiple for the first two polynomials is $3 x {(x - 2)}^{2}$
The LCM of three polynomials is LCM of $3 x {(x - 2)}^{2}$ and $(x - 2) (x - 3)$
$\Rightarrow L C M = 3 x ({x - 2}^{2}) (x + 3)$
The highest common factor of first two polynomials is $(x - 2)$
The HCF of all the three polynomials is HCF of $(x - 2)$ and $(x - 2) (x - 3)$
$\Rightarrow H C F = (x - 2)$

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