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LCM of Polynomials

Find the LCM ...

Question

Find the LCM of the pair of the given polynomials.
$(x - 1) (x - 2) (x^{2} - 3 x + 3), (x - 1), (x^{3} - 4 x^{2} + 6 x - 3)$

A

(x - 2) (x - 1)

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B

(x - 1)

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C

(x - 3) (x - 1)

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D

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

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Solution

The correct option is D $(x - 1) (x - 2) (x^{2} - 3 x + 3)$
We know that the least common multiple (LCM) is the smallest number or expression that is a common multiple of two or more numbers or algebraic terms.

We first factorize the given polynomials as shown below:

$(x - 1) (x - 2) (x^{2} - 3 x + 3) = (x - 1) (x - 2) (x^{2} - 3 x + 3)$
$(x - 1) = (x - 1)$
$(x^{3} - 4 x^{2} + 6 x - 3) = (x - 1) (x^{2} - 3 x + 3)$

Now multiply all the factors, using each common factor only once, therefore, the LCM is:
$= (x - 1) (x - 2) (x^{2} - 3 x + 3)$

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(viii) $x^{3} - 4 x^{2} - x + 1 = {(x - 2)}^{3}$

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