Answer :

u(x) = (x – 1)^{2}

= (x – 1) (x – 1)

v(x) = (x^{2} – 1)

= (x + 1) (x – 1)

LCM of u(x) and v(x) = (x – 1)^{2} (x + 1)

HCF of u(x) and v(x) = (x – 1)

u(x) × v(x) = (x – 1) (x – 1) × (x^{2} – 1)

= (x^{2} – 2x + 1) × (x^{2} – 1)

= x^{4} – 2x^{3} + x^{2} - x^{2} + 2x – 1

= x^{4} – 2x^{3} + 2x – 1

HCF × LCM = (x – 1)^{2} (x + 1) × (x – 1)

= (x^{2} – 2x + 1) (x^{2} – 1)

= x^{4} – 2x^{3} + x^{2} - x^{2} + 2x – 1

= x^{4} – 2x^{3} + 2x – 1

So it is observed that HCF × LCM = u(x) × v(x).

Hence Proved.

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