If u(x) = (x – 1)2 and v(x) = (x2 – 1) then, verify the relatio

1.	If u(x) = (x – 1)2 and v(x) = (x2 – 1) then, verify the relation L.C.M. × H.C.F. = u(x) × v(x).
Answer» u(x) = (x – 1)² ⇒ u(x) = (x – 1)(x – 1) and v(x) = (x² – 1) = (x – 1)(x + 1) Product of common least powers = (x – 1) H.C.F. = (x – 1) Product of highest power of prime factors = (x – 1)²(x + 1) L.C.M. = (x – 1)²(x + 1) Test: u(x) × v(x) = (x – 1)² × (x² – 1) = (x – 1)²(x² – 1) and H.C.F. × L.C.M. = (x – 1)(x – 1)²(x + 1) = (x – 1)²(x² – 1) So, L.C.M. × H.C.F. = w(x) × v(x).

If u(x) = (x – 1)2 and v(x) = (x2 – 1) then, verify the relation L.C.M. × H.C.F. = u(x) × v(x).

Answer»

u(x) = (x – 1)²

⇒ u(x) = (x – 1)(x – 1)

and v(x) = (x² – 1)

= (x – 1)(x + 1)

Product of common least powers

= (x – 1)

H.C.F. = (x – 1)

Product of highest power of prime factors

= (x – 1)²(x + 1)

L.C.M. = (x – 1)²(x + 1)

Test:

u(x) × v(x) = (x – 1)² × (x² – 1)

= (x – 1)²(x² – 1)

and H.C.F. × L.C.M.

= (x – 1)(x – 1)²(x + 1)

= (x – 1)²(x² – 1)

So, L.C.M. × H.C.F. = w(x) × v(x).