Q. 5 D4.9( 7 Votes )

# Let’s find the G.C.D and L.C.M. of the following expressions:

a^{2} + b^{2}-c^{2} + 2ab, a^{2} + c^{2}-b^{2} + 2ca, b^{2} + c^{2}-a^{2} + 2bc

Answer :

factors of a^{2} + b^{2}-c^{2} + 2ab

=(a + b)^{2}-c^{2}

=(a + b + c)(a + b-c)

Factors of a^{2} + c^{2}-b^{2} + 2ca

=(a + c)^{2}-b^{2}

=(a + c + b)(a + c-b)

Factors of b^{2} + c^{2}-a^{2} + 2bc

=(b + c)^{2}-a^{2}

=(b + c + a)(b + c-a)

LCM is the lowest common multiple, which is the product of all the common factors taken once and the remaining factors as it is.

∴ the LCM of a^{2} + b^{2}-c^{2} + 2ab, a^{2} + c^{2}-b^{2} + 2ca, b^{2} + c^{2}-a^{2} + 2bc is (a + b-c)(a + c-b)(b + c-a)(a + b + c)

GCD is the greatest common divisor, which is equal to the product of all the common divisors.

∴ the GCD is (a + b + c)

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