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# Let's find the G.C.D. of the algebraic expressions x(x^{2}– 9), x^{2} – x – 12.

Answer :

Let us understand what a G.C.D, Greatest Common Divisor is.

The greatest common divisor of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers.

To find G.C.D of x(x^{2} – 9) and x^{2} – x – 12, let us write down factors of each term.

Factorization of x(x^{2} – 9) = x(x – 3)(x + 3)

[∵, by algebraic identity, a^{2} – b^{2} = (a – b)(a + b)]

For factorization of x^{2} – x – 12,

x^{2} – x – 12 = x^{2} – 4x + 3x – 12 [∵, Sum of -4 and 3 is -1 and multiplication is -12]

⇒ x^{2} – x – 12 = x(x – 4) + 3(x – 4) [∵, common from the first two terms is x and last two terms is 3]

⇒ x^{2} – x – 12 = (x – 4)(x + 3) [∵, common from the two terms is (x – 4)]

So, factorization of x^{2} – x – 12 = (x – 4)(x + 3)

Find the factors that these two lists share in common.

Factors of x(x^{2} – 9) = x, (x – 3), (x + 3)

Factors of x^{2} – x – 12 = (x – 4), (x + 3)

Common factor that is found in these two terms is (x + 3).

Thus, the gcd of x(x^{2} – 9) and x^{2} – x – 12 is (x + 3).

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