To find the GCD by Factorisation
(i) Each expression is to be resolved into factors fi rst.
(ii) Th e product of factors having the highest common powers in those factors will be the GCD.
(iii) If the expression have numerical coefficient, find their GCD separately and then prefix it as a coefficient to the GCD for the given expressions.
Example 1 :
Find the GCD for the following:
(i) p5, p11, p9
Solution :
The minimum term of given terms,
= p5
Hence the required GCD is p5
(ii) 4x3, y3, z3
Solution :
There is not common term for the given three terms.
Hence GCD is 1.
(iii) 9 a2 b2 c3, 15 a3 b2 c4
Solution :
9 a2 b2 c3 = 32 a2 ⋅ b2 ⋅ c3
15 a3 b2 c4 = 3 ⋅ 5 a3 ⋅ b2 ⋅ c4
G.C.D = 3a2 b2 c3
(iv) 64x8, 240x6
Solution :
64 = 26 and 240 = 24 ⋅ 5 ⋅ 3
G.C.D = 24 x6 = 16x6
Hence the required G.C.D is 16x6.
(v) ab2c3, a2b3c, a3bc2
Solution :
ab2c3, a2b3c, a3bc2
Hence GCD is abc.
(vi) 35 x5 y3 z4, 49 x2 y z3, 14 x y2 z2
Solution :
35 = 5 ⋅ 7
49 = 72
14 = 2 ⋅ 7
= 7 x2 y z3
Hence the GCD is 7 x2 y z3.
(vii) 25ab3c, 100 a2bc, 125 ab
Solution :
25ab3c = 52ab3c
100 a2bc = 22 ⋅ 52 a2bc
125 ab = 53 ab
GCD = 52 ab = 25ab
Hence GCD is 25ab.
(viii) 3 abc, 5 xyz, 7 pqr
Solution :
There is no common terms, hence the GCD is 1.
Example 2 :
Find the GCD of the following:
(i) (2x +5), (5x +2)
Solution :
The given polynomials are different, since they have no common terms.
G.C.D is 1.
(ii) am + 1, am + 2, am + 3
Solution :
The common term is am + 1
Hence GCD is am + 1.
(iii) 2a2 + a, 4a2 - 1
Solution :
2a2 + a = a(2a + 1)
4a2 - 1 = (2a)2 - 12
= (2a + 1)(2a - 1)
Hence GCD is 2a + 1.
(iv) 3a2, 5b3, 7c4
Solution :
Hence the GCD is 1.
(v) x4 - 1, x2 - 1
Solution :
x4 - 1 = (x2)2 - (12)2
= (x2 + 1)(x2 - 1)
= (x2 + 1)(x + 1)(x - 1)
x2 - 1 = (x + 1) (x - 1)
Hence GCD is (x + 1) and (x - 1).
(vi) a3 - 9ax2, (a - 3x)2
Solution :
a3 - 9ax2 = a(a2 - 9x2)
= a(a2 - (3x)2)
= a (a + 3x) (a - 3x)
(a - 3x)2 = (a - 3x)(a - 3x)
Hence the GCD is (a - 3x).
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