**Finding GCD of Polynomials Examples :**

Here we are going to see some example problems on finding GCD of polynomials.

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.

**Question 1 :**

Find the GCD for the following:

(i) p^{5}, p^{11}, p^{9}

**Solution :**

The minimum term of given terms,

= p^{5}

Hence the required GCD is p^{5}

(ii) 4x^{3}, y^{3}, z^{3}

**Solution :**

There is not common term for the given three terms.

Hence GCD is 1.

(iii) 9 a^{2 }b^{2 }c^{3}, 15 a^{3 }b^{2 }c^{4}

**Solution :**

9 a^{2 }b^{2 }c^{3 }= 3^{2} a^{2 }⋅ ^{ }b^{2 }⋅ c^{3}

15 a^{3 }b^{2 }c^{4 }= 3^{ }⋅ 5 a^{3 }⋅ ^{ }b^{2 }⋅ c^{4}

G.C.D = 3a^{2} b^{2 }c^{3}

(iv) 64x^{8}, 240x^{6}

**Solution :**

64 = 2^{6 } and 240 = 2^{4 }⋅ 5 ⋅ 3

G.C.D = 2^{4} x^{6 } = 16x^{6}

Hence the required G.C.D is 16x^{6}.

(v) ab^{2}c^{3}, a^{2}b^{3}c, a^{3}bc^{2}

**Solution :**

ab^{2}c^{3}, a^{2}b^{3}c, a^{3}bc^{2}

Hence GCD is abc.

(vi) 35 x^{5} y^{3} z^{4}, 49 x^{2 }y z^{3}, 14 x y^{2 }z^{2}

**Solution :**

35 = 5 ⋅ 7

49 = 7^{2}

14 = 2 ⋅ 7

= 7 x^{2 }y z^{3}

Hence the GCD is 7 x^{2 }y z^{3}.

(vii) 25ab^{3}c, 100 a^{2}bc, 125 ab

**Solution :**

25ab^{3}c = 5^{2}ab^{3}c

100 a^{2}bc = 2^{2} ⋅ 5^{2} a^{2}bc

125 ab = 5^{3} ab

GCD = 5^{2} ab = 25ab

Hence GCD is 25ab.

(viii) 3 abc, 5 xyz, 7 pqr

**Solution :**

There is no common terms, hence the GCD is 1.

**Question 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) a^{m + 1}, a^{m + 2}, a^{m + 3}

**Solution :**

The common term is a^{m + 1}

Hence GCD is a^{m + 1}.

(iii) 2a^{2} + a, 4a^{2} - 1

**Solution :**

2a^{2} + a = a(2a + 1)

4a^{2} - 1 = (2a)^{2} - 1^{2}

= (2a + 1)(2a - 1)

Hence GCD is 2a + 1.

(iv) 3a^{2}, 5b^{3}, 7c^{4}

**Solution :**

Hence the GCD is 1.

(v) x^{4} - 1, x^{2} - 1

**Solution :**

x^{4} - 1 = (x^{2})^{2} - (1^{2})^{2}

= (x^{2} + 1)(x^{2} - 1)

= (x^{2} + 1)(x + 1)(x - 1)

x^{2} - 1 = (x + 1) (x - 1)

Hence GCD is (x + 1) and (x - 1).

(vi) a^{3} - 9ax^{2}, (a - 3x)^{2}

**Solution :**

a^{3} - 9ax^{2 } = a(a^{2} - 9x^{2})

= a(a^{2} - (3x)^{2})

= a (a + 3x) (a - 3x)

(a - 3x)^{2} = (a - 3x)(a - 3x)

Hence the GCD is (a - 3x).

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