FINDING GCD OF POLYNOMIALS EXAMPLES

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)  a^{m + 1}, a^{m + 2}, a^{m + 3}

Solution :

The common term is a^{m + 1}

Hence GCD is a^{m + 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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