To find the GCD by Factorisation

**Greatest Common Divisor (GCD)**

The Greatest Common Divisor, abbreviated as GCD, of two or more polynomials is a polynomial, of
the highest common possible degree, that is a factor of the given two or more polynomials.
It is also known as the Highest Common Factor (HCF).

This concept
is similar to the greatest common divisor of two integers.

For example, Consider the expressions 14*xy*^{2} and 42*xy*. The common
divisors of 14 and 42 are 2, 7 and 14. Their
GCD is thus 14. The only common divisors of *xy*^{2} and *xy are*
*x, y *and* xy*; their GCD is thus* xy.*

14*xy*^{2}
= 1 ×
2 ×
7 ×
*x* × *y* × *y*

42*xy*
= 1 ×
2 ×
3 ×
7 ×
*x* × *y*

Therefore
the requried GCD of 14*xy*^{2} and 42*xy* is 14*xy.*

(i) Each
expression is to be resolved into factors first.

(ii) The
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 3.41**

Find
GCD of the following:

(i) 16*x*^{3}*y*^{2},
24*xy*^{3}*z*

(ii) (*y*^{3}
+
1) and (*y*^{2} −1)

(iii) 2*x*^{2}
− 18 and *x*^{2} -2*x* –
3

(iv) (*a*
− *b* )^{2} , (*b* -*c*)^{3}, (*c* -*a*)^{4}

*Solutions*

(i) 16*x*
^{3}*y*^{2} = 2 × 2 × 2 × 2 × *x* ^{3}*y*^{2} =
2^{4} × *x*^{3} ×*y*^{2} = 2^{3}
×
2 ×*x*^{2} × *x*
×*y*^{2}

24*xy*^{3}*z*
=
2 ×
2 ×
2 ×3×
*x* ×*y*^{3} ×*z* = 2 ^{3} ×3×
*x* ×*y*^{3} ×*z* = 2^{3} ×
3 ×*x* ×
*y* ×*y*^{2} ×*z*

Therefore,
*GCD* = 2^{3} *xy*^{2}

(ii)
*y *^{3}* *+* *1* *=* y*^{3}* *+1^{3}* *=*
*(*y** *+1)(*y *^{2}*
*−*y
*+1)

*y*^{2}* *-1* *=*
y *^{2}* *−1^{2}* *=*
*(*y** *+1)(*y *−1)

Therefore,
*GCD* = (*y* +1)

(iii)
2*x* ^{2} -18 = 2(*x*^{2} −9)
=
2(*x*^{2} − 3^{2} ) =
2(*x* + 3)(*x* − 3)

*x*^{2} − 2*x* − 3 = *x*^{2} − 3*x* + *x* – 3

* = x *(*x
*−*
*3)* *+*
*1(*x *−*
*3)

= (*x*
− 3)(*x* +1)

Therefore,
*GCD* = (*x* − 3)

(iv)
(*a* − *b* )^{2} , (*b* -*c*)^{3} , (*c* -*a*)^{4}

There
is no common factor other than one.

Therefore,
*GCD* = 1

Tags : To find the GCD by Factorisation, Example Solved Problems | Algebra | Maths , 9th Maths : UNIT 3 : Algebra

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9th Maths : UNIT 3 : Algebra : Greatest Common Divisor (GCD) | To find the GCD by Factorisation, Example Solved Problems | Algebra | Maths

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