Greatest Common Factor (GCF) of two numbers is the greatest factor that is common to both of them
To find the greatest common divisor of the given numbers or for algebraic expressions we have to follow the steps.
Step 1 :
List the prime factors of each of the given number. For algebraic expression we have to find factors of them.
Step 2 :
List the common factors of the given numbers or common factors.
Step 3 :
Multiply those common factors.
Example :
Find the GCD of the following
(i) c2-d2, c(c-d)
(ii) x4-27a3x, (x-3a)2
(iii) m2-3m-18, m2+5m+6
(iv) x2+14x+33, x3+10x2-11x
(v) x2+3xy+2y2, x2+5xy+6y2
(vi) 2x2-x-1, 4x2+8x+3
(vii) x2-x-2, x2+x-6, 3x2-13x+14
(viii) x3-x2+x-1, x4 - 1
(ix) 24 (6x4-x3-2x2), 20(2x6+3x5+x4)
(x) (a-1)5 (a+3)2, (a-2)2 (a-1)3 (a+3)4
(i) Answer :
c2-d2, c(c-d)
By using algebraic identity
a2-b2 = (a+b)(a-b)
c2-d2= (c+d) (c-d)
c(c - d) = c(c - d)
Common factor is (c-d).
So, greatest common divisor is (c - d).
(ii) Answer :
x4-27a3x, (x-3a)2
x4-27a3x = x(x3-27a3)
= x(x3-(3a)3)
By using algebraic identity
a3-b3 = (a-b)(a2+ab+b2)
x3-(3a)3 = (x-3a)(x2+3ax+9a2)
x(x3-(3a)3) = x(x-3a)(x2+3ax+9a2) ----(1)
(x-3a)2 = (x-3a)(x-3a) ----(2)
Common factors of (1) and (2) are (x-3a).
So, greatest common divisor is (x-3a).
(iii) Answer :
m2-3m-18, m2+5m+6
By doing factorization, we get
m2-3m-18 = (m-6) (m+3)
m2+5m+6 = (m+2) (m+3)
Common factor is (m+3).
So, greatest common divisor is (m+3).
(iv) Answer :
x2+14x+33, x3+10x2-11x
x2+14x+33 = (x+11)(x+3) ---(1)
x3+10x2-11x = x(x2+10x-11)
= x(x+11)(x-1) ----(2)
Common factor is (x+11).
So, greatest common divisor is (x+11).
(v) Answer :
x2+3xy+2y2, x2+5xy+6y2
x2+3xy+2y2 = x2+xy+2xy+2y2
= x(x+y)+2y(x+y)
= (x+2y)(x+y) ---(1)
x2+5xy+6y2 = x2+2xy+3xy+6y2
= x(x+2y)+3y(x+2y)
= (x+2y) (x+3y)----(2)
Greatest common factor of (1) and (2) is (x+2y).
(vi) Answer :
2x2-x-1, 4x2+8x+3
2x2-x-1 = (2x+1) (x-1)
4x2+8x+3 = (2x+3)(2x+1)
Common factor is (2x+1).
So, the greatest common divisor is (2x+1).
(vii) Answer :
x2-x-2, x2+x-6, 3x2-13x+14
x2-x-2 = (x-2) (x+1)
x2+x-6 = (x+3)(x-2)
3x2-13x+14 = (x-2)(3x-7)
Common factor is (x-2).
So, the greatest common divisor is (x - 2).
(viii) Answer :
x3-x2+x-1, x4 - 1
= (x-1)(x2+1) ----(1)
x4-1 = (x2)2-(12)2
= (x2+1) (x2-1)
= (x2+1)(x+1)(x-1) ----(2)
Common factors in (1) and (2) are (x-1) and (x2+1)
So, the greatest common divisor is (x-1)(x2+1).
(ix) Answer :
24 (6x4-x3-2x2), 20(2x6+3x5+x4)
24 (6x4-x3-2x2) = 23 ⋅ 3 x2 (6x2-x-2)
= 23 ⋅ 3 x2 (3x-2) (2x+1)
20(2x6+3x5+x4) = 5 ⋅ 22 x4 (2x2+3x+1)
= 5 ⋅ 22 x4 (x+1) (2x+1)
Common factors are 22 x2 (2x+1)
So, the greatest common divisor is 4x2(2x + 1).
(x) Answer :
(a-1)5 (a+3)2, (a-2)2 (a-1)3 (a+3)4
The common factors are (a-1)3(a+3)2
So, the greatest common divisor is (a-1)3(a+3)2.
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