Let’s find G.C.D of the following algebraic expressions:
4a2b2, 20ab2
4a2b2, 20ab2
Factors of 4a2b2=4×a×a×b×b
Factors of 20ab2=4×5×a×b×b
GCD is the greatest common divisor, which is equal to the product of all the common divisors.
∴ The GCD of 4a2b2, 20ab2 is 4ab2
Let’s find G.C.D of the following algebraic expressions:
5p2q2, 10p2q2, 25p4q3
factors of 5p2q2=5×p×p×q×q
factors of 10p2q2=2×5×p×p×q×q
factors of 25p4q3=2×2×5×5×p×q×q×q
GCD is the greatest common divisor, which is equal to the product of all the common divisors.
∴the GCD of 5p2q2, 10p2q2, 25p4q3 is 5pq2
Let’s find G.C.D of the following algebraic expressions:
7y3z6, 21y2, 14z2
factors of 7y3z6=7×y×y×y×z×z×z×z×z×z
Factors of 21y2=3×7×y×y
Factors of 14z2=2×7×z×z
GCD is the greatest common divisor, which is equal to the product of all the common divisors.
∴ the GCD of 7y3z6, 21y2, 14z2 is 7
Let’s find G.C.D of the following algebraic expressions:
3a2b2c, 12a2b4c2, 9a5b4
factors of 3a2b2c=3×a×a×b×b×c
Factors of 12a2b4c2= 2×2×3×a×a×b×b×b×b×c×c
Factors of 9a5b4=3×3×a×a×a×a×a×b×b×b×b
GCD is the greatest common divisor, which is equal to the product of all the common divisors.
∴the GCD of 3a2b2c, 12a2b4c2, 9a5b4=3a2b2
Let’s find the L.C.M. of the following algebraic expressions:
2x2y3, 10x3y
2x2y3, 10x3y
Factors of 2x2y3=2× x × x × y × y × y
Factors of 10x3y=2×5 × x × x × x × y
LCM is the lowest common multiple, which is the product of all the common factors taken once and the remaining factors as it is.
∴the LCM of 2x2y3, 10x3y is 10x3y3
Let’s find the L.C.M. of the following algebraic expressions:
7p2q3, 35p3q, 42pq4
factors of 7p2q3=7×p×p×q×q×q
Factors of 35p3q=5×7××p×p×p×q
Factors of 42pq4=2×3×7×p×q×q×q×q
LCM is the lowest common multiple, which is the product of all the common factors taken once and the remaining factors as it is.
∴ the LCM of 7p2q3, 35p3q, 42pq4 is 210p3q4
Let’s find the L.C.M. of the following algebraic expressions:
5a5b, 15ab2c, 25a2b2c2
factors of 5a5b=5×a×a×a×a×a×b
Factors of 15ab2c=3×5×a×b×b×c
Factors of 25a2b2c2=5×5×a×a×b×b×c×c
LCM is the lowest common multiple, which is the product of all the common factors taken once and the remaining factors as it is.
∴the LCM of 5a5b, 15ab2c, 25a2b2c2=75a5b2c2
Let’s find the L.C.M. of the following algebraic expressions:
11a2bc2, 33a2b2c, 55a2bc2
Factors of 11a2bc2=11×a×a×b×c×c
Factors of 33a2b2c=3×11×a×a×b×b×c
Factors of 55a2bc2=5×11×a×a×b×c×c
LCM is the lowest common multiple, which is the product of all the common factors taken once and the remaining factors as it is.
∴the LCM of 11a2bc2, 33a2b2c, 55a2bc2=165 a2b2c2
Let’s find G.C.D of the following algebraic expression:
5x(x + y),x3-xy2
5x(x + y), x3-xy2
Factors of 5x(x + y)=5×x×(x + y)
x3-xy2=x(x2-y2)
Apply the formula a2 – b2 = (a + b)(a-b)
=x(x + y)(x-y)
Factors of x3-xy2= x(x + y)(x-y)
GCD is the greatest common divisor, which is equal to the product of all the common divisors.
∴the GCD 5x(x + y), x3-xy2 is x(x + y)
Let’s find G.C.D of the following algebraic expression:
x3-3x2y, x2-9y2
x3-3x2y=x2(x-3y)
∴Factors of x3-3x2y= x2(x-3y)
x2-9y2=(x + 3y)(x-3y)
∴Factors of x2-9y2=(x + 3y)(x-3y)
GCD is the greatest common divisor, which is equal to the product of all the common divisors.
∴ the GCD of x3-3x2y, x2-9y2 is (x-3y)
Let’s find G.C.D of the following algebraic expression:
2ax(a-x)2, 4a2x(a-x)3
Factors of 2ax(a-x)2=2×a×x×(a-x) ×(a-x)
Factors of 4a2x(a-x)3=2×2×a×a×x×(a-x) ×(a-x) ×(a-x)
GCD is the greatest common divisor, which is equal to the product of all the common divisors.
∴the GCD of 2ax(a-x)2, 4a2x(a-x)3 is 2ax(a-x)2
Let’s find G.C.D of the following algebraic expression:
x2-1, x2-2x + 1, x3 + x2-2x
Factors of x2-1=(x + 1)(x-1)
Factors of x2-2x + 1
x2-2x + 1=x2-x-x + 1
=x(x-1)-1(x-1)
=(x-1)2
=(x-1)×(x-1)
Factors of x3 + x2-2x
x3 + x2-2x=x(x2 + x-2)
=x(x2 + 2x-x-2)
=x[{x(x + 2)-1(x + 2)}]
=x[{(x + 2)(x-1)}]
=x×(x + 2)×(x-1)
GCD is the greatest common divisor, which is equal to the product of all the common divisors.
∴ the GCD of x2-1, x2-2x + 1, x3 + x2-2x is (x-1)
Let’s find G.C.D of the following algebraic expression:
a2-1, a3-1, a2 + a-2
Factors of a2-1=(a + 1)(a-1)
Factors of a3-1=(a-1)(a2 + a + 1) … (from properties)
Factors of a2 + a-2
=(a2 + a-2)
=(a2 + 2a-a-2)
=[{a(a + 2)-1(a + 2)}]
=[{(a + 2)(a-1)}]
=(a + 2)×(a-1)
GCD is the greatest common divisor, which is equal to the product of all the common divisors.
∴the GCD of a2-1, a3-1, a2 + a-2 is (a-1)
Let’s find G.C.D of the following algebraic expression:
x2 + 3x + 2, x2 + 4x + 3, x2 + 5x + 6
Factors of x2 + 3x + 2
= x2 + 2x + x + 2
=x(x + 2) + 1(x + 2)
=(x + 1)(x + 2)
Factors of x2 + 4x + 3
= x2 + 3x + x + 3
=x(x + 3) + 1(x + 3)
=(x + 1)(x + 3)
Factors of x2 + 5x + 6
= x2 + 2x + 3x + 6
=x(x + 2) + 3(x + 2)
=(x + 2)(x + 3)
GCD is the greatest common divisor, which is equal to the product of all the common divisors.
∴ the GCD of x2 + 3x + 2, x2 + 4x + 3, x2 + 5x + 6 is 1 because no other factor is common to all the three and only 1 which is a universal factor of every number is the answer.
Let’s find G.C.D of the following algebraic expression:
x2 + xy,xz + yz, x2 + 2xy + y2
Factors of x2 + xy=x(x + y)
Factors of xz + yz=z(x + y)
Factors of x2 + 2xy + y2=(x + y)2 … (from properties of (a + b)2)
GCD is the greatest common divisor, which is equal to the product of all the common divisors.
∴ the GCD of x2 + xy,xz + yz, x2 + 2xy + y2 is (x + y)
Let’s find G.C.D of the following algebraic expression:
8(x2-4),12(x3 + 8),36(x2-3x-10)
Factors of 8(x2-4)=2×2×2×(x-2)×(x + 2)
Factors of 12(x3 + 8)
=2×2×3×(x + 2)×(x2-2x + 4)
… ((x3 + 8)= (x + 2)×(x2-2x + 4) from properties)
Factors of 36(x2-3x-10)
Factors of (x2-3x-10)
=(x2-5x + 2x-10)
=x(x-5) + 2(x-5)
=(x + 2)(x-5)
Factors of 36(x2-3x-10)=2×2×3×3×(x + 2)×(x-5)
GCD is the greatest common divisor, which is equal to the product of all the common divisors.
∴the GCD of 8(x2-4),12(x3 + 8),36(x2-3x-10) is 4(x + 2)
Let’s find G.C.D of the following algebraic expression:
a2-b2-c2 + 2bc, b2-c2-a2 + 2ac, c2- a2-b2 + 2ab
Factors of a2-b2-c2 + 2bc
= a2-(b2 + c2-2bc)
= a2-(b-c)2
=(a + b-c)(a-b + c) … (Using a2-b2=(a + b)(a-b))
Factors of b2-c2-a2 + 2ac
= b2-(c2 + a2-2ac)
=b2-(c-a)2
=(b + c-a)(b-c + a) … (Using a2-b2=(a + b)(a-b))
Factors of c2-a2-b2 + 2ab
= c2-(a2 + b2-2ab)
=c2-(a-b)2
=(c + a-b)(c-a + b) … (Using a2-b2=(a + b)(a-b))
GCD is the greatest common divisor, which is equal to the product of all the common divisors.
∴the GCD of a2-b2-c2 + 2bc, b2-c2-a2 + 2ac, c2- a2-b2 + 2ab is 1 as no other factor is common to all the three terms
Let’s find G.C.D of the following algebraic expression:
x3-16x, 2x3 + 9x2 + 4x, 2x3 + x2-28x
Factors of x3-16x
=x(x2-16)
=x(x + 4)(x-4) … (Using a2-b2=(a + b)(a-b))
Factors of 2x3 + 9x2 + 4x
=x(2x2 + 9x + 4)
=x(2x2 + 8x + x + 4)
=x{2x(x + 4) + 1(x + 4)}
=x[{(2x + 1)(x + 4)}]
=x×(2x + 1)×(x + 4)
Factors of 2x3 + x2-28x
=x(2x2 + x-28)
=x(2x2 + 8x-7x-28)
=x{2x(x + 4)-7(x + 4)}
=x[{(2x-7)(x + 4)}]
=x×(2x-7)×(x + 4)
GCD is the greatest common divisor, which is equal to the product of all the common divisors.
∴ the GCD of x3-16x, 2x3 + 9x2 + 4x, 2x3 + x2-28x is x(x + 4)
Let’s find G.C.D of the following algebraic expression:
4x2-1, 8x3-1, 4x2-4x + 1
Factors of 4x2-1
=(2x + 1)(2x-1) … (Using a2-b2=(a + b)(a-b))
Factors of 8x3-1
=(2x-1)(4x2 + 2x + 1) … (using a3-b3=(a-b)(a2 + ab + b2)
Factors of 4x2-4x + 1
=(2x-1)2 … (from properties)
GCD is the greatest common divisor, which is equal to the product of all the common divisors.
∴ the GCD of 4x2-1, 8x3-1, 4x2-4x + 1 is (2x-1)
Let’s find G.C.D of the following algebraic expression:
x3-3x2-10x, x3 + 6x2 + 8x, x4-5x3-14x2
Factors of x3-3x2-10x
=x(x2-3x-10)
=x(x2-5x + 2x-10)
=[x{x(x-5) + 2(x-5)}]
=x(x + 2)(x-5)
Factors of x3 + 6x2 + 8x
=x(x2 + 6x + 8)
=x(x2 + 4x + 2x + 8)
=x[x(x + 4) + 2(x + 4)]
=x(x + 2)(x + 4)
Factors of x4-5x3-14x2
=x2(x2-5x-14)
= x2(x2-7x + 2x-14)
= x2{x(x-7) + 2(x-7)}
= x2 � �(x + 2)(x-7)
GCD is the greatest common divisor, which is equal to the product of all the common divisors.
∴ the GCD of x3-3x2-10x, x3 + 6x2 + 8x, x4-5x3-14x2 is x(x + 2)
Let’s find G.C.D of the following algebraic expression:
6x2-13xa + 6a2, 6x2 + 11xa-10a2, 6x2 + 2xa-4a2
Factors of 6x2-13xa + 6a2
=6x2-9xa-4xa + 6a2
=3x(2x-3a)-2a(2x-3a)
= (3x-2a)(2x-3a)
Factors of 6x2 + 11xa-10a2
= 6x2 + 15xa-4xa-10a2
=3x(2x + 5a)-2a(2x + 5a)
=(3x-2a)(2x + 5a)
Factors of 6x2 + 2xa-4a2
=6x2 + 6xa-4xa-4a2
=6x(x + a)-4a(x + a)
=(6x-4a)(x + a)
=2(3x-2a)(x + a)
GCD is the greatest common divisor, which is equal to the product of all the common divisors.
∴ the GCD of 6x2-13xa + 6a2, 6x2 + 11xa-10a2, 6x2 + 2xa-4a2is 3x-2a
Let’s find the L.C.M. of the following algebraic expressions:
p2-q2,(p + q)2
p2-q2,(p + q)2
Factors of p2-q2
=(p + q)(p-q)
Factors of (p + q)2
=(p + q) (p + q)
LCM is the lowest common multiple, which is the product of all the common factors taken once and the remaining factors as it is.
∴ the LCM of p2-q2,(p + q)2 is (p-q) (p + q)2
Let’s find the L.C.M. of the following algebraic expressions:
(x2y2-x2);(xy2-2xy + x)
Factors of (x2y2-x2)
= x2 (y2-1)
= x2(y-1)(y + 1)
Factors of (xy2-2xy + x)
=x(y2-2y + 1)
=x(y-1) (y-1)
LCM is the lowest common multiple, which is the product of all the common factors taken once and the remaining factors as it is.
∴ the LCM of (x2y2-x2);(xy2-2xy + x) is x2(y-1)2(y + 1)
Let’s find the L.C.M. of the following algebraic expressions:
(p + q) (p + r), (q + r) (r + p), (r + p) (p + q)
Factors of (p + q) (p + r)
=(p + q)×(p + r)
Factors of (q + r)(r + p)
= (q + r)×(r + p)
Factors of (r + p) (p + q)
=(r + p)×(p + q)
LCM is the lowest common multiple, which is the product of all the common factors taken once and the remaining factors as it is.
∴ the LCM of (p + q)(p + r), (q + r)(r + p), (r + p)(p + q) is (p + q)(q + r)(p + r)
Let’s find the L.C.M. of the following algebraic expressions:
ab4-8ab, a2b4 + 8a2b, ab4-4ab2
factors of ab4-8ab
=ab(b3-8)
=ab(b-2)(b2 + 2b + 4)
Factors of a2b4 + 8a2b
=a2b(b3 + 8)
= a2b(b + 2)( b2-2b + 4)
Factors of ab4-4ab2
=ab2(b2-4)
= ab2(b + 2)(b-2)
LCM is the lowest common multiple, which is the product of all the common factors taken once and the remaining factors as it is.
∴ the LCM of ab4-8ab, a2b4 + 8a2b, ab4-4ab2 is
a2b2(b + 2)(b-2)(b2-2b + 4)(b2 + 2b + 4)
= a2b2(b3 + 8) (b3-8)
Let’s find the L.C.M. of the following algebraic expressions:
x4 + x2y2 + y4, x3y + y4, (x2-xy)3
factors of x4 + x2y2 + y4:
Add and subtract x2y2 to get,
x4 + x2y2 + y4 = x4 + x2y2 + y4 + x2y2 - x2y2
= x4 + 2x2y2 + y4 - x2y2
As we know (a + b)2 = a2 + b2 + 2ab,
So,
x4 + x2y2 + y4 = (x2 + y2)2 - x2y2
= (x2 + y2)2 – (xy)2
Apply the formula a2 – b2 = (a + b) (a-b) in (x2 + y2)2 – (xy)2to get,
= (x2 + y2-xy) (x2 + y2+xy)
Now factorise x3y + y4,
= y (x3 + y3)
Apply a3 + b3 = (a + b)(a2 + b2 – ab) in (x3 + y3)
= y (x+y)(x2 + y2 – xy)
Now factorise (x2-xy)3,
= (x2-xy)2(x2-xy)
Apply (a-b)2 = a2 + b2 – 2ab in (x2-xy)2 to get,
= [(x2)2 + (xy)2 – 2x2xy][x(x-y)]
= x2 [x2 + y2 – 2xy][x(x-y)]
As we know (a - b)2 = a2 + b2 - 2ab in ,
= x3 (x-y)2 (x-y)
= x3 (x-y)3
So the LCM of x4 + x2y2 + y4, x3y + y4, (x2-xy)3 is:
x3y (x-y)3(x+y)(x2 + y2 – xy) (x2 + y2+xy)
Let’s find the L.C.M. of the following algebraic expressions:
p2 + 2p, 2p4 + 3p3-2p2, 2p3-3p2-14p
Factors of p2 + 2p
=p(p + 2)
Factors of 2p4 + 3p3-2p2
=p2(2p2 + 3p-2)
= p2(2p2 + 4p-p-2)
=p2{2p(p + 2)-1(p + 2)}
=p2{(2p-1)(p + 2)}
Factors of 2p3-3p2-14p
=p(2p2-3p-14)
=p(2p2-7p + 4p-14)
=p{p(2p-7) + 2(2p-7)}
=p(p + 2)(2p-7)
LCM is the lowest common multiple, which is the product of all the common factors taken once and the remaining factors as it is.
∴ the LCM of p2 + 2p, 2p4 + 3p3-2p2, 2p3-3p2-14p is
p2(2p-1)(p + 2) (2p-7)
Let’s find the L.C.M. of the following algebraic expressions:
x2-y2 + z2-2xz, x2-y2-z2 + 2yz, xy + zx + y2-z2
Factors of x2-y2 + z2-2xz
= x2 + z2-2xz-y2
=(x-z)2-y2
=(x-z + y)(x-z-y)
Factors of x2-y2-z2 + 2yz
= x2-(y2 + z2-2yz)
= x2-(y-z)2
=(x + y-z)(x-y + z)
Factors of xy + zx + y2-z2
=x(y + z) + (y + z)(y-z)
=(x + y-z)(y + z)
LCM is the lowest common multiple, which is the product of all the common factors taken once and the remaining factors as it is.
∴ the LCM of x2-y2 + z2-2xz, x2-y2-z2 + 2yz, xy + zx + y2-z2 Is
(x-z + y)(x-z-y)(x-y + z)(y + z)
Let’s find the L.C.M. of the following algebraic expressions:
x2-xy-2y2, 2x2-5xy + 2y2, 2x2 + xy-y2
Factors of x2-xy-2y2
= x2-2xy + xy-2y2
=x(x-2y) + y(x-2y)
=(x + y)(x-2y)
Factors of 2x2-5xy + 2y2
=2x2-4xy-xy + 2y2
=2x(x-2y)-y(x-2y)
=(2x-y)(x-2y)
Factors of 2x2 + xy-y2
=2x2 + 2xy-xy-y2
=2x(x + y)-y(x + y)
=(2x-y) (x + y)
LCM is the lowest common multiple, which is the product of all the common factors taken once and the remaining factors as it is.
∴ the LCM of x2-xy-2y2, 2x2-5xy + 2y2, 2x2 + xy-y2 is
(x + y)(x-2y) (2x-y)
Let’s find the L.C.M. of the following algebraic expressions:
3x2-15x + 18, 2x2 + 2x-24, 4x2 + 36x + 80
factors of 3x2-15x + 18
=3x2-9x-6x + 18
=3x(x-3)-6(x-3)
=(3x-6) (x-3)
=3(x-2) (x-3)
Factors of 2x2 + 2x-24
=2(x2 + x-12)
=2(x2 + 4x-3x-12)
=2(x(x + 4)-3(x + 4))
=2(x-3) (x + 4)
Factors of 4x2 + 36x + 80
=4(x2 + 9x + 20)
=4(x2 + 5x + 4x + 20)
=4{x(x + 5) + 4(x + 5)}
=4(x + 5) (x + 4)
LCM is the lowest common multiple, which is the product of all the common factors taken once and the remaining factors as it is.
∴ the LCM of 3x2-15x + 18, 2x2 + 2x-24, 4x2 + 36x + 80 is
12(x-2) (x-3) (x + 4) (x + 5)
Let’s find the L.C.M. of the following algebraic expressions:
(a2 + 2a)2 , 2a3 + 3a2-2a, 2a4-3a3-12a2
factors of (a2 + 2a)2
= a4 + 4a2 + 4a3
= a2(a2 + 4 + 4a)
= a2(a2 + 2a + 2a + 4)
= a2(a(a + 2) + 2(a + 2))
= a2(a + 2)(a + 2)
= a2(a + 2)2
Factors of 2a3 + 3a2-2a
=a(2a2 + 3a-2)
=a(2a2 + 4a-a-2)
=a{2a(a + 2)-1(a + 2)}
=a(a + 2) (2a-1)
Factors of 2a4-3a3-14a2
=a2(2a2-3a-14)
=a2(2a2 + 4a-7a-14)
= a2(2a(a + 2)-7(a + 2))
= a2(2a-7) (a + 2)
LCM is the lowest common multiple, which is the product of all the common factors taken once and the remaining factors as it is.
∴ the LCM of (a2 + 2a)2, 2a3 + 3a2-2a, 2a4-3a3-12a2 is a2(a + 2)2(2a-1)(2a-7).
Let’s find the L.C.M. of the following algebraic expressions:
3a2-5ab-12b2, a5-27a2b3, 9a2 + 24ab + 16b2
factors of 3a2-5ab-12b2
=3a2-9ab + 4ab-12b2
=3a(a-3b) + 4b(a-3b)
=(3a + 4b)(a-3b)
Factors of a5-27a2b3
=a2(a3-27b3)
=a2(a-3b)(a2 + 3ab + 9b2)
Factors of 9a2 + 24ab + 16b2
=9a2 + 12ab + 12ab + 16b2
=3a(3a + 4b) + 4b(3a + 4b)
=(3a + 4b)2
LCM is the lowest common multiple, which is the product of all the common factors taken once and the remaining factors as it is.
∴ the LCM of 3a2-5ab-12b2, a5-27a2b3, 9a2 + 24ab + 16b2is
a2(a-3b)(a2 + 3ab + 9b2) (3a + 4b)2
Let’s find the G.C.D and L.C.M. of the following expressions:
x3-8, x2 + 3x-10, x3 + 2x2-8x
Factors of x3-8
=(x-2)(x2 + 2x + 4)
Factors of x2 + 3x-10
= x2 + 5x-2x-10
=x(x + 5)-2(x + 5)
=(x + 5)(x-2)
Factors of x3 + 2x2-8x
=x(x2 + 2x-8)
=x(x2 + 4x-2x-8)
=x{x(x + 4)-2(x + 4)}
=x(x + 4)(x-2)
LCM is the lowest common multiple, which is the product of all the common factors taken once and the remaining factors as it is.
∴ the LCM of x3-8, x2 + 3x-10, x3 + 2x2-8x is x(x-2)(x2 + 2x + 4) (x + 4)(x + 5)
GCD is the greatest common divisor, which is equal to the product of all the common divisors.
∴ the GCD is (x-2)
Let’s find the G.C.D and L.C.M. of the following expressions:
3y2-15y + 18, 2y2 + 2y-24, 4y2 + 36y + 80
factors of 3y2-15y + 18
=3y2-9y-6y + 18
=3y(y-3)-6(y-3)
=(3y-6) (y-3)
=3(y-2) (y-3)
Factors of 2y2 + 2y-24
=2(y2 + y-12)
=2(y2 + 4y-3y-12)
=2(y(y + 4)-3(y + 4))
=2(y-3)(y + 4)
Factors of 4y2 + 36y + 80
=4(y2 + 9y + 20)
=4(y2 + 5y + 4y + 20)
=4{y(y + 5) + 4(y + 5)}
=4(y + 5)(y + 4)
LCM is the lowest common multiple, which is the product of all the common factors taken once and the remaining factors as it is.
∴ the LCM of 3y2-15y + 18, 2y2 + 2y-24, 4y2 + 36y + 80 is 12(y-2)(y + 5)(y + 4)(y-3)
GCD is the greatest common divisor, which is equal to the product of all the common divisors.
∴ the GCD is 1 as no other factor is common.
Let’s find the G.C.D and L.C.M. of the following expressions:
a3-4a2 + 4a, a2 + a-6, a3-8
Factors of a3-4a2 + 4a
=a(a2-4a + 4)
=a(a-2)2
Factors of a2 + a-6
= a2 + 3a-2a-6
=a(a + 3)-2(a + 3)
=(a + 3)(a-2)
Factors of a3-8
=(a-2)(a2 + 2a + 4)
LCM is the lowest common multiple, which is the product of all the common factors taken once and the remaining factors as it is.
∴ the LCM of a3-4a2 + 4a, a2 + a-6, a3-8 is a(a-2)2(a2 + 2a + 4) (a + 3)
GCD is the greatest common divisor, which is equal to the product of all the common divisors.
∴ the GCD is (a-2)
Let’s find the G.C.D and L.C.M. of the following expressions:
a2 + b2-c2 + 2ab, a2 + c2-b2 + 2ca, b2 + c2-a2 + 2bc
factors of a2 + b2-c2 + 2ab
=(a + b)2-c2
=(a + b + c)(a + b-c)
Factors of a2 + c2-b2 + 2ca
=(a + c)2-b2
=(a + c + b)(a + c-b)
Factors of b2 + c2-a2 + 2bc
=(b + c)2-a2
=(b + c + a)(b + c-a)
LCM is the lowest common multiple, which is the product of all the common factors taken once and the remaining factors as it is.
∴ the LCM of a2 + b2-c2 + 2ab, a2 + c2-b2 + 2ca, b2 + c2-a2 + 2bc is (a + b-c)(a + c-b)(b + c-a)(a + b + c)
GCD is the greatest common divisor, which is equal to the product of all the common divisors.
∴ the GCD is (a + b + c)
Let’s find the G.C.D and L.C.M. of the following expressions:
x3-4x, 4(x2-5x + 6),(x2-4x + 4)
factors of x3-4x
=x(x2-4)
=x(x-2)(x + 2)
Factors of 4(x2-5x + 6)
=4(x2-3x-2x + 6)
=4{x(x-3)-2(x-3)}
=4(x-2)(x-3)
Factors of (x2-4x + 4)
=(x-2)2
LCM is the lowest common multiple, which is the product of all the common factors taken once and the remaining factors as it is.
∴ the LCM of x3-4x, 4(x2-5x + 6),(x2-4x + 4) is 4x(x-3)(x + 2)(x-2)2
GCD is the greatest common divisor, which is equal to the product of all the common divisors.
∴ the GCD is (x-2)
Let's find the G.C.D. of ax2, a2x3 & a4x.
Let us understand what a G.C.D, Greatest Common Divisor is.
The greatest common divisor of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers.
To find G.C.D of ax2, a2x3 and a4x, let us first write down factors of each term.
Factorization of ax2 = a × x × x
Factorization of a2x3 = a × a × x × x × x
Factorization of a4x = a × a × a × a × x
Find the factors that these three lists share in common.
Factors of ax2 = a, x, x
Factors of a2x3 = a, a, x, x, x
Factors of a4x = a, a, a, a, x
Common factors that are found in these three terms is a and x.
By multiplying these factors, we get
a × x = ax
Thus, the gcd of ax2, a2x3 and a4x is ax.
Let's find the G.C.D. of the algebraic expressions x(x2– 9), x2 – x – 12.
Let us understand what a G.C.D, Greatest Common Divisor is.
The greatest common divisor of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers.
To find G.C.D of x(x2 – 9) and x2 – x – 12, let us write down factors of each term.
Factorization of x(x2 – 9) = x(x – 3)(x + 3)
[∵, by algebraic identity, a2 – b2 = (a – b)(a + b)]
For factorization of x2 – x – 12,
x2 – x – 12 = x2 – 4x + 3x – 12 [∵, Sum of -4 and 3 is -1 and multiplication is -12]
⇒ x2 – x – 12 = x(x – 4) + 3(x – 4) [∵, common from the first two terms is x and last two terms is 3]
⇒ x2 – x – 12 = (x – 4)(x + 3) [∵, common from the two terms is (x – 4)]
So, factorization of x2 – x – 12 = (x – 4)(x + 3)
Find the factors that these two lists share in common.
Factors of x(x2 – 9) = x, (x – 3), (x + 3)
Factors of x2 – x – 12 = (x – 4), (x + 3)
Common factor that is found in these two terms is (x + 3).
Thus, the gcd of x(x2 – 9) and x2 – x – 12 is (x + 3).
Let's find L.C.M. of 4a2b4c, 12a3bc5 and 18a2b3c2
Let us understand what L.C.M, Least Common Multiple is.
Least common multiple of two or more integers, is the smallest positive integer that is divisible by these two or more integers.
In order to find L.C.M, we need to find factors of each terms.
Factorization of 4a2b4c = 2 × 2 × a × a × b × b × b × b × c
Or, Factorization of 4a2b4c = 22 × a2 × b4 × c
Factorization of 12a3bc5 = 2 × 2 × 3 × a × a × a × b × c × c × c × c × c
Or, Factorization of 12a3bc5 = 22 × 3 × a3 × b × c5
Factorization of 18a2b3c2 = 2 × 3 × 3 × a × a × b × b × b × c × c
Or, Factorization of 18a2b3c2 = 2 × 32 × a2 × b3 × c2
Now, we need to find the factor of highest power.
Factors of 4a2b4c = 22, a2, b4, c
Factors of 12a3bc5 = 22, 3, a3, b, c5
Factors of 18a2b3c2 = 2, 32, a2, b3, c2
The factors are 22, 32, a3, b4 and c5.
By multiplying these factors, we get
22 × 32 × a3 × b4 × c5 = 4 × 9 a3b4c5
⇒ 22 × 32 × a3 × b4 × c5 = 36 a3b4c5
Thus, lcm of 4a2b4c, 12a3bc5 and 18a2b3c2 is 36 a3b4c5.
Let's find the L.C.M. of 2(x – 4) and (x2 – 3x + 2)
Let us understand what L.C.M, Least Common Multiple is.
Least common multiple of two or more integers, is the smallest positive integer that is divisible by these two or more integers.
In order to find L.C.M, we need to find factors of each terms.
Let us factorize 2(x – 4).
2(x – 4) = 2 × (x – 4)
Let us factorize (x2 – 3x + 2).
x2 – 3x + 2 = x2 – (2x + x) + 2
[∵, Sum of -2 and -1 is -3, and its multiplication is 2]
⇒ x2 – 3x + 2 = x2 – 2x – x + 2
⇒ x2 – 3x + 2 = x(x – 2) – (x – 2)
[∵, common from first two terms is x and last two terms is -1]
⇒ x2 – 3x + 2 = (x – 2)(x – 1)
[∵, common from the two terms is (x – 2)]
Now, we need to find the factor of highest power.
Factors of 2(x – 4) = 2, (x – 4)
Factors of x2 – 3x + 2 = (x – 2), (x – 1)
The factors are 2, (x – 4), (x – 2) and (x – 1).
Multiplying these factors, we get
2 × (x – 4) × (x – 2) × (x – 1) = 2(x – 4)(x – 2)(x – 1)
Thus, the lcm of 2(x – 4) and (x2 – 3x + 2) is 2(x – 4)(x – 2)(x – 1).
Let's find G.C.D. and L.C.M. of (y3 – 8), (y3 – 4y2 + 4y) and (y2 + y – 6)
Let us understand what a G.C.D, Greatest Common Divisor and L.C.M, Least Common Multiple is.
The greatest common divisor of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers.
Least common multiple of two or more integers, is the smallest positive integer that is divisible by these two or more integers.
To find G.C.D of (y3 – 8), (y3 – 4y2 + 4y) and (y2 + y – 6), let us write down factors of each term.
For factorization of y3 – 8,
Factorization of y3 – 8 = Factorization of y3 – 23
[∵, y3 – 8 = y3 – 23, as 23 = 2 × 2 × 2 = 8]
⇒ y3 – 8 = (y – 2)(y2 + 22 + (2)(y))
⇒ y3 – 8 = (y – 2)(y2 + 4 + 2y)
⇒ y3 – 8 = (y – 2)(y2 + 2y + 4)
[∵, by algebraic identity, a3 – b3 = (a – b)(a2 + b2 + ab)]
For factorization of y3 – 4y2 + 4y,
⇒ y3 – 4y2 + 4y = y(y2 – 4y + 4)
⇒ y3 – 4y2 + 4y = y(y2 – (2y + 2y) + 4) [∵, Sum of -2 and -2 is -4 and multiplication is 4]
⇒ y3 – 4y2 + 4y = y(y2 – 2y – 2y + 4)
⇒ y3 – 4y2 + 4y = y(y(y – 2) – 2(y – 2)) [∵, common from the first two terms is y and last two terms is -2]
⇒ y3 – 4y2 + 4y = y((y – 2)(y – 2)) [∵, common from the two terms is (y – 2)]
⇒ y3 – 4y2 + 4y = y(y – 2)(y – 2)
⇒ y3 – 4y2 + 4y = y(y – 2)2
For factorization of y2 + y – 6,
⇒ y2 + y – 6 = y2 + 3y – 2y – 6 [∵, Sum of 3 and -2 is 1 and multiplication is -6]
⇒ y2 + y – 6 = y(y + 3) – 2(y + 3) [∵, common from the first two terms is y and last two terms is -2]
⇒ y2 + y – 6 = (y + 3)(y – 2) [∵, common from the two terms is (y + 3)]
Find the factors that these two lists share in common.
Factors of y3 – 8 = (y – 2), (y2 + 2y + 4)
Factors of y3 – 4y2 + 4y = y, (y – 2), (y – 2)
Factors of y2 + y – 6 = (y + 3), (y – 2)
Common factor that is found in these three terms is (y – 2).
So, gcd is (y – 2).
To find L.C.D of (y3 – 8), (y3 – 4y2 + 4y) and (y2 + y – 6), let us write down factors of each term.
Then, we need to find the factor of highest power.
Factors of y3 – 8 = (y – 2), (y2 + 2y + 4)
Factors of y3 – 4y2 + 4y = y, (y – 2)2
Factors of y2 + y – 6 = (y + 3), (y – 2)
The factors are y, (y – 2)2, (y + 3) and (y2 + 2y + 4).
Multiplying these factors, we get
y × (y – 2)2 × (y + 3) × (y2 + 2y + 4) = y(y – 2)2(y + 3)(y2 + 2y + 4)
So, lcm is y(y – 2)2(y + 3)(y2 + 2y + 4).