To find the least common multiple of algebraic expressions, we have to follow the steps given below.
Step 1 :
Break down the coefficients and write down it as product of prime factors.
Step 2 :
Write down the factors in exponential form.
Step 3 :
In common term, choose the highest exponents and if we find some thing extra, include that also.
Step 4 :
The product of these will be the least common multiple.
Finding the LCM of polynomials
Problem 1 :
4xy2, 2x2y3
Solution:
4xy2 = 22 × x × y × y
2x2y3 = 2 × x × x × y × y × y
LCM = 22 ⋅ x2 ⋅ y3
= 4x2y3
Problem 2 :
-9a3b, 12a2bc
Solution:
-9a3b = -32 × a × a × a × b
12a2bc = 22 × 3 × a × a × b × c
LCM = 32 ⋅ 22 ⋅ a3 ⋅ b ⋅ c
= 36a3bc
Problem 3 :
5xy, 15x2z, 10y2
Solution:
5xy = 5 × x × y
15x2z = 5 × 3 × x × x × z
10y2 = 5 × 2 × y × y
LCM = 5 ⋅ 3 ⋅ 2 ⋅ x2 ⋅ y2 ⋅ z
= 30x2y2z
Problem 4 :
16m, -12m2n, 8n2
Solution:
16m = 22 × 22 × m
-12m2n = -22 × 3 × m × m × n
8n2 = 22 × 2 × n × n
LCM = 22 × 22 × 3 × m2 × n2
= 48m2n2
Problem 5 :
x, x - 2
Solution:
We don't see anything in common. So, the least common multiple will be the product of the given terms.
LCM = x(x - 2)
Problem 6 :
y2, y + 3
Solution:
We don't see anything in common. So, the least common multiple will be the product of the given terms.
LCM = y2(y + 3)
Problem 7 :
x - 1, x + 4
Solution:
We don't see anything in common. So, the least common multiple will be the product of the given terms.
LCM = (x - 1) (x + 4)
Problem 8 :
z + 8, z + 2
Solution:
We don't see anything in common. So, the least common multiple will be the product of the given terms.
LCM = (z + 8) (z + 2)
Problem 9 :
x(x - 1), x2, (x - 1)2
Solution:
By comparing x(x - 1) and (x - 1)2, the highest term is (x - 1)2.
The extra term is x2.
So, the least common multiple is x2(x - 1)2
Problem 10 :
(y - 2)(y + 2), (y + 2)2
Solution:
By comparing (y + 2) and (y + 2)2, the highest term is (y + 2)2.
The extra term is (y - 2).
So, the least common multiple is (y - 2)(y + 2)2.
Problem 11 :
x2 - 1, x2 + 2x + 1
Solution:
x2 - 1 = (x + 1) (x - 1)
x2 + 2x + 1 = (x + 1)2
So, the least common multiple is (x - 1)(x + 1)2.
Problem 12 :
n2 - 3n + 2, n2 - 4
Solution:
n2 - 3n + 2 = (n - 1)(n - 2)
n2 - 4 = (n + 2)(n - 2)
So, the least common multiple is (n - 2)(n - 1)(n + 2).
Problem 13 :
t, t2 - 1, t2 + 5t - 6
Solution:
t2 - 1 = (t + 1)(t - 1)
t2 + 5t - 6 = (t - 1)(t + 6)
So, the least common multiple is (t - 1)(t + 1)(t + 6).
Problem 14 :
w2 - 9, 9w2, w2 - 6w + 9
Solution:
w2 - 9 = (w + 3)(w - 3)
w2 - 6w + 9 = (w - 3)2
The extra term is 9w2.
So, the least common multiple is 9w2 (w - 3)2(w + 3).
Problem 15 :
8x - 4, 6x2 + x - 2
Solution:
8x - 4 = 4(2x - 1) ---> (1)
6x2 + x - 2 = 6x2 - 3x + 4x - 2
= 3x(2x - 1) + 2 (2x - 1)
= (3x + 2)(2x - 1) ---> (2)
By comparing (1) and (2), we get
4(2x - 1), (3x + 2)(2x - 1)
So, the least common multiple is 4(2x - 1)(3x + 2).
Problem 16 :
x3 - y3, x2 - xy + y2, x2 - 2xy + y2
Solution:
x3 - y3 = (x - y)(x2 + xy + y2)
x2 - 2xy + y2 = (x - y)2
The extra term is x2 - xy + y2.
So, the least common multiple is (x - y)2(x2 - xy + y2)
May 21, 24 08:51 PM
May 21, 24 08:51 AM
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