Consider the equality 15 = 5 * 3. Here, 15 is the multiple of both 5 and 3.
In the case of polynomials, this holds true. For example, if f(x)=p(x) q(x), then f(x) is called a multiple of both p(x) and of q(x).
Consider the multiples of 8 =16, 24, 32, 40, 48, 56, … and of 12 =24, 36, 48, 60, 72, … Of these, 24 and 48 are both common multiples of 8 and 12.
However, 24 is the least common multiple of 2.
Alternately, we can find the least common multiple or LCM by prime factorization as follows:
8=23
12=22*3
We now take all of the prime factor occurring in the factorization of 8 and 12 and raise each to the highest or greatest exponent occurring in the factorization.
This gives us 23 and 31. Their product 23*3=8*3=24 gives us the required LCM.
Now consider the following example.
Example 1
Consider the polynomials
(x-2) (x^{2}-3x+2) and x^{2}-5x+6
Let f(x)=(x-2) (x^{2}-3x+2)
g(x)=x^{2}-5x+6
Factorizing
f(x)=(x-2) [x^{2}-2x-x+2]
=(x-2) [x(x-2)-1(x-2)]
=(x-2) (x-1) (x-2)
f(x)= (x-1) (x-2x^{2}
g(x)=x^{2}-5x+6
=x^{2}-3x-2x+6
=x(x-3)-2(x-3)
=(x-2)(x-3)
Any multiple of f(x) should have (x-1) and (x-2x^{2} as factors.
Any multiple of g(x) should have (x-2) and (x-3) as factors.
Of these, (x-2) is ignored because if (x-2x^{2} is a factor then so is (x-2).
∴the LCM=(x-2x^{2}(x-1)(x-3)
To find the LCM of two or more polynomials, we follow these steps.
Step 1: Express each polynomial as a product of powers of irreducible factors.
Step 2: List all of the irreducible factors occurring only once in the given polynomials. For each of these factors, find the greatest exponent in the factorized form of the given polynomials.
Step 3: Raise each irreducible factor to the greatest exponent in Step 2 and multiply to get the LCM.
Consider this example.
Example 2
Find the LCM of the polynomials
f(x)=4(x-1x^{2}(x^{2}+6x+5)
g(x)=10(x-1)(x+2)(x^{2}+7x+10)
Factorizing
Step 1:
f(x)=22(x-1x^{2}[x^{2}+5x+x+5]
=22(x-1x^{2}[x(x+5)+1(x+5)]
f(x)=22(x-1x^{2}(x+1)(x+5)
g(x)=2*5*(x-1)(x+2)[(x^{2}+5x+2x+10]
=2*5*(x-1)(x+2)[(x(x+5)+2(x+5)]
=2*5*(x-1)(x+2)(x+2)(x+5)
=2*5*(x-1)(x+2x^{2}(x+5)
Step 2:
Irreducible Factor |
Greatest Exponent |
2 |
2 |
5 |
1 |
(x-1) |
2 |
(x+1) |
1 |
(x+2) |
2 |
(x+5) |
1 |
Step 3:
LCM=22*5*(x-1x^{2}*(x+1)(x+2x^{2}*(x+5)
=20(x-1x^{2} (x+1)(x+2x^{2} (x+5)
Example 3
Find the LCM of the polynomials 11x^{3} (x+1)^{3} and
121x(2x^{2}+3x+1).
Let p(x)=11x^{3} (x+1) 3
q(x)=121x(2x^{2}+3x+1)
Step 1: Factorizing q(x) we get
q(x)=112x[2x^{2}+2x+x+1]
=112x[2x(x+1)+1(x+1)]
q(x)=112x[2x+1][x+1]
p(x)=11x^{3} (x+1)^{3}
Step 2: Irreducible factors 11, x, x+1, 2x+1
The highest exponents are 2, 3, 3, 1 respectively.
Step 3: LCM=112*x^{3}*(x+1)^{3}*2x+1
=121x^{3}(x+1)^{3}(2x+1)
Try these questions
Find out the least common multiple of the following polynomials
- (2x+3) (3x+7x^{2} and (2x+3x^{2} (4x+5)
- 2(x^{2}+x) and 4x^{2}(x+1)
- 2(x+1)4(x–1x^{2} (x+2) and 8(x+1x^{2} (x–1)^{3} (x+2x^{2}
- (x+1) (x^{2}+x+1) and (x–1)
- x(8x^{3}+27) and 2x^{2}(2x^{2}+9x+9)
- 6(x+2x^{2} (x^{2}–x+1) and 15(x+1) (x+2)^{3}
Answers
- To find the LCM of
p(x)=(2x+3)(x+7x^{2}
q(x)= (2x+3x^{2} (4x+5)
Irreducible factors are 2x+3, x+7, 4x+5
Highest exponent of factors 2, 2, 1 respectively.
LCM=(2x+3x^{2} (x+7x^{2} (4x+5)
- To find the LCM of
p(x)=2(x^{2}+x)
q(x)=4x^{2}(x+1)
Factorizing
p(x)=2x(x+1)
q(x)= 22x^{2}(x+1)
Irreducible factors are 2, x, x+1
Highest exponent of factors are 2, 2, 1 respectively.
LCM = 22 * x^{2} * (x+1)
=4x^{2}(x+1)
- To find the LCM of
p(x)=2(x+1)4 (x–1x^{2} (x+2)
q(x)= 8(x+1x^{2}(x–1)^{3} (x+2x^{2}
Irreducible factors are 2, x+1, x–1, x+2
Highest exponent of factors are 3, 4, 3, 2 respectively.
LCM=23 (x+1)4 (x–1)^{3} (x+2x^{2}
=8(x+1)4(x–1)^{3}(x+2x^{2}
- To find the LCM of
p(x)=(x+1) (x^{2}+x+1)
q(x)=(x+1)
Irreducible factors are x+1, x^{2}+x+1, (x–1)
Highest exponent of factors are 1, 1, 1 respectively.
LCM=(x+1) (x^{2}+x+1) (x–1)
=(x+1) (x^{3}–1)
Since (x-1) (x^{2}+x+1)=x^{3}+1
So the LCM = (x+1)(x^{3}-1)
- To find the LCM of
p(x)=(8x^{3}+27)
q(x)=2x^{2}(2x^{2}+9x+9)
Factorizing
p(x)=x[(2x)^{3}+(3)^{3}]
=x(2x+3)(4x^{2}-6x+9)
q(x)=2x^{2}[2x^{2}+6x+3x+9]
=2x^{2}[2x(x+3)+3(x+3)]
q(x)=2x^{2}(2x+3) (x+3)
Irreducible factors are 2, x, 2x+3, x+3, 4x^{2}–6x+9
Highest exponent of factors are 1, 2, 1, 1, 1 respectively.
LCM=2x^{2}(2x+3) (4x^{2}-6x+9) (x+3)
- To find the LCM of
p(x)=6(x+2x^{2} (x^{2}–x+1)
q(x)=15(x+1) (x+2)^{3}
p(x)=2 * 3(x+2x^{2} (x^{2}–x+1)
q(x)=5 * 3(x+1) (x+2)^{3}
Irreducible factors are 2, 3, 5, x+1, x+2, x^{2}–x+1
Highest exponent of factors are 1,1, 1, 1, 3, 1
LCM=2 * 3 * 5 * (x+1) * (x+2)^{3} * (x^{2}–x+1)
=30(x+1) (x+2)^{3 }(x^{2}-x+1)