Answer :
### Problem Statement
We need to find the Least Common Multiple (LCM) of several sets of polynomial expressions. Here, we'll address sets \( (d) \) and \( (e) \) in detail.
### d) \(a^2 x + a b x, a b x^2 + b^2 x^2\)
To find the LCM of the polynomials \(a^2 x + a b x\) and \(a b x^2 + b^2 x^2\):
1. Factorize each polynomial:
- \( a^2 x + a b x = a x (a + b) \)
- \( a b x^2 + b^2 x^2 = b x^2 (a + b) \)
2. Identify common and unique factors:
- Common factor: \( (a + b) \)
- Unique factors: \(a x\) and \(b x^2\)
3. Form the LCM:
- LCM should include the highest power of each distinct factor:
- Multiply the highest powers: \(a\), \(b\), \(x^2\), and \((a + b)\)
Therefore, the LCM is:
[tex]\[ a^2 b x^2 + a b^2 x^2 \][/tex]
### e) \(3 x^2 + 6 x, 2 x^3 + 4 x^2\)
To find the LCM of the polynomials \(3 x^2 + 6 x\) and \(2 x^3 + 4 x^2\):
1. Factorize each polynomial:
- \( 3 x^2 + 6 x = 3 x (x + 2) \)
- \( 2 x^3 + 4 x^2 = 2 x^2 (x + 2) \)
2. Identify common and unique factors:
- Common factor: \(x + 2\)
- Unique factors: \(3 x\) and \(2 x^2\)
3. Form the LCM:
- LCM should include the highest power of each distinct factor:
- Multiply the highest powers: \(3\), \(2\), \(x^3\), and \((x + 2)\)
Therefore, the LCM is:
[tex]\[ 6 x^4 + 18 x^3 + 12 x^2 \][/tex]
### Summary
The LCM of the given polynomial sets are:
- For set (d): \(a^2 b x^2 + a b^2 x^2\)
- For set (e): \(6 x^4 + 18 x^3 + 12 x^2\)
These results give us the least common multiples of the polynomials in sets [tex]\(d\)[/tex] and [tex]\(e\)[/tex] as required.
We need to find the Least Common Multiple (LCM) of several sets of polynomial expressions. Here, we'll address sets \( (d) \) and \( (e) \) in detail.
### d) \(a^2 x + a b x, a b x^2 + b^2 x^2\)
To find the LCM of the polynomials \(a^2 x + a b x\) and \(a b x^2 + b^2 x^2\):
1. Factorize each polynomial:
- \( a^2 x + a b x = a x (a + b) \)
- \( a b x^2 + b^2 x^2 = b x^2 (a + b) \)
2. Identify common and unique factors:
- Common factor: \( (a + b) \)
- Unique factors: \(a x\) and \(b x^2\)
3. Form the LCM:
- LCM should include the highest power of each distinct factor:
- Multiply the highest powers: \(a\), \(b\), \(x^2\), and \((a + b)\)
Therefore, the LCM is:
[tex]\[ a^2 b x^2 + a b^2 x^2 \][/tex]
### e) \(3 x^2 + 6 x, 2 x^3 + 4 x^2\)
To find the LCM of the polynomials \(3 x^2 + 6 x\) and \(2 x^3 + 4 x^2\):
1. Factorize each polynomial:
- \( 3 x^2 + 6 x = 3 x (x + 2) \)
- \( 2 x^3 + 4 x^2 = 2 x^2 (x + 2) \)
2. Identify common and unique factors:
- Common factor: \(x + 2\)
- Unique factors: \(3 x\) and \(2 x^2\)
3. Form the LCM:
- LCM should include the highest power of each distinct factor:
- Multiply the highest powers: \(3\), \(2\), \(x^3\), and \((x + 2)\)
Therefore, the LCM is:
[tex]\[ 6 x^4 + 18 x^3 + 12 x^2 \][/tex]
### Summary
The LCM of the given polynomial sets are:
- For set (d): \(a^2 b x^2 + a b^2 x^2\)
- For set (e): \(6 x^4 + 18 x^3 + 12 x^2\)
These results give us the least common multiples of the polynomials in sets [tex]\(d\)[/tex] and [tex]\(e\)[/tex] as required.
