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Find the L.C.M. of the following polynomial expressions.

a) [tex]a x^2 + a x, a^2 x^2 + a^2 x[/tex]
b) [tex]2 x^2 + 4 x, x^3 + 2 x^2[/tex]
c) [tex]3 a^2 b + 6 a b^2, 2 a^3 + 4 a^2 b[/tex]
d) [tex]a^2 x + a b x, a b x^2 + b^2 x^2[/tex]
e) [tex]3 x^2 + 6 x, 2 x^3 + 4 x^2[/tex]
f) [tex]2 a + 4, a^2 - 4[/tex]
g) [tex]3 a^2 + 3 a, 6 a^2 - 6[/tex]
h) [tex]x^2 y - 5 x y, x^2 - 25[/tex]
i) [tex]a^4 x^2 - a^2 x^4, 7 a^2 x - 7 a x^2[/tex]
j) [tex]x^2 - x y, x^3 y - x y^3[/tex]
k) [tex]4 x^2 - 2 x, 8 x^3 - 2 x[/tex]
l) [tex]x^2 - 4, x^2 + 5 x + 6[/tex]
m) [tex]x^2 + x - 6, x^2 - 9[/tex]
n) [tex]2 x^3 - 50 x, 2 x^2 + 7 x - 15[/tex]
o) [tex]4 a^3 - 9 a, 2 a^2 + 3 a - 9[/tex]
p) [tex]x^2 + 8 x + 15, x^2 + 7 x + 12[/tex]
q) [tex]a^2 - 9 a + 20, a^2 - 2 a - 15[/tex]
r) [tex]a^2 + 5 a - 14, a^2 - 8 a + 12[/tex]
s) [tex]2 a^2 + 5 a + 2, 2 a^2 - 3 a - 2[/tex]
t) [tex]3 x^2 + 8 x - 16, 3 x^2 - 16 x + 16[/tex]
u) [tex]4 x^2 - x - 3, 3 x^2 - 2 x - 1[/tex]
v) [tex]2 x^2 + 3 x - 9, 4 x^2 - 12 x + 9[/tex]



Answer :

GinnyAnswer
### 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.

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