Answer :

To find the least common multiple (LCM) of the polynomials [tex]\( f(x) = 18x^4 - 36x^3 + 18x^2 \)[/tex] and [tex]\( g(x) = 45x^6 - 45x^3 \)[/tex], we follow these steps:

### Step 1: Factor each polynomial
#### For [tex]\( f(x) \)[/tex]:
[tex]\[ f(x) = 18x^4 - 36x^3 + 18x^2 \][/tex]

We start by factoring out the greatest common divisor (GCD) of the coefficients and the common power of [tex]\( x \)[/tex]:

[tex]\[ f(x) = 18x^2(x^2 - 2x + 1) \][/tex]

Next, we recognize that [tex]\( x^2 - 2x + 1 \)[/tex] is a perfect square trinomial:

[tex]\[ x^2 - 2x + 1 = (x - 1)^2 \][/tex]

Thus,

[tex]\[ f(x) = 18x^2(x - 1)^2 \][/tex]

#### For [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = 45x^6 - 45x^3 \][/tex]

We start by factoring out the greatest common divisor of the coefficients and the common power of [tex]\( x \)[/tex]:

[tex]\[ g(x) = 45x^3(x^3 - 1) \][/tex]

Next, we recognize that [tex]\( x^3 - 1 \)[/tex] is a difference of cubes, which can be factored using the formula [tex]\( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \)[/tex]:

[tex]\[ x^3 - 1 = (x - 1)(x^2 + x + 1) \][/tex]

Thus,

[tex]\[ g(x) = 45x^3(x - 1)(x^2 + x + 1) \][/tex]

### Step 2: Find the LCM of the factored terms
The LCM of two polynomials is found by taking the highest power of each distinct factor appearing in the factorizations of the polynomials. We compare the factored forms:

- For [tex]\( 18x^2(x - 1)^2 \)[/tex]:
- Coefficient: [tex]\( 18 \)[/tex]
- [tex]\( x \)[/tex]: [tex]\( x^2 \)[/tex]
- [tex]\( (x - 1) \)[/tex]: [tex]\( (x - 1)^2 \)[/tex]

- For [tex]\( 45x^3(x - 1)(x^2 + x + 1) \)[/tex]:
- Coefficient: [tex]\( 45 \)[/tex]
- [tex]\( x \)[/tex]: [tex]\( x^3 \)[/tex]
- [tex]\( (x - 1) \)[/tex]: [tex]\( (x - 1) \)[/tex]
- [tex]\( (x^2 + x + 1) \)[/tex]: [tex]\( (x^2 + x + 1) \)[/tex]

We take the highest powers of each factor:
- Coefficient: LCM of 18 and 45 is 90.
- [tex]\( x \)[/tex]: [tex]\( x^3 \)[/tex] (highest power)
- [tex]\( (x - 1) \)[/tex]: [tex]\( (x - 1)^2 \)[/tex] (highest power from [tex]\( f(x) \)[/tex])
- [tex]\( (x^2 + x + 1) \)[/tex]: appears in [tex]\( g(x) \)[/tex]

Thus, the LCM expression is:

[tex]\[ \text{LCM}(f(x), g(x)) = 90x^3(x - 1)^2(x^2 + x + 1) \][/tex]

Therefore, the LCM of the polynomials [tex]\( f(x) = 18x^4 - 36x^3 + 18x^2 \)[/tex] and [tex]\( g(x) = 45x^6 - 45x^3 \)[/tex] is:

[tex]\[ 90x^3(x - 1)^2(x^2 + x + 1) \][/tex]

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