To find the least common multiple (LCM) of the two polynomials [tex]\(27x^3 - 54x^2 + 36x - 8\)[/tex] and [tex]\(9x^2 + 9x - 10\)[/tex], follow these steps:
### Step 1: Identify the given polynomials
We have two polynomials:
[tex]\[ P(x) = 27x^3 - 54x^2 + 36x - 8 \][/tex]
[tex]\[ Q(x) = 9x^2 + 9x - 10 \][/tex]
### Step 2: Find the LCM of the polynomials
To determine the LCM of two polynomials, we need to determine the polynomial that is the smallest degree polynomial that both original polynomials will divide without leaving a remainder.
### Step 3: Determine the LCM result
The LCM of the polynomials [tex]\(P(x)\)[/tex] and [tex]\(Q(x)\)[/tex] is calculated to be:
[tex]\[ \text{LCM}(P(x), Q(x)) = 81x^4 - 27x^3 - 162x^2 + 156x - 40 \][/tex]
### Step 4: Verify the coefficients and degree
The resulting LCM polynomial is:
[tex]\[ 81x^4 - 27x^3 - 162x^2 + 156x - 40 \][/tex]
This polynomial has been determined by aligning the factors and coefficients correctly and ensuring it is divisible by both of the original polynomials.
### Conclusion
The least common multiple of the polynomials [tex]\(27x^3 - 54x^2 + 36x - 8\)[/tex] and [tex]\(9x^2 + 9x - 10\)[/tex] is:
[tex]\[ \boxed{81x^4 - 27x^3 - 162x^2 + 156x - 40} \][/tex]
