If `P (x) = (x ^(3) - 8) (x +1) and Q (x) = (x ^(3) +1) (x - 2),` then find the LCM of `P (x) and Q (x).`

Options

A. `(x - 2) (x +1) `
B. `(x ^(2) + 2x + 4) ( x ^(2) + 4x +1)`
C. `(x +1) ^(2) (x -2) ^(2) (x ^(2) + 2x + 4) (x ^(2) + 4x +1)`
D. `(x +1) (x -2) (x ^(2) +2x + 4) (x ^(2) -x +1)`

Answer

To find the LCM of the polynomials \( P(x) \) and \( Q(x) \), we will first factor both polynomials using the relevant algebraic identities.

### Step 1: Factor \( P(x) \)

Given:
\[ P(x) = (x^3 - 8)(x + 1) \]
We can factor \( x^3 - 8 \) using the difference of cubes formula:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
Here, \( a = x \) and \( b = 2 \):
\[ x^3 - 8 = (x - 2)(x^2 + 2x + 4) \]
Thus, we can rewrite \( P(x) \):
\[ P(x) = (x - 2)(x^2 + 2x + 4)(x + 1) \]

### Step 2: Factor \( Q(x) \)

Given:
\[ Q(x) = (x^3 + 1)(x - 2) \]
We can factor \( x^3 + 1 \) using the sum of cubes formula:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
Here, \( a = x \) and \( b = 1 \):
\[ x^3 + 1 = (x + 1)(x^2 - x + 1) \]
Thus, we can rewrite \( Q(x) \):
\[ Q(x) = (x + 1)(x^2 - x + 1)(x - 2) \]

### Step 3: Find the LCM of \( P(x) \) and \( Q(x) \)

To find the LCM, we take the highest power of each factor from both polynomials:
- From \( P(x) \): \( (x - 2), (x^2 + 2x + 4), (x + 1) \)
- From \( Q(x) \): \( (x + 1), (x^2 - x + 1), (x - 2) \)

The LCM will be:
\[ \text{LCM} = (x - 2)(x + 1)(x^2 + 2x + 4)(x^2 - x + 1) \]

### Final LCM Expression
Thus, the LCM of \( P(x) \) and \( Q(x) \) is:
\[ \text{LCM}(P(x), Q(x)) = (x - 2)(x + 1)(x^2 + 2x + 4)(x^2 - x + 1) \]