The LCM and HCF of two polyonomials P (x) and Q (x) are `2 (x ^(4) - 1) and (x +1) (x ^(2) +1),` respectively. If
`P (x) = x ^(3) + x ^(2) + x +1,` then find Q (x).

Options

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

Answer

To find the polynomial \( Q(x) \) given the LCM and HCF of two polynomials \( P(x) \) and \( Q(x) \), we can use the relationship:

\[ P(x) \cdot Q(x) = \text{LCM}(P(x), Q(x)) \cdot \text{HCF}(P(x), Q(x)) \]

### Step 1: Write down the given values
- \( P(x) = x^3 + x^2 + x + 1 \)
- \( \text{LCM}(P(x), Q(x)) = 2(x^4 - 1) \)
- \( \text{HCF}(P(x), Q(x)) = (x + 1)(x^2 + 1) \)

### Step 2: Factor \( P(x) \)
We can factor \( P(x) \):
\[ P(x) = x^3 + x^2 + x + 1 = (x + 1)(x^2 + 1) \]

### Step 3: Substitute into the formula
Using the relationship:
\[ P(x) \cdot Q(x) = \text{LCM}(P(x), Q(x)) \cdot \text{HCF}(P(x), Q(x)) \]
Substituting the known values:
\[ (x + 1)(x^2 + 1) \cdot Q(x) = 2(x^4 - 1) \cdot (x + 1)(x^2 + 1) \]

### Step 4: Simplify the equation
We can cancel \( (x + 1)(x^2 + 1) \) from both sides (assuming \( x \neq -1 \)):
\[ Q(x) = 2(x^4 - 1) \]

### Step 5: Factor \( x^4 - 1 \)
We can factor \( x^4 - 1 \) using the difference of squares:
\[ x^4 - 1 = (x^2 - 1)(x^2 + 1) = (x - 1)(x + 1)(x^2 + 1) \]

### Step 6: Substitute back into \( Q(x) \)
Thus,
\[ Q(x) = 2((x - 1)(x + 1)(x^2 + 1)) = 2(x^3 - 1) \]

### Final Answer
So, the polynomial \( Q(x) \) is:
\[ Q(x) = 2(x^4 - 1) \]