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

Options

A. `(x-1)`
B. `(x^(2) -x +1)`
C. `(x-1) (x^(2) -x+1)`
D. `(x^(3) +1) (x^(2) -3x +2)`

Answer

To find the highest common factor (HCF) of the polynomials \( P(x) \) and \( Q(x) \), we will first factor both polynomials completely.

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

Given:
\[ P(x) = (x^3 + 1)(x - 1) \]

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 \( P(x) \):
\[ P(x) = (x + 1)(x^2 - x + 1)(x - 1) \]

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

Given:
\[ Q(x) = (x^2 - x + 1)(x^2 - 3x + 2) \]

Now we will factor \( x^2 - 3x + 2 \):
\[ x^2 - 3x + 2 = (x - 1)(x - 2) \]

So we can rewrite \( Q(x) \):
\[ Q(x) = (x^2 - x + 1)(x - 1)(x - 2) \]

### Step 3: Identify Common Factors

Now we have the factorizations:
\[ P(x) = (x + 1)(x^2 - x + 1)(x - 1) \]
\[ Q(x) = (x^2 - x + 1)(x - 1)(x - 2) \]

The common factors in \( P(x) \) and \( Q(x) \) are:
1. \( x^2 - x + 1 \)
2. \( x - 1 \)

### Step 4: Find the HCF

The highest common factor (HCF) of \( P(x) \) and \( Q(x) \) is:
\[ \text{HCF} = (x^2 - x + 1)(x - 1) \]

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

---