The HCF of the polynomials `(x^(3)-9^(3))(x+3) and (x^(2)-9)(x^(2)-3)` is ____________.

Options

A. `x+sqrt(3)`
B. `x-9`
C. `x+3`
D. `x-3`

Answer

To find the HCF (Highest Common Factor) of the polynomials \((x^3 - 9^3)(x + 3)\) and \((x^2 - 9)(x^2 - 3)\), we will follow these steps:

### Step 1: Factor the first polynomial
The first polynomial is \((x^3 - 9^3)(x + 3)\). We can factor \(x^3 - 9^3\) using the difference of cubes formula:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
Here, \(a = x\) and \(b = 9\).
So,
\[ x^3 - 9^3 = (x - 9)(x^2 + 9x + 81) \]
Thus, the first polynomial becomes:
\[ (x - 9)(x^2 + 9x + 81)(x + 3) \]

### Step 2: Factor the second polynomial
The second polynomial is \((x^2 - 9)(x^2 - 3)\). We can factor \(x^2 - 9\) using the difference of squares formula:
\[ a^2 - b^2 = (a - b)(a + b) \]
Here, \(a = x\) and \(b = 3\).
So,
\[ x^2 - 9 = (x - 3)(x + 3) \]
The second polynomial becomes:
\[ (x - 3)(x + 3)(x^2 - 3) \]

### Step 3: Write down the complete factorizations
Now we have:
1. First polynomial: \((x - 9)(x^2 + 9x + 81)(x + 3)\)
2. Second polynomial: \((x - 3)(x + 3)(x^2 - 3)\)

### Step 4: Identify the common factors
Now, we will identify the common factors in both factorizations:
- From the first polynomial, we have \(x + 3\).
- From the second polynomial, we also have \(x + 3\).

### Step 5: Determine the HCF
The only common factor between the two polynomials is \(x + 3\). Therefore, the HCF of the given polynomials is:
\[ \text{HCF} = x + 3 \]

### Final Answer
The HCF of the polynomials \((x^3 - 9^3)(x + 3)\) and \((x^2 - 9)(x^2 - 3)\) is \(\boxed{x + 3}\).

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