Find the HCF of the polynomials `6(x^(2)-36) and 36(x+6)`.

Options

A. `6(x+6)`
B. `6(x-6)`
C. `(x+6)`
D. `(x-6)`

Answer

To find the HCF (Highest Common Factor) of the polynomials \(6(x^2 - 36)\) and \(36(x + 6)\), we will follow these steps:

### Step 1: Factor the first polynomial
The first polynomial is \(6(x^2 - 36)\). We can factor \(x^2 - 36\) using the difference of squares formula, which states that \(a^2 - b^2 = (a + b)(a - b)\).

\[ x^2 - 36 = (x + 6)(x - 6) \]

Thus, we can rewrite the first polynomial as:

\[ 6(x^2 - 36) = 6((x + 6)(x - 6)) = 6(x + 6)(x - 6) \]

### Step 2: Factor the second polynomial
The second polynomial is \(36(x + 6)\). We can factor \(36\) as \(6 \times 6\):

\[ 36(x + 6) = 6 \cdot 6(x + 6) \]

### Step 3: Identify the common factors
Now we have the factored forms of both polynomials:

1. \(6(x + 6)(x - 6)\)
2. \(6 \cdot 6(x + 6)\)

Next, we identify the common factors in both expressions.

- The common factor in both polynomials is \(6\).
- The term \((x + 6)\) is also common in both.

### Step 4: Write the HCF
The HCF is the product of the common factors:

\[ \text{HCF} = 6(x + 6) \]

### Final Answer
Thus, the HCF of the polynomials \(6(x^2 - 36)\) and \(36(x + 6)\) is:

\[ \text{HCF} = 6(x + 6) \]

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