The HCF of the polynomials `10(a-1)(a-2)^(3),120(a-3)(a-2)^(3)`, and `135(a+3)(a-2)^(3)` is ___________.

Options

A. `25(a-3)(a-2)`
B. `5(a-3)(a+2)`
C. `5(a-2)^(3)`
D. `5(a-3)(a-2)(a+3)`

Answer

To find the HCF (Highest Common Factor) of the polynomials \(10(a-1)(a-2)^{3}\), \(120(a-3)(a-2)^{3}\), and \(135(a+3)(a-2)^{3}\), we will follow these steps:

### Step 1: Factor the coefficients
First, we need to factor the coefficients of each polynomial.

1. **For \(10\)**:
\[ 10 = 2 \times 5 \]

2. **For \(120\)**:
\[ 120 = 12 \times 10 = 3 \times 4 \times 10 = 3 \times 2^3 \times 5 = 2^3 \times 3 \times 5 \]

3. **For \(135\)**:
\[ 135 = 27 \times 5 = 3^3 \times 5 \]

### Step 2: Identify the common factors
Now, we will identify the common factors from the coefficients:

- The common prime factors among \(10\), \(120\), and \(135\) are:
- \(5\) (present in all three)

### Step 3: Factor the polynomial parts
Next, we will consider the polynomial parts:

1. **For \(10(a-1)(a-2)^{3}\)**:
- The polynomial part is \((a-1)(a-2)^{3}\)

2. **For \(120(a-3)(a-2)^{3}\)**:
- The polynomial part is \((a-3)(a-2)^{3}\)

3. **For \(135(a+3)(a-2)^{3}\)**:
- The polynomial part is \((a+3)(a-2)^{3}\)

### Step 4: Identify the common polynomial factors
Now, we will check for common polynomial factors:

- The common polynomial factor is \((a-2)^{3}\) (present in all three).

### Step 5: Combine the common factors
Now, we combine the common numerical factor and the common polynomial factor:

\[ \text{HCF} = 5 \times (a-2)^{3} \]

### Final Answer
Thus, the HCF of the given polynomials is:
\[ \boxed{5(a-2)^{3}} \]

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