The LCM of the polynomials `(x-3)(x+5)^(2),(x+5)(x-7)^(2)and(x-7)(x-3)^(2)` is ___________.

Options

A. `(x-3)(x-7)(x-5)`
B. 1
C. `(x-3)^(2)(x-7)^(2)(x+5)^(2)`
D. `(x-3)^(2)(x-7)^(2)`

Answer

To find the LCM of the polynomials \((x-3)(x+5)^{2}\), \((x+5)(x-7)^{2}\), and \((x-7)(x-3)^{2}\), we will follow these steps:

### Step 1: Identify the Polynomials
We have the following three polynomials:
1. \( P_1 = (x-3)(x+5)^{2} \)
2. \( P_2 = (x+5)(x-7)^{2} \)
3. \( P_3 = (x-7)(x-3)^{2} \)

### Step 2: Factor Each Polynomial
Let's rewrite each polynomial in its factored form:
- \( P_1 = (x-3)^{1} (x+5)^{2} \)
- \( P_2 = (x+5)^{1} (x-7)^{2} \)
- \( P_3 = (x-3)^{2} (x-7)^{1} \)

### Step 3: Determine the Highest Power of Each Factor
Next, we will find the highest power of each factor present in the polynomials:
- For \( (x-3) \):
- From \( P_1 \): \( 1 \)
- From \( P_2 \): \( 0 \) (not present)
- From \( P_3 \): \( 2 \)
- **Maximum power**: \( 2 \)

- For \( (x+5) \):
- From \( P_1 \): \( 2 \)
- From \( P_2 \): \( 1 \)
- From \( P_3 \): \( 0 \) (not present)
- **Maximum power**: \( 2 \)

- For \( (x-7) \):
- From \( P_1 \): \( 0 \) (not present)
- From \( P_2 \): \( 2 \)
- From \( P_3 \): \( 1 \)
- **Maximum power**: \( 2 \)

### Step 4: Write the LCM
Now, we can write the LCM using the maximum powers of each factor:
\[ \text{LCM} = (x-3)^{2} (x+5)^{2} (x-7)^{2} \]

### Final Answer
Thus, the LCM of the given polynomials is:
\[ \text{LCM} = (x-3)^{2} (x+5)^{2} (x-7)^{2} \]

---