HCF of the polynomials `20x^2 y(x^2 - y^2)` and `35xy^2 (x - y)` is

Options

A. `5x^2 y^2 (x- y)`
B. `5xy (x - y)`
C. `5x^2 y^2 (x + y)`
D. `5xy (x^2 - y^2)`

Answer

To find the HCF (Highest Common Factor) of the polynomials \(20x^2y(x^2 - y^2)\) and \(35xy^2(x - y)\), we will follow these steps:

### Step 1: Factor each polynomial completely.

1. **For the first polynomial \(20x^2y(x^2 - y^2)\)**:
- Factor \(20\): \(20 = 2 \times 2 \times 5\)
- Factor \(x^2\): \(x^2 = x \times x\)
- Factor \(y\): \(y = y\)
- Factor \(x^2 - y^2\) using the difference of squares: \(x^2 - y^2 = (x - y)(x + y)\)

So, we can write:
\[ 20x^2y(x^2 - y^2) = 2 \times 2 \times 5 \times x \times x \times y \times (x - y)(x + y) \]

This gives us:
\[ = 2^2 \times 5 \times x^2 \times y \times (x - y)(x + y) \]

2. **For the second polynomial \(35xy^2(x - y)\)**:
- Factor \(35\): \(35 = 5 \times 7\)
- Factor \(x\): \(x = x\)
- Factor \(y^2\): \(y^2 = y \times y\)
- Factor \(x - y\): \(x - y = (x - y)\)

So, we can write:
\[ 35xy^2(x - y) = 5 \times 7 \times x \times y \times y \times (x - y) \]

This gives us:
\[ = 5 \times 7 \times x \times y^2 \times (x - y) \]

### Step 2: Identify the common factors.

Now we have:
- From \(20x^2y(x^2 - y^2)\): \(2^2 \times 5 \times x^2 \times y \times (x - y)(x + y)\)
- From \(35xy^2(x - y)\): \(5 \times 7 \times x \times y^2 \times (x - y)\)

**Common factors**:
- The common numerical factor is \(5\).
- The common variable factor is \(x\) (minimum power is \(x^1\)).
- The common variable factor is \(y\) (minimum power is \(y^1\)).
- The common polynomial factor is \((x - y)\).

### Step 3: Multiply the common factors to find the HCF.

Thus, the HCF is:
\[ HCF = 5 \times x^1 \times y^1 \times (x - y) = 5xy(x - y) \]

### Final Answer:
The HCF of the polynomials \(20x^2y(x^2 - y^2)\) and \(35xy^2(x - y)\) is:
\[ \boxed{5xy(x - y)} \]