Find HCF of `x^(3) + 3x^(2) y + 2xy^(2) and x^(4) + 6x^(3)y + 8x^(2) y^(2)`

Answer

To find the Highest Common Factor (HCF) of the two polynomials \( P(x) = x^3 + 3x^2y + 2xy^2 \) and \( Q(x) = x^4 + 6x^3y + 8x^2y^2 \), we will follow these steps:

### Step 1: Factor the first polynomial \( P(x) \)

1. Start with the polynomial:
\[ P(x) = x^3 + 3x^2y + 2xy^2 \]

2. Group the terms:
\[ P(x) = (x^3 + 3x^2y) + (2xy^2) \]

3. Factor out the common factors from the first group:
\[ = x^2(x + 3y) + 2y^2(x) \]

4. Now, we can factor out \( x \) from the entire expression:
\[ = x^2(x + 3y) + 2y^2(x) = x^2(x + 2y) + 2y(x)(x + 2y) \]

5. Combine the common factors:
\[ = (x + 2y)(x^2 + 2xy) \]

### Step 2: Factor the second polynomial \( Q(x) \)

1. Start with the polynomial:
\[ Q(x) = x^4 + 6x^3y + 8x^2y^2 \]

2. Group the terms:
\[ Q(x) = (x^4 + 6x^3y) + (8x^2y^2) \]

3. Factor out the common factors from the first group:
\[ = x^3(x + 6y) + 8y^2(x^2) \]

4. Now, we can factor out \( x^2 \) from the entire expression:
\[ = x^3(x + 2y) + 4y(x^2)(x + 2y) \]

5. Combine the common factors:
\[ = (x + 2y)(x^3 + 4y(x^2)) \]

### Step 3: Finding the HCF

1. From the factored forms:
- \( P(x) = (x + 2y)(x^2 + 2xy) \)
- \( Q(x) = (x + 2y)(x^3 + 4xy) \)

2. The common factor is \( (x + 2y) \).

3. Now, we need to find the common factors of the remaining parts:
- From \( P(x) \): \( x^2 + 2xy \)
- From \( Q(x) \): \( x^3 + 4xy \)

4. The common factor from the remaining parts is \( x \) (the lowest power of \( x \) in both).

5. Therefore, the HCF of the two polynomials is:
\[ HCF = (x + 2y)(x) \]

### Final Answer
The HCF of \( P(x) \) and \( Q(x) \) is:
\[ HCF = x(x + 2y) \]