HCF of polynomials `x^(3)+3x^(2)y +2xy^(2) and x^(4) +6x^(3)y +8x^(2)y^(2)` is ____.

Options

A. A) `x(x+2y)`
B. B) `x(x+3y)`
C. C) `x+2y`
D. D) None of these

Answer

To find the HCF (Highest Common Factor) of the given polynomials \( P_1 = x^3 + 3x^2y + 2xy^2 \) and \( P_2 = x^4 + 6x^3y + 8x^2y^2 \), we will factor both polynomials step by step.

### Step 1: Factor the first polynomial \( P_1 \)

The first polynomial is:
\[ P_1 = x^3 + 3x^2y + 2xy^2 \]

We can factor out \( x \) from the first term:
\[ P_1 = x(x^2 + 3xy + 2y^2) \]

### Step 2: Factor the quadratic \( x^2 + 3xy + 2y^2 \)

Next, we need to factor the quadratic expression \( x^2 + 3xy + 2y^2 \). We look for two numbers that multiply to \( 2y^2 \) (the constant term) and add to \( 3y \) (the coefficient of the middle term). The numbers \( 1y \) and \( 2y \) work:
\[ x^2 + 3xy + 2y^2 = (x + 2y)(x + y) \]

Thus, we can rewrite \( P_1 \) as:
\[ P_1 = x(x + 2y)(x + y) \]

### Step 3: Factor the second polynomial \( P_2 \)

Now, let's factor the second polynomial:
\[ P_2 = x^4 + 6x^3y + 8x^2y^2 \]

We can factor out \( x^2 \):
\[ P_2 = x^2(x^2 + 6xy + 8y^2) \]

### Step 4: Factor the quadratic \( x^2 + 6xy + 8y^2 \)

Next, we factor the quadratic \( x^2 + 6xy + 8y^2 \). We need two numbers that multiply to \( 8y^2 \) and add to \( 6y \). The numbers \( 2y \) and \( 4y \) work:
\[ x^2 + 6xy + 8y^2 = (x + 2y)(x + 4y) \]

Thus, we can rewrite \( P_2 \) as:
\[ P_2 = x^2(x + 2y)(x + 4y) \]

### Step 5: Identify the HCF

Now we have:
\[ P_1 = x(x + 2y)(x + y) \]
\[ P_2 = x^2(x + 2y)(x + 4y) \]

The common factors in both polynomials are \( x \) and \( (x + 2y) \).

Thus, the HCF of the two polynomials is:
\[ \text{HCF} = x(x + 2y) \]

### Final Answer:
The HCF of the polynomials \( x^3 + 3x^2y + 2xy^2 \) and \( x^4 + 6x^3y + 8x^2y^2 \) is \( x(x + 2y) \).

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