Factorisation of the polynomial `sqrt(3)x^(2)+11x+6sqrt(3)`

Options

A. `(sqrt(3)x+2)(x-3sqrt(3))`
B. `(sqrt(3)x+2)(x+3sqrt(3))`
C. `(sqrt(2)x+3)(x+2sqrt(3))`
D. `(sqrt(2)x-2)(x+3sqrt(2))`

Answer

To factor the polynomial \( \sqrt{3}x^2 + 11x + 6\sqrt{3} \), we will use the middle term splitting method. Here’s a step-by-step solution:

### Step 1: Identify the coefficients
The given polynomial is \( \sqrt{3}x^2 + 11x + 6\sqrt{3} \).
- The coefficient of \( x^2 \) (a) is \( \sqrt{3} \).
- The coefficient of \( x \) (b) is \( 11 \).
- The constant term (c) is \( 6\sqrt{3} \).

### Step 2: Multiply the first and last coefficients
We need to multiply the coefficient of \( x^2 \) and the constant term:
\[ a \cdot c = \sqrt{3} \cdot 6\sqrt{3} = 6 \cdot 3 = 18 \]

### Step 3: Find two numbers that multiply to \( 18 \) and add up to \( 11 \)
We need to find two numbers that multiply to \( 18 \) (from step 2) and add up to \( 11 \) (the coefficient of \( x \)):
- The numbers \( 9 \) and \( 2 \) satisfy this condition because:
\[ 9 \cdot 2 = 18 \quad \text{and} \quad 9 + 2 = 11 \]

### Step 4: Rewrite the middle term
Now, we can rewrite the polynomial by splitting the middle term using the two numbers we found:
\[ \sqrt{3}x^2 + 9x + 2x + 6\sqrt{3} \]

### Step 5: Group the terms
Next, we group the terms:
\[ (\sqrt{3}x^2 + 9x) + (2x + 6\sqrt{3}) \]

### Step 6: Factor out the common terms in each group
Now, we factor out the common factors from each group:
1. From the first group \( \sqrt{3}x^2 + 9x \), we can factor out \( 3x \):
\[ 3x(\frac{\sqrt{3}}{3}x + 3) = 3x(\sqrt{3}x + 3) \]
2. From the second group \( 2x + 6\sqrt{3} \), we can factor out \( 2 \):
\[ 2(x + 3\sqrt{3}) \]

### Step 7: Combine the factored groups
Now, we can combine the factored groups:
\[ 3x(\sqrt{3}x + 3) + 2(x + 3\sqrt{3}) \]

### Step 8: Factor out the common binomial
Notice that both terms contain the common binomial \( x + 3\sqrt{3} \):
\[ (x + 3\sqrt{3})(\sqrt{3}x + 2) \]

### Final Answer
Thus, the factorization of the polynomial \( \sqrt{3}x^2 + 11x + 6\sqrt{3} \) is:
\[ (\sqrt{3}x + 2)(x + 3\sqrt{3}) \]