Find the HCF of `f(x)=33 (2x + 3)^(2) (3x-4)^(3) (4x-5)^(4)` and `g(x)= 22 (x+1) (2x + 3) (4x-5)^(2) (4x^(2)-9)`

Answer

To find the HCF (Highest Common Factor) of the functions \( f(x) \) and \( g(x) \), we will follow these steps:

### Step 1: Factor the coefficients
The coefficients of \( f(x) \) and \( g(x) \) are 33 and 22, respectively. We can factor these coefficients:
- \( 33 = 3 \times 11 \)
- \( 22 = 2 \times 11 \)

### Step 2: Identify common factors in coefficients
The common factor in the coefficients is \( 11 \).

### Step 3: Factor the polynomial parts
Now we will factor the polynomial parts of both functions:

1. **For \( f(x) \)**:
\[ f(x) = 33 (2x + 3)^{2} (3x - 4)^{3} (4x - 5)^{4} \]
- The factors are \( (2x + 3)^{2} \), \( (3x - 4)^{3} \), and \( (4x - 5)^{4} \).

2. **For \( g(x) \)**:
\[ g(x) = 22 (x + 1) (2x + 3) (4x - 5)^{2} (4x^{2} - 9) \]
- We can factor \( 4x^{2} - 9 \) as \( (2x - 3)(2x + 3) \) using the difference of squares:
\[ g(x) = 22 (x + 1) (2x + 3) (4x - 5)^{2} (2x - 3)(2x + 3) \]
- The factors are \( (x + 1) \), \( (2x + 3) \), \( (4x - 5)^{2} \), \( (2x - 3) \).

### Step 4: Identify common polynomial factors
Now we will identify the common polynomial factors between \( f(x) \) and \( g(x) \):
- **Common factors**:
- \( (2x + 3) \): The minimum power is \( 2 \) (from \( f(x) \)).
- \( (4x - 5) \): The minimum power is \( 2 \) (from \( g(x) \)).

- **Non-common factors**:
- \( (3x - 4) \) from \( f(x) \) is not in \( g(x) \).
- \( (x + 1) \) and \( (2x - 3) \) from \( g(x) \) are not in \( f(x) \).

### Step 5: Combine common factors
Now we can combine the common factors:
\[ \text{HCF} = 11 \cdot (2x + 3)^{2} \cdot (4x - 5)^{2} \]

### Final Answer
Thus, the HCF of \( f(x) \) and \( g(x) \) is:
\[ \text{HCF} = 11 (2x + 3)^{2} (4x - 5)^{2} \]

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