Find the HCF and LCM of `(3+13x-30x^(2))` and `(25x^(2)-30x+9)`.

Answer

To find the HCF (Highest Common Factor) and LCM (Least Common Multiple) of the polynomials \( P_1 = 3 + 13x - 30x^2 \) and \( P_2 = 25x^2 - 30x + 9 \), we will follow these steps:

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

The first polynomial is:
\[ P_1 = 3 + 13x - 30x^2 \]

Rearranging it gives:
\[ P_1 = -30x^2 + 13x + 3 \]

To factor this, we can look for two numbers that multiply to \( -30 \times 3 = -90 \) and add to \( 13 \). The numbers \( 18 \) and \( -5 \) work since \( 18 \times (-5) = -90 \) and \( 18 + (-5) = 13 \).

Now, we can rewrite the polynomial:
\[ P_1 = -30x^2 + 18x - 5x + 3 \]

Grouping the terms:
\[ P_1 = (-30x^2 + 18x) + (-5x + 3) \]

Factoring out the common terms:
\[ P_1 = -6x(5x - 3) - 1(5x - 3) \]

This gives us:
\[ P_1 = (5x - 3)(-6x + 1) \]

### Step 2: Factor the second polynomial \( P_2 \)

The second polynomial is:
\[ P_2 = 25x^2 - 30x + 9 \]

This can be recognized as a perfect square:
\[ P_2 = (5x - 3)^2 \]

### Step 3: Find the HCF of \( P_1 \) and \( P_2 \)

The factored forms are:
\[ P_1 = (5x - 3)(-6x + 1) \]
\[ P_2 = (5x - 3)^2 \]

The common factor is \( (5x - 3) \). The highest common factor (HCF) is:
\[ \text{HCF} = 5x - 3 \]

### Step 4: Find the LCM of \( P_1 \) and \( P_2 \)

The least common multiple (LCM) can be found using the formula:
\[ \text{LCM}(P_1, P_2) = \frac{P_1 \cdot P_2}{\text{HCF}(P_1, P_2)} \]

Calculating this gives:
\[ \text{LCM} = \frac{(5x - 3)(-6x + 1) \cdot (5x - 3)^2}{(5x - 3)} \]

This simplifies to:
\[ \text{LCM} = (5x - 3)(-6x + 1)(5x - 3) = (5x - 3)^2(-6x + 1) \]

### Final Results

Thus, the HCF and LCM of the polynomials are:
- **HCF**: \( 5x - 3 \)
- **LCM**: \( (5x - 3)^2(-6x + 1) \)