The HCF of two polynomials is x+ 3 and their LCM is `x^(3) - 7x + 6`. If one of the polynomials is `x^(2) + 2x - 3`. Then, the other polynomial is

Options

A. `x^(2) - x + 6`
B. `x^(2) + x - 6`
C. `x^(2) + x + 2`
D. `x^(2) + x + 3`

Answer

To find the other polynomial given the HCF and LCM of two polynomials, we can use the relationship between HCF, LCM, and the polynomials themselves. The relationship states that:

\[ \text{HCF}(P, Q) \times \text{LCM}(P, Q) = P \times Q \]

Where \( P \) and \( Q \) are the two polynomials.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- HCF = \( x + 3 \)
- LCM = \( x^3 - 7x + 6 \)
- One polynomial \( P = x^2 + 2x - 3 \)

2. **Apply the Relationship:**
Using the relationship mentioned above, we can express the other polynomial \( Q \) as follows:
\[ Q = \frac{\text{HCF} \times \text{LCM}}{P} \]

3. **Calculate the Product of HCF and LCM:**
Substitute the values of HCF and LCM into the equation:
\[ \text{HCF} \times \text{LCM} = (x + 3)(x^3 - 7x + 6) \]

4. **Expand the Product:**
We need to expand \( (x + 3)(x^3 - 7x + 6) \):
\[ = x(x^3 - 7x + 6) + 3(x^3 - 7x + 6) \]
\[ = x^4 - 7x^2 + 6x + 3x^3 - 21x + 18 \]
\[ = x^4 + 3x^3 - 7x^2 - 15x + 18 \]

5. **Substitute \( P \) into the Equation:**
Now we need to divide the expanded product by \( P \):
\[ Q = \frac{x^4 + 3x^3 - 7x^2 - 15x + 18}{x^2 + 2x - 3} \]

6. **Perform Polynomial Long Division:**
We will perform polynomial long division of \( x^4 + 3x^3 - 7x^2 - 15x + 18 \) by \( x^2 + 2x - 3 \).

- Divide the leading term: \( x^4 \div x^2 = x^2 \)
- Multiply \( x^2 \) by \( x^2 + 2x - 3 \) and subtract:
\[ x^4 + 2x^3 - 3x^2 \]
- Subtract:
\[ (3x^3 - 2x^3) + (-7x^2 + 3x^2) - 15x + 18 = x^3 - 4x^2 - 15x + 18 \]

- Repeat the process:
- Divide \( x^3 \div x^2 = x \)
- Multiply and subtract:
\[ x^3 + 2x^2 - 3x \]
- Resulting in:
\[ (-4x^2 - 2x + 18) \]

- Finally:
- Divide \( -4x^2 \div x^2 = -4 \)
- Multiply and subtract:
\[ -4x^2 - 8x + 12 \]
- Resulting in:
\[ (-15x + 8x + 18 - 12) = -7x + 6 \]

- The final result of the division is:
\[ Q = x^2 + x - 4 + \frac{-7x + 6}{x^2 + 2x - 3} \]

7. **Final Result:**
The other polynomial \( Q \) is:
\[ Q = x^2 - 4 \]