The LCM of two polynomials p(x) and q(x) is `x^(3) - 7x + 6`. If `p(x) = (x^(2) + 2x -3)` and `q(x) = (x^(2) + x - 6)`, then the HCF is

Options

A. x + 3
B. x+1
C. `(x+2)(x+1)`
D. None of these

Answer

To find the HCF of the two polynomials \( p(x) \) and \( q(x) \) given their LCM, we can use the relationship between HCF and LCM:

\[ \text{HCF}(p(x), q(x)) \times \text{LCM}(p(x), q(x)) = p(x) \times q(x) \]

### Step 1: Identify the given polynomials and LCM
We have:
- \( p(x) = x^2 + 2x - 3 \)
- \( q(x) = x^2 + x - 6 \)
- \( \text{LCM}(p(x), q(x)) = x^3 - 7x + 6 \)

### Step 2: Calculate the product \( p(x) \times q(x) \)
Now we will multiply \( p(x) \) and \( q(x) \):

\[ p(x) \times q(x) = (x^2 + 2x - 3)(x^2 + x - 6) \]

Using the distributive property (FOIL method):

1. \( x^2 \cdot x^2 = x^4 \)
2. \( x^2 \cdot x = x^3 \)
3. \( x^2 \cdot -6 = -6x^2 \)
4. \( 2x \cdot x^2 = 2x^3 \)
5. \( 2x \cdot x = 2x^2 \)
6. \( 2x \cdot -6 = -12x \)
7. \( -3 \cdot x^2 = -3x^2 \)
8. \( -3 \cdot x = -3x \)
9. \( -3 \cdot -6 = 18 \)

Now, combining all these results:

\[ p(x) \times q(x) = x^4 + (x^3 + 2x^3) + (-6x^2 + 2x^2 - 3x^2) + (-12x - 3x) + 18 \]

This simplifies to:

\[ = x^4 + 3x^3 - 7x^2 - 15x + 18 \]

### Step 3: Use the relationship to find HCF
Now we can use the relationship:

\[ \text{HCF}(p(x), q(x)) = \frac{p(x) \times q(x)}{\text{LCM}(p(x), q(x))} \]

Substituting the values we have:

\[ \text{HCF}(p(x), q(x)) = \frac{x^4 + 3x^3 - 7x^2 - 15x + 18}{x^3 - 7x + 6} \]

### Step 4: Perform polynomial long division
Now we will divide \( x^4 + 3x^3 - 7x^2 - 15x + 18 \) by \( x^3 - 7x + 6 \).

1. Divide the leading term: \( \frac{x^4}{x^3} = x \)
2. Multiply \( x \) by \( x^3 - 7x + 6 \):
- \( x^4 - 7x^2 + 6x \)
3. Subtract this from \( x^4 + 3x^3 - 7x^2 - 15x + 18 \):
- \( (x^4 - x^4) + (3x^3) + (7x^2 - 7x^2) + (-15x - 6x) + 18 \)
- This gives \( 3x^3 - 21x + 18 \)

4. Now divide \( 3x^3 \) by \( x^3 \):
- \( 3 \)
5. Multiply \( 3 \) by \( x^3 - 7x + 6 \):
- \( 3x^3 - 21x + 18 \)
6. Subtract:
- \( (3x^3 - 3x^3) + (0) + (0) = 0 \)

Since the remainder is 0, we conclude that:

\[ \text{HCF}(p(x), q(x)) = x + 3 \]

### Final Answer
The HCF of the polynomials \( p(x) \) and \( q(x) \) is:

\[ \text{HCF}(p(x), q(x)) = x + 3 \]