The HCF of the polynomials `9(x +a)^(p) (x -b)^(q) (x + c)^(r) and 12(x + a)^(p +3) (x -b)^(q-3) (x +c)^(r +2) " is " 3(x +a)^(6) (x -b)^(6) (x + c)^(6)`, then the value of `p +q -r` is

Options

A. 21
B. 9
C. 15
D. 6

Explanation

To solve the problem step by step, we need to analyze the given polynomials and their highest common factor (HCF).

### Given:
1. First polynomial: \( 9(x + a)^{p} (x - b)^{q} (x + c)^{r} \)
2. Second polynomial: \( 12(x + a)^{p + 3} (x - b)^{q - 3} (x + c)^{r + 2} \)
3. HCF: \( 3(x + a)^{6} (x - b)^{6} (x + c)^{6} \)

### Step 1: Factor out the coefficients
- The coefficients of the first polynomial are 9 and the second polynomial is 12.
- The HCF of the coefficients \(9\) and \(12\) is \(3\).

### Step 2: Analyze the powers of \(x + a\)
- For \(x + a\):
- From the first polynomial: \(p\)
- From the second polynomial: \(p + 3\)
- HCF power: \(6\)

To find the HCF, we take the minimum power:
\[ \text{HCF power} = \min(p, p + 3) = 6 \]
This implies:
\[ p = 6 \]

### Step 3: Analyze the powers of \(x - b\)
- For \(x - b\):
- From the first polynomial: \(q\)
- From the second polynomial: \(q - 3\)
- HCF power: \(6\)

To find the HCF, we take the minimum power:
\[ \text{HCF power} = \min(q, q - 3) = 6 \]
This implies:
\[ q - 3 = 6 \implies q = 9 \]

### Step 4: Analyze the powers of \(x + c\)
- For \(x + c\):
- From the first polynomial: \(r\)
- From the second polynomial: \(r + 2\)
- HCF power: \(6\)

To find the HCF, we take the minimum power:
\[ \text{HCF power} = \min(r, r + 2) = 6 \]
This implies:
\[ r = 6 \]

### Step 5: Calculate \(p + q - r\)
Now that we have the values:
- \(p = 6\)
- \(q = 9\)
- \(r = 6\)

We can substitute these values into the expression \(p + q - r\):
\[ p + q - r = 6 + 9 - 6 = 9 \]

### Final Answer:
The value of \(p + q - r\) is \(9\).

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