If unity is double repeated root of `px^3+ g(x^2 + x) +r=0`, then

Answer

To solve the problem step-by-step, we start by understanding the given polynomial and the condition of having a double repeated root.

### Step 1: Define the polynomial
Let \( f(x) = px^3 + gx^2 + x + r \). We know that \( x = 1 \) is a double repeated root of this polynomial.

### Step 2: Express the polynomial using its roots
Since \( x = 1 \) is a double root, we can express the polynomial as:
\[ f(x) = p(x - 1)^2(x - a) \]
where \( a \) is the third root.

### Step 3: Expand the polynomial
Expanding \( f(x) \):
\[ f(x) = p((x - 1)^2)(x - a) = p(x^2 - 2x + 1)(x - a) \]
Now, expand this:
\[ = p[(x^2 - 2x + 1)(x - a)] = p[x^3 - ax^2 - 2x^2 + 2ax + x - a] \]
Combining like terms gives:
\[ = p[x^3 + (-a - 2)x^2 + (2a + 1)x - a] \]

### Step 4: Compare coefficients
Now, we compare this with the original polynomial \( f(x) = px^3 + gx^2 + x + r \):
- Coefficient of \( x^3 \): \( p \)
- Coefficient of \( x^2 \): \( g = -p(a + 2) \)
- Coefficient of \( x \): \( 1 = p(2a + 1) \)
- Constant term: \( r = -pa \)

### Step 5: Solve for \( a \)
From the equation \( 1 = p(2a + 1) \), we can express \( p \):
\[ p = \frac{1}{2a + 1} \]

### Step 6: Substitute \( p \) into the equation for \( g \)
Substituting \( p \) into \( g \):
\[ g = -\frac{1}{2a + 1}(a + 2) \]

### Step 7: Find \( a \) using the relationship between \( g \) and \( r \)
From the constant term \( r = -pa \):
\[ r = -\frac{1}{2a + 1}a \]

### Step 8: Set up the equations
Now we have:
1. \( g = -\frac{a + 2}{2a + 1} \)
2. \( r = -\frac{a}{2a + 1} \)

### Step 9: Find conditions for \( p \), \( g \), and \( r \)
We need to find the conditions on \( p \), \( g \), and \( r \) based on the values we have derived.

### Step 10: Calculate \( p \cdot g \) and \( p \cdot r \)
1. \( p \cdot g = \frac{1}{2a + 1} \cdot -\frac{a + 2}{2a + 1} = -\frac{a + 2}{(2a + 1)^2} \)
2. \( p \cdot r = \frac{1}{2a + 1} \cdot -\frac{a}{2a + 1} = -\frac{a}{(2a + 1)^2} \)

### Step 11: Analyze the signs
- If \( a < -2 \), then \( p \cdot g < 0 \).
- If \( a = -2 \), then \( p \cdot g = 0 \).
- If \( a > -2 \), then \( p \cdot g > 0 \).

### Conclusion
From the analysis, we can conclude:
- The correct option is that \( p \cdot g < 0 \) when \( a \) is in the appropriate range.