If `x=1a n dx=2` are solutions of equations `x^3+a x^2+b x+c=0a n da+b=1,` then find the value of `bdot`

Explanation

To solve the problem step by step, we start with the given information:

1. The cubic equation is \( x^3 + ax^2 + bx + c = 0 \).
2. The roots of the equation are \( x = 1 \) and \( x = 2 \).
3. We also have the condition \( a + b = 1 \).

### Step 1: Substitute \( x = 1 \) into the equation
Since \( x = 1 \) is a root, we substitute \( x = 1 \) into the cubic equation:

\[ 1^3 + a(1^2) + b(1) + c = 0 \]

This simplifies to:

\[ 1 + a + b + c = 0 \]

### Step 2: Substitute \( x = 2 \) into the equation
Next, we substitute \( x = 2 \) into the cubic equation:

\[ 2^3 + a(2^2) + b(2) + c = 0 \]

This simplifies to:

\[ 8 + 4a + 2b + c = 0 \]

### Step 3: Set up the equations
Now we have two equations from our substitutions:

1. \( 1 + a + b + c = 0 \) (Equation 1)
2. \( 8 + 4a + 2b + c = 0 \) (Equation 2)

We also have the condition:

\[ a + b = 1 \quad (Equation 3) \]

### Step 4: Solve for \( c \) using Equation 1
From Equation 1, we can express \( c \):

\[ c = -1 - a - b \]

Using Equation 3, \( a + b = 1 \):

\[ c = -1 - 1 = -2 \]

### Step 5: Substitute \( c \) back into Equation 2
Now, substitute \( c = -2 \) into Equation 2:

\[ 8 + 4a + 2b - 2 = 0 \]

This simplifies to:

\[ 6 + 4a + 2b = 0 \]

### Step 6: Substitute \( b = 1 - a \) into the equation
Using Equation 3 again, substitute \( b = 1 - a \):

\[ 6 + 4a + 2(1 - a) = 0 \]

This simplifies to:

\[ 6 + 4a + 2 - 2a = 0 \]

Combining like terms gives:

\[ 8 + 2a = 0 \]

### Step 7: Solve for \( a \)
Now, solve for \( a \):

\[ 2a = -8 \implies a = -4 \]

### Step 8: Find \( b \)
Now that we have \( a \), we can find \( b \) using Equation 3:

\[ b = 1 - a = 1 - (-4) = 1 + 4 = 5 \]

### Final Answer
Thus, the value of \( b \) is:

\[ \boxed{5} \]