What number should be added to `27 x ^(3) - 54 x ^(2) + 36 x - 11` so that resulting polynomial becomes divisible by `3x - 2` ?

Answer

To solve the problem of finding the number that should be added to the polynomial \(27x^3 - 54x^2 + 36x - 11\) so that it becomes divisible by \(3x - 2\), we can follow these steps:

### Step 1: Identify the polynomial and the divisor
Let the polynomial be:
\[ f(x) = 27x^3 - 54x^2 + 36x - 11 \]
We want to find a number \(c\) such that \(f(x) + c\) is divisible by \(3x - 2\).

### Step 2: Find the root of the divisor
The divisor \(3x - 2\) can be set to zero to find its root:
\[ 3x - 2 = 0 \implies x = \frac{2}{3} \]

### Step 3: Substitute the root into the polynomial
We need to evaluate \(f\left(\frac{2}{3}\right) + c = 0\). First, calculate \(f\left(\frac{2}{3}\right)\):
\[ f\left(\frac{2}{3}\right) = 27\left(\frac{2}{3}\right)^3 - 54\left(\frac{2}{3}\right)^2 + 36\left(\frac{2}{3}\right) - 11 \]

### Step 4: Calculate each term
Calculate each term separately:
1. \(27\left(\frac{2}{3}\right)^3 = 27 \cdot \frac{8}{27} = 8\)
2. \(-54\left(\frac{2}{3}\right)^2 = -54 \cdot \frac{4}{9} = -24\)
3. \(36\left(\frac{2}{3}\right) = 36 \cdot \frac{2}{3} = 24\)
4. The constant term is \(-11\).

Now, substitute these values back into the polynomial:
\[ f\left(\frac{2}{3}\right) = 8 - 24 + 24 - 11 \]

### Step 5: Simplify the expression
Combine the terms:
\[ f\left(\frac{2}{3}\right) = 8 - 24 + 24 - 11 = 8 - 11 = -3 \]

### Step 6: Set up the equation
Now we have:
\[ f\left(\frac{2}{3}\right) + c = 0 \implies -3 + c = 0 \]

### Step 7: Solve for \(c\)
Solving for \(c\):
\[ c = 3 \]

### Final Answer
The number that should be added is:
\[ \boxed{3} \]