Find the number that must be subtracted from the polynomial `3y^3+ y^2 -22y + 15,` so that the resulting polynomial is completely divisible by y +3.

Answer

To solve the problem of finding the number that must be subtracted from the polynomial \(3y^3 + y^2 - 22y + 15\) so that the resulting polynomial is completely divisible by \(y + 3\), we can follow these steps:

### Step-by-Step Solution:

1. **Define the Polynomial**:
Let \(f(y) = 3y^3 + y^2 - 22y + 15\).

2. **Identify the Value of \(y\)**:
Since we want \(f(y)\) to be divisible by \(y + 3\), we set \(y + 3 = 0\) which gives us \(y = -3\).

3. **Substitute \(y = -3\) into the Polynomial**:
We need to find \(f(-3)\):
\[ f(-3) = 3(-3)^3 + (-3)^2 - 22(-3) + 15 \]

4. **Calculate Each Term**:
- Calculate \(3(-3)^3\):
\[ 3 \times (-27) = -81 \]
- Calculate \((-3)^2\):
\[ 9 \]
- Calculate \(-22(-3)\):
\[ 66 \]
- The constant term is \(15\).

5. **Combine All Terms**:
Now combine all the calculated values:
\[ f(-3) = -81 + 9 + 66 + 15 \]

6. **Simplify**:
- First, combine \(-81 + 9\):
\[ -72 \]
- Then combine \(-72 + 66\):
\[ -6 \]
- Finally, combine \(-6 + 15\):
\[ 9 \]

7. **Set the Result Equal to Zero**:
Since we want the polynomial to be divisible by \(y + 3\), we need:
\[ f(-3) - k = 0 \]
Thus, we have:
\[ 9 - k = 0 \]
This implies:
\[ k = 9 \]

### Conclusion:
The number that must be subtracted from the polynomial \(3y^3 + y^2 - 22y + 15\) so that the resulting polynomial is completely divisible by \(y + 3\) is \(9\).