What number should be subtracted from `2x^3 - 5x^2 + 5x + 8` so that the resulting polynomial has a factor 2x - 3?

Answer

To solve the problem of finding what number should be subtracted from the polynomial \(2x^3 - 5x^2 + 5x + 8\) so that the resulting polynomial has a factor of \(2x - 3\), we can use polynomial long division. Here’s the step-by-step solution:

### Step 1: Set up the division
We need to divide the polynomial \(2x^3 - 5x^2 + 5x + 8\) by the factor \(2x - 3\).

### Step 2: Divide the leading terms
Divide the leading term of the polynomial \(2x^3\) by the leading term of the divisor \(2x\):
\[ \frac{2x^3}{2x} = x^2 \]
This means we will multiply \(2x - 3\) by \(x^2\).

### Step 3: Multiply and subtract
Now, multiply \(2x - 3\) by \(x^2\):
\[ (2x - 3) \cdot x^2 = 2x^3 - 3x^2 \]
Now, subtract this from the original polynomial:
\[ (2x^3 - 5x^2 + 5x + 8) - (2x^3 - 3x^2) = -5x^2 + 3x^2 + 5x + 8 = -2x^2 + 5x + 8 \]

### Step 4: Repeat the process
Now, we need to divide the new polynomial \(-2x^2 + 5x + 8\) by \(2x - 3\). Divide the leading term \(-2x^2\) by \(2x\):
\[ \frac{-2x^2}{2x} = -x \]
Multiply \(2x - 3\) by \(-x\):
\[ (2x - 3)(-x) = -2x^2 + 3x \]
Now, subtract this from \(-2x^2 + 5x + 8\):
\[ (-2x^2 + 5x + 8) - (-2x^2 + 3x) = 5x - 3x + 8 = 2x + 8 \]

### Step 5: Repeat again
Now, divide \(2x + 8\) by \(2x - 3\). Divide the leading term \(2x\) by \(2x\):
\[ \frac{2x}{2x} = 1 \]
Multiply \(2x - 3\) by \(1\):
\[ (2x - 3)(1) = 2x - 3 \]
Now, subtract this from \(2x + 8\):
\[ (2x + 8) - (2x - 3) = 8 + 3 = 11 \]

### Step 6: Conclusion
The remainder we obtained is \(11\). For \(2x - 3\) to be a factor of the polynomial \(2x^3 - 5x^2 + 5x + 8\), we need to subtract this remainder from the polynomial. Therefore, the number that should be subtracted is:
\[ \boxed{11} \]