Write the quotient and remainder when we divide : `(2x^(3) - 5x^(2) + 8x - 5)` by `(2x^(2) - 3x + 5)`

Answer

To find the quotient and remainder when dividing \(2x^3 - 5x^2 + 8x - 5\) by \(2x^2 - 3x + 5\), we will use polynomial long division. Here’s the step-by-step solution:

### Step 1: Set up the division
We will divide \(2x^3 - 5x^2 + 8x - 5\) (the dividend) by \(2x^2 - 3x + 5\) (the divisor).

### Step 2: Divide the leading terms
Divide the leading term of the dividend \(2x^3\) by the leading term of the divisor \(2x^2\):
\[ \frac{2x^3}{2x^2} = x \]
This gives us the first term of the quotient.

### Step 3: Multiply and subtract
Now, multiply the entire divisor \(2x^2 - 3x + 5\) by \(x\):
\[ x(2x^2 - 3x + 5) = 2x^3 - 3x^2 + 5x \]
Next, subtract this result from the original polynomial:
\[ (2x^3 - 5x^2 + 8x - 5) - (2x^3 - 3x^2 + 5x) = (-5x^2 + 3x^2) + (8x - 5x) - 5 \]
This simplifies to:
\[ -2x^2 + 3x - 5 \]

### Step 4: Repeat the process
Now, we will divide the new leading term \(-2x^2\) by the leading term of the divisor \(2x^2\):
\[ \frac{-2x^2}{2x^2} = -1 \]
This gives us the next term of the quotient.

### Step 5: Multiply and subtract again
Multiply the entire divisor by \(-1\):
\[ -1(2x^2 - 3x + 5) = -2x^2 + 3x - 5 \]
Subtract this from the current polynomial:
\[ (-2x^2 + 3x - 5) - (-2x^2 + 3x - 5) = 0 \]

### Step 6: Conclusion
Since the remainder is \(0\), we conclude that the division is exact.

Thus, the quotient is:
\[ \text{Quotient} = x - 1 \]
And the remainder is:
\[ \text{Remainder} = 0 \]

### Final Answer:
- Quotient: \(x - 1\)
- Remainder: \(0\)