Find the quotient and the remainder (if any), when, `2x^(3) - 8x^(2) + 5x - 8` is divided by x - 2.

Answer

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

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

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

### Step 3: Multiply and subtract
Now, multiply \(2x^2\) by the entire divisor \(x - 2\):
\[ 2x^2 \cdot (x - 2) = 2x^3 - 4x^2 \]
Subtract this from the original polynomial:
\[ (2x^3 - 8x^2 + 5x - 8) - (2x^3 - 4x^2) = -8x^2 + 4x^2 + 5x - 8 = -4x^2 + 5x - 8 \]

### Step 4: Repeat the process
Now, repeat the process with the new polynomial \(-4x^2 + 5x - 8\).

Divide the leading term \(-4x^2\) by \(x\):
\[ \frac{-4x^2}{x} = -4x \]
This is the next term of our quotient.

### Step 5: Multiply and subtract again
Multiply \(-4x\) by the divisor \(x - 2\):
\[ -4x \cdot (x - 2) = -4x^2 + 8x \]
Subtract this from \(-4x^2 + 5x - 8\):
\[ (-4x^2 + 5x - 8) - (-4x^2 + 8x) = 5x - 8 - 8x = -3x - 8 \]

### Step 6: Continue the process
Now, divide the leading term \(-3x\) by \(x\):
\[ \frac{-3x}{x} = -3 \]
This is the next term of our quotient.

### Step 7: Multiply and subtract one last time
Multiply \(-3\) by the divisor \(x - 2\):
\[ -3 \cdot (x - 2) = -3x + 6 \]
Subtract this from \(-3x - 8\):
\[ (-3x - 8) - (-3x + 6) = -8 - 6 = -14 \]

### Conclusion
Now we have completed the division. The quotient is:
\[ 2x^2 - 4x - 3 \]
And the remainder is:
\[ -14 \]

### Final Answer
- **Quotient:** \(2x^2 - 4x - 3\)
- **Remainder:** \(-14\)

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