Divide `(2x^(2) -x +3)` by ( 2-x) and write the quotient and the remainder.

Answer

To divide the polynomial \(2x^2 - x + 3\) by \(2 - x\), we can follow these steps:

### Step 1: Rewrite the divisor
First, we can rewrite the divisor \(2 - x\) as \(-x + 2\).

### Step 2: Set up the long division
We will set up the long division with \(2x^2 - x + 3\) as the dividend and \(-x + 2\) as the divisor.

```
______________________
-x + 2 | 2x^2 - x + 3
```

### Step 3: Divide the leading terms
Divide the leading term of the dividend \(2x^2\) by the leading term of the divisor \(-x\):

\[ \frac{2x^2}{-x} = -2x \]

### Step 4: Multiply and subtract
Now, multiply \(-2x\) by the entire divisor \(-x + 2\):

\[ -2x \cdot (-x + 2) = 2x^2 - 4x \]

Now, subtract this result from the original polynomial:

\[ (2x^2 - x + 3) - (2x^2 - 4x) = (-x + 4x) + 3 = 3x + 3 \]

### Step 5: Repeat the process
Now, we will divide the new leading term \(3x\) by the leading term of the divisor \(-x\):

\[ \frac{3x}{-x} = -3 \]

### Step 6: Multiply and subtract again
Multiply \(-3\) by the entire divisor \(-x + 2\):

\[ -3 \cdot (-x + 2) = 3x - 6 \]

Now, subtract this from \(3x + 3\):

\[ (3x + 3) - (3x - 6) = 3 + 6 = 9 \]

### Step 7: Write the final result
Now we have completed the division. The quotient is:

\[ Q(x) = -2x - 3 \]

And the remainder is:

\[ R(x) = 9 \]

### Final Answer
Thus, when dividing \(2x^2 - x + 3\) by \(2 - x\), the quotient is \(-2x - 3\) and the remainder is \(9\).

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