Question

Divide using ling division. Check your answer.
(3x^2 -4x+1)/(x-2)
The quotient is ____ with remainder ____

Answer 1

To divide \(3x^2 - 4x + 1\) by \(x - 2\) using long division, we can follow these steps:

### Step 1: Set up the division
We set up the long division as follows:

```
_______________
x - 2 | 3x^2 - 4x + 1
```

### Step 2: Divide the first term
Divide the leading term of the dividend (\(3x^2\)) by the leading term of the divisor (\(x\)):
\[
\frac{3x^2}{x} = 3x
\]
Write \(3x\) above the division line.

### Step 3: Multiply and subtract
Now we will multiply \(3x\) by the entire divisor \((x - 2)\) and subtract this from the original polynomial:
\[
3x(x - 2) = 3x^2 - 6x
\]
Now, perform the subtraction:
\[
(3x^2 - 4x + 1) - (3x^2 - 6x) = (-4x + 6x) + 1 = 2x + 1
\]
The result now becomes:
```
3x
_______________
x - 2 | 3x^2 - 4x + 1
-(3x^2 - 6x)
_______________
2x + 1
```

### Step 4: Repeat the process
Now, divide the leading term \(2x\) by the leading term of the divisor (\(x\)):
\[
\frac{2x}{x} = 2
\]
Write \(2\) above the division line:
```
3x + 2
_______________
x - 2 | 3x^2 - 4x + 1
-(3x^2 - 6x)
_______________
2x + 1
-(2x - 4)
___________
5
```

### Step 5: Multiply and subtract again
Now multiply \(2\) by the entire divisor \((x - 2)\):
\[
2(x - 2) = 2x - 4
\]
Now, perform the subtraction:
\[
(2x + 1) - (2x - 4) = 1 + 4 = 5
\]

### Step 6: Final result
The final result of the division is as follows:
\[
\text{Quotient} = 3x + 2 \quad \text{Remainder} = 5
\]

### Summary
Thus, we have:

- The quotient is \(3x + 2\)
- The remainder is \(5\)

Final Answer:
**The quotient is \(3x + 2\) with remainder \(5\)**.