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 \(5\).
1 answer