Question

Use polynomial long division
6x^3-5x^2+13x-4 by 3x-1

Answer 1

To perform polynomial long division with \(6x^3 - 5x^2 + 13x - 4\) divided by \(3x - 1\), we follow these steps:

1. **Set up the division**: Write the dividend \(6x^3 - 5x^2 + 13x - 4\) under the long division symbol and the divisor \(3x - 1\) to the left.

2. **Perform the division step by step**:

- **Step 1**: Divide the leading term of the dividend by the leading term of the divisor:
\[
\frac{6x^3}{3x} = 2x^2
\]

- **Step 2**: Multiply \(2x^2\) by the entire divisor \(3x - 1\):
\[
2x^2(3x - 1) = 6x^3 - 2x^2
\]

- **Step 3**: Subtract this result from the original polynomial:
\[
(6x^3 - 5x^2 + 13x - 4) - (6x^3 - 2x^2) = -5x^2 + 2x^2 + 13x - 4 = -3x^2 + 13x - 4
\]

- **Step 4**: Repeat the process with the new polynomial \(-3x^2 + 13x - 4\):
Divide \(-3x^2\) by \(3x\):
\[
\frac{-3x^2}{3x} = -x
\]

- **Step 5**: Multiply \(-x\) by the divisor \(3x - 1\):
\[
-x(3x - 1) = -3x^2 + x
\]

- **Step 6**: Subtract this from \(-3x^2 + 13x - 4\):
\[
(-3x^2 + 13x - 4) - (-3x^2 + x) = 13x - 4 - x = 12x - 4
\]

- **Step 7**: Repeat the process again with the new polynomial \(12x - 4\):
Divide \(12x\) by \(3x\):
\[
\frac{12x}{3x} = 4
\]

- **Step 8**: Multiply \(4\) by the divisor \(3x - 1\):
\[
4(3x - 1) = 12x - 4
\]

- **Step 9**: Subtract this from \(12x - 4\):
\[
(12x - 4) - (12x - 4) = 0
\]

3. **Conclusion**: Because the final remainder is 0, we conclude that:
\[
\frac{6x^3 - 5x^2 + 13x - 4}{3x - 1} = 2x^2 - x + 4
\]

Thus, the result of the polynomial long division is:
\[
\boxed{2x^2 - x + 4}
\]