Divide \( 3 x^{2}-7 x+2 \) by \( 3 x-1 \)

To divide \( 3x^2 - 7x + 2 \) by \( 3x - 1 \), we can use polynomial long division. 1. Divide the leading term of the dividend \( 3x^2 \) by the leading term of the divisor \( 3x \) to get \( x \). 2. Multiply the entire divisor \( 3x - 1 \) by \( x \): \( x(3x - 1) = 3x^2 - x \). 3. Subtract this result from the original polynomial: \((3x^2 - 7x + 2) - (3x^2 - x) = -6x + 2\). 4. Now, divide the leading term of the new polynomial \( -6x \) by the leading term of the divisor \( 3x \) to get \( -2 \). 5. Multiply \( -2 \) by the entire divisor: \(-2(3x - 1) = -6x + 2\). 6. Subtract again: \((-6x + 2) - (-6x + 2) = 0\). So, \( 3x^2 - 7x + 2 \) divided by \( 3x - 1 \) gives you \( x - 2 \) with a remainder of \( 0 \). Thus, \( \frac{3x^2 - 7x + 2}{3x - 1} = x - 2 \).