14. (6x^3-16x^2+17x-6)div (3x-2)

To divide the polynomial \(6x^3 - 16x^2 + 17x - 6\) by \(3x - 2\), we can use polynomial long division or synthetic division. Here, we'll use polynomial long division.<br /><br />1. **Divide the leading term of the dividend by the leading term of the divisor:**<br /> \[<br /> \frac{6x^3}{3x} = 2x^2<br /> \]<br /><br />2. **Multiply the entire divisor by this result and subtract from the original polynomial:**<br /> \[<br /> (6x^3 - 16x^2 + 17x - 6) - (2x^2 \cdot (3x - 2)) = (6x^3 - 16x^2 + 17x - 6) - (6x^3 - 4x^2)<br /> \]<br /> \[<br /> = -12x^2 + 17x - 6<br /> \]<br /><br />3. **Repeat the process with the new polynomial:**<br /> \[<br /> \frac{-12x^2}{3x} = -4x<br /> \]<br /> \[<br /> (-12x^2 + 17x - 6) - (-4x \cdot (3x - 2)) = (-12x^2 + 17x - 6) - (-12x^2 + 8x)<br /> \]<br /> \[<br /> = 9x - 6<br /> \]<br /><br />4. **Continue the process:**<br /> \[<br /> \frac{9x}{3x} = 3<br /> \]<br /> \[<br /> (9x - 6) - (3 \cdot (3x - 2)) = (9x - 6) - (9x - 6)<br /> \]<br /> \[<br /> = 0<br /> \]<br /><br />So, the quotient is:<br />\[<br />2x^2 - 4x + 3<br />\]<br /><br />Therefore, the result of \((6x^3 - 16x^2 + 17x - 6) \div (3x - 2)\) is:<br />\[<br />2x^2 - 4x + 3<br />\]