If \((x + 2)\) and \((3x - 1)\) are factors of \(6x^{3} + x^{2} - 19x + 6\), find the third factor.

Answer Details

If \((x+2)\) and \((3x-1)\) are factors of \(6x^3+x^2-19x+6\), then by the factor theorem, if we divide the given polynomial by these factors, the remainder should be equal to zero. Therefore, we can write: \[\begin{aligned} 6x^3 + x^2 - 19x + 6 &= (x+2)(3x^2+ax+b) + c_1 \\ &= (3x-1)(2x^2+cx+d) + c_2 \end{aligned}\] where \(c_1\) and \(c_2\) are the remainders obtained after the division. By equating the coefficients of the corresponding powers of \(x\) in both the equations, we can obtain a system of linear equations. Solving this system will help us to find the coefficients of the required polynomial. After solving the system, we get the third factor as \(\boxed{2x-3}\). Therefore, is the correct answer.