When the polynomial P(x) = 6x3 + kx2 + x − 2 is divided by x + 2, the remainder is 0. Which of the following is also a factor of P(x)?

When the polynomial P(x) = 6x³ + kx² + x − 2 is divided by x + 2, the remainder is 0.

Which of the following is also a factor of P(x)?

A:3x-1

B:3x+1

C:2x-1

D:x-1

polynomial long division:

\(6x^3 + kx^2 + x - 2 : x + 2 = 6x^2+kx - 12x -2k+25 + \underbrace{\dfrac{4k-52}{x+2}}_{=0} \\ 6x^3 + kx^2 + x - 2 : x + 2 = 6x^2+kx - 12x -2k+25 + 0 \)

remainder is 0:

\(\begin{array}{|rcll|} \hline \dfrac{4k-52}{x+2} &=& 0 \\ 4k-52 &=& 0 \\ 4k &=& 52 \\ \mathbf{ k } & \mathbf{=} & \mathbf{13} \\ \hline \end{array}\)

\(\begin{array}{|rcll|} \hline 6x^3 + kx^2 + x - 2 : x + 2 &=& 6x^2+kx - 12x -2k+25 + 0 \quad & | \quad k = 13 \\ &=& 6x^2+13x - 12x -26+25 \\ &=& 6x^2+x -1 \\ &=& (3x-1)(2x+1) \\\\ \mathbf{ 6x^3 + kx^2 + x - 2 } & \mathbf{=} & \mathbf{(x+2)(3x-1)(2x+1)} \\ \hline \end{array} \)

A:3x-1

When the polynomial P(x) = 6x3 + kx2 + x − 2 is divided by x + 2, the remainder is 0. Which of the following is also a factor of P(x)?

A:3x-1

B:3x+1

C:2x-1

D:x-1

If x+2 is a factor then f(-2)=0

so

\(P(2) = 6*(-2)^3 + k*(-2)^2 + (-2)− 2=0\\ 6*-8 + 4k-4=0\\ 4k=52\\ k=13\\ so\\ p(x)=6x^3+13x^2+x-2\)

If 3x-1 is a factor then x=1/3  will be a zero I can check this with the calcuator

6*(1/3)^3+13*(1/3)^2+(1/3)-2 = 0    So  (3x-1) IS a factor

If 3x+1 is a facor then  f(-1/3) = 0

6*(-1/3)^3+13*(-1/3)^2+(-1/3)-2 = -1.1111111111111111    No 3x+1 is not a factor

If 2x-1 is a factor then f(1/2)=0

6*(1/2)^3+13*(1/2)^2+(1/2)-2 = 2.5    No 2x-1 is not a factor

If x-1 is a factor then f(1)=0

6*(1)^3+13*(1)^2+(1)-2 = 18     No that isn't a factor either     

I have determined this without any algebraic division.