It is known that two roots of the nonlinear equation x3 – 6×2 + 11x – 6 = are 1 and 3. The third root will be A. j B. -j C. 2 D. 4

The correct answer is $\boxed{-2}$.

To solve for the third root, we can use Vieta’s formulas. Vieta’s formulas state that the sum of the roots of a polynomial equation is equal to the negative of the coefficient of the $x^2$ term, and the product of the roots is equal to the constant term. In this case, the coefficient of the $x^2$ term is $-6$, and the constant term is $-6$. Therefore, the sum of the roots is $-6$, and the product of the roots is $-6$.

We know that two of the roots are $1$ and $3$. Therefore, the third root must be $-2$, since $1 + 3 – 2 = -6$ and $1 \cdot 3 \cdot -2 = -6$.

Here is a brief explanation of each option:

• Option A: $j$ is an imaginary number. The sum of the roots of a polynomial equation must be a real number, so $j$ cannot be the third root.
• Option B: $-j$ is also an imaginary number. The sum of the roots of a polynomial equation must be a real number, so $-j$ cannot be the third root.
• Option C: $2$ is a real number. The sum of the roots of a polynomial equation must be a real number, so $2$ cannot be the third root.
• Option D: $4$ is a real number. The sum of the roots of a polynomial equation must be a real number, so $4$ cannot be the third root.