Let p(z) = z³ + (1 + j) z² + (2 + j) z + 3, where z is a complex number. Which one of the following is true?

To determine which statement is true for the polynomial $p\left(z\right)=z^3+(1+j)z^2+(2+j)z+3$, let us analyze the properties of its roots and coefficients.

The correct option is: All the roots cannot be real

Let us prove this by contradiction. Suppose all three roots of the cubic equation $p\left(z\right)=0$ are real numbers. Let these roots be $r_1$, $r_2$, and $r_3$, where $r_1,r_2,r_3\in ℝ$.

According to Vieta's formulas, the sum of the roots of a polynomial of the form $z^3+a_2{z}^2+a_{1}z+a_0=0$ is equal to negative of the coefficient of $z^2$:
$r_1+r_2+r_3=-a_2$
For the given polynomial, the coefficient of $z^2$ is $a_2=1+j$.

Therefore, the sum of the roots is:
$r_1+r_2+r_3=-(1+j)=-1-j$

Since we assumed $r_1,r_2,r_3$ are all real numbers, their sum $r_1+r_2+r_3$ must also be a real number. However, the sum is $-1-j$, which has a non-zero imaginary part ($-1$). This is a contradiction.

Hence, the assumption that all the roots are real must be false. Therefore, all the roots cannot be real.