Complex roots of cubic and quartic equations Quiz

1. $1+\mathrm{i}$ is a root of $x^3-4x^2+6x-4=0$. Find the real root.

• A. $2$
• B. $-2$
• C. $4$
• D. $1$

2. A quartic with real coefficients has roots $3+\mathrm{i}$ and $1-2\mathrm{i}$. The other two roots are:

• A. $-3-\mathrm{i}$ and $-1+2\mathrm{i}$
• B. $3-\mathrm{i}$ and $-1-2\mathrm{i}$
• C. $-3+\mathrm{i}$ and $1+2\mathrm{i}$
• D. $3-\mathrm{i}$ and $1+2\mathrm{i}$

3. Which cubic with integer coefficients has roots $3$ and $1\pm\mathrm{i}$?

• A. $z^3-5z^2+8z+6$
• B. $z^3-5z^2+8z-6$
• C. $z^3-3z^2+2z-6$
• D. $z^3+5z^2+8z+6$

4. Divide $x^3+2x^2+x+2$ by $x^2+1$.

• A. $x-2$
• B. $x^2+2$
• C. $x+2$
• D. $x+1$

5. What is the least number of real roots of a cubic with real coefficients?

• A. $0$
• B. $1$
• C. $2$
• D. $3$