Complex conjugates and polynomial rootsAQA A-Level Further Maths: Subtopic test

1. 1
   The cubic equation $z^3-5z^2+17z-13=0$ has a real root $z=1$.

   (a)

   Which quadratic is the factor that remains after dividing the cubic by $(z-1)$?

   [1 mark]

   • A$z^2-6z+13$
   • B$z^2+4z+13$
   • C$z^2-4z-13$
   • D$z^2-4z+13$

   (b)

   Which are the two non-real roots of the cubic?

   [1 mark]

   • A$-2\pm3i$
   • B$4\pm6i$
   • C$2\pm3i$
   • D$2\pm\sqrt{13}$

   (c)

   Explain why the cubic has exactly one real root.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The cubic equation $z^3-4z^2+6z-4=0$ has real coefficients and a root $z=1+i$.

   (a)

   Which other root must the equation have?

   [1 mark]

   • A$1-i$
   • B$-1+i$
   • C$-1-i$
   • D$1-2i$

   (b)

   Find the real root of the equation.

   [1 mark]

   • A$-2$
   • B$2$
   • C$4$
   • D$1$

   (c)

   Show by substitution that $1-i$ is a root of the equation.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The quartic $f(z)=z^4-2z^3+9z^2-8z+20$ has $z^2-2z+5$ as a factor.

   (a)

   Find the other quadratic factor of $f(z)$.

   [3 marks]

   (b)

   Solve $f(z)=0$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The cubic equation $z^3+az^2+bz-10=0$, where $a$ and $b$ are real constants, has a root $z=2+i$.

   (a)

   (i) Write down a second root of the equation, and find the real quadratic factor of the cubic that has these two roots.
   (ii) Find the third root, and the values of $a$ and $b$.

   [6 marks]

   (b)

   (i) Verify that $2+i$ is a root of $z^3-6z^2+13z-10=0$.
   (ii) Using your answer to part (a), solve $(z^3-6z^2+13z-10)(z^2+4z+8)=0$.

   [6 marks]

   Total for question 4: 12 marks

End of questions