Complex roots of cubic and quartic equationsEdexcel International A Level Further Maths: Subtopic test

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

   (a)

   Which of the following is a root of the equation?

   [1 mark]

   • A$z=-1$
   • B$z=13$
   • C$z=1$
   • D$z=5$

   (b)

   When $z^3-5z^2+17z-13$ is divided by $(z-1)$, the quotient is:

   [1 mark]

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

   (c)

   Given that $z=1$ is a root, find the other two roots.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The cubic equation $z^3+z^2-z+15=0$, which has a root $1-2\mathrm{i}$.

   (a)

   Which of the following must also be a root of the equation?

   [1 mark]

   • A$-1+2\mathrm{i}$
   • B$-1-2\mathrm{i}$
   • C$1-4\mathrm{i}$
   • D$1+2\mathrm{i}$

   (b)

   Find the real root of the equation.

   [1 mark]

   • A$-3$
   • B$3$
   • C$15$
   • D$-5$

   (c)

   Find the quadratic factor of the cubic with real coefficients.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The quartic function $\text{g}(x)=x^4-6x^3+18x^2-30x+25$, which has a root $x=1+2\mathrm{i}$.

   (a)

   Write down a second root of $\text{g}(x)=0$ and hence find a quadratic factor of $\text{g}(x)$ with real coefficients.

   [3 marks]

   (b)

   Find the other two roots of $\text{g}(x)=0$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The quartic function $\text{h}(x)=x^4-4x^3+6x^2-4x-15$.

   (a)

   (i) Show that $x=3$ and $x=-1$ are roots of $\text{h}(x)=0$.
   (ii) Hence solve $\text{h}(x)=0$ completely.

   [6 marks]

   (b)

   The four roots of $\text{h}(x)=0$ are represented by points on an Argand diagram. Show that these points are the vertices of a square and find its area.

   [6 marks]

   Total for question 4: 12 marks

End of questions