Roots of polynomialsEdexcel A-Level Further Maths: Subtopic test

1. 1
   The cubic equation $2x^3-6x^2+3x+5=0$ has roots $\alpha$, $\beta$ and $\gamma$.

   (a)

   Write down the value of $\alpha+\beta+\gamma$.

   [1 mark]

   • A$-3$
   • B$3$
   • C$\frac32$
   • D$-\frac52$

   (b)

   Find the value of $\alpha^2+\beta^2+\gamma^2$.

   [1 mark]

   • A$9$
   • B$12$
   • C$6$
   • D$\frac{15}{2}$

   (c)

   Find the value of $\frac1\alpha+\frac1\beta+\frac1\gamma$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The quartic equation $x^4+2x^3-7x^2+4x-3=0$ has roots $\alpha$, $\beta$, $\gamma$ and $\delta$.

   (a)

   Write down the value of $\alpha\beta\gamma\delta$.

   [1 mark]

   • A$-3$
   • B$3$
   • C$4$
   • D$-7$

   (b)

   Find the value of $\frac1\alpha+\frac1\beta+\frac1\gamma+\frac1\delta$.

   [1 mark]

   • A$-\frac43$
   • B$\frac34$
   • C$-\frac34$
   • D$\frac43$

   (c)

   Find the value of $\alpha^2+\beta^2+\gamma^2+\delta^2$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The cubic equation $x^3+3x^2-5x+2=0$ has roots $\alpha$, $\beta$ and $\gamma$.

   (a)

   Find the value of $(3+\alpha)(3+\beta)(3+\gamma)$.

   [3 marks]

   (b)

   Find the value of $\alpha^3+\beta^3+\gamma^3$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The quartic equation $x^4+3x^2+6x+10=0$ has roots $\alpha$, $\beta$, $\gamma$ and $\delta$.

   (a)

   (i) Show that $\alpha^2+\beta^2+\gamma^2+\delta^2=-6$.
   (ii) Hence explain why the equation cannot have four real roots.
   (iii) Find the quartic equation, with integer coefficients, whose roots are $\alpha+1$, $\beta+1$, $\gamma+1$ and $\delta+1$.

   [6 marks]

   (b)

   (i) Show that the equation with roots $\frac1\alpha$, $\frac1\beta$, $\frac1\gamma$ and $\frac1\delta$ is $10y^4+6y^3+3y^2+1=0$.
   (ii) Hence find the value of $\frac1{\alpha^2}+\frac1{\beta^2}+\frac1{\gamma^2}+\frac1{\delta^2}$.

   [6 marks]

   Total for question 4: 12 marks

End of questions