Roots and coefficients of polynomialsAQA A-Level Further Maths: Subtopic test

1. 1
   The roots of the equation $x^3-4x^2+5x-7=0$ are $\alpha$, $\beta$ and $\gamma$.

   (a)

   What is the value of $\alpha\beta\gamma$?

   [1 mark]

   • A$-7$
   • B$5$
   • C$4$
   • D$7$

   (b)

   What is the value of $\alpha^2+\beta^2+\gamma^2$?

   [1 mark]

   • A$26$
   • B$6$
   • C$11$
   • D$21$

   (c)

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

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The roots of the equation $x^4-6x^3+3x^2+2x-5=0$ are $\alpha$, $\beta$, $\gamma$ and $\delta$.

   (a)

   What is the value of $\alpha\beta\gamma\delta$?

   [1 mark]

   • A$-5$
   • B$5$
   • C$2$
   • D$-2$

   (b)

   What is the sum of the products of the roots taken three at a time, $\alpha\beta\gamma+\alpha\beta\delta+\alpha\gamma\delta+\beta\gamma\delta$?

   [1 mark]

   • A$2$
   • B$3$
   • C$-2$
   • D$6$

   (c)

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

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The roots of the equation $x^3+3x^2-5x+1=0$ are $\alpha$, $\beta$ and $\gamma$.

   (a)

   Show that $\alpha^2+\beta^2+\gamma^2=19$.

   [3 marks]

   (b)

   Find a cubic equation with integer coefficients whose roots are $2\alpha+1$, $2\beta+1$ and $2\gamma+1$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The roots of the equation $x^3-14x^2+kx-64=0$, where $k$ is a constant, are in geometric progression.

   (a)

   Find the roots of the equation and the value of $k$.

   [6 marks]

   (b)

   Using your value of $k$ and without finding the individual roots, use the relationships between the roots and the coefficients to find the exact value of $\frac1{\alpha^2}+\frac1{\beta^2}+\frac1{\gamma^2}$, where $\alpha$, $\beta$, $\gamma$ are the roots.

   [6 marks]

   Total for question 4: 12 marks

End of questions