Write down the numerical value of the sum and of the product of the roots of this equation.

The roots of this equation are three consecutive terms of an arithmetic sequence.

Taking the roots to be $\alpha {\text{ , }}\alpha  \pm \beta$ , solve the equation.

${\text{sum}} = \frac{{45}}{9},{\text{ product}} = \frac{{40}}{9}$     A1

[1 mark]

it follows that $3\alpha = \frac{{45}}{9}$ and $\alpha ({\alpha ^2} - {\beta ^2}) = \frac{{40}}{9}$     A1A1

solving, $\alpha = \frac{5}{3}$     A1

$\frac{5}{3}\left( {\frac{{25}}{9} - {\beta ^2}} \right) = \frac{{40}}{9}$     M1

$\beta = ( \pm )\frac{1}{3}$     A1

the other two roots are 2, $\frac{4}{3}$     A1

[6 marks]