(a)     Show that the complex number i is a root of the equation $${x^4} - 5{x^3} + 7{x^2} - 5x + 6 = 0{\text{ }}.$$ (b)     Find the other roots of this equation.

(a)     ${{\text{i}}^4} - 5{{\text{i}}^3} + 7{{\text{i}}^2} - 5{\text{i}} + 6 = 1 + 5{\text{i}} - 7 - 5{\text{i}} + 6$     M1A1

$= 0$     AG     N0

(b)     ${\text{i}}$ root $\Rightarrow$ $- {\text{i}}$ is second root     (M1)A1

moreover, ${x^4} - 5{x^3} + 7{x^2} - 5x + 6 = \left( {x - {\text{i}}} \right)\left( {x + {\text{i}}} \right)q(x)$

where $q(x) = {x^2} - 5x + 6$

finding roots of $q(x)$

the other two roots are $2$ and $3$     A1A1

Note: Final A1A1 is independent of previous work.

[6 marks]

A surprising number of candidates solved the question by dividing the expression by $1 - i$ rather than substituting $l$ into the expression. Many students were not aware that complex roots occur in conjugate pairs, and many did not appreciate the difference between a factor and a root. Generally the question was well done.