Write down the sum and the product of the roots of $P(z) = 0$.

Show that $(z - 1)$ is a factor of $P(z)$.

Find the value of $b$ and the value of $c$.

Hence find the complex roots of $P(z) = 0$.

Show that the graph of $y = q(x)$ is concave up for $x > 1$.

Sketch the graph of $y = q(x)$ showing clearly any intercepts with the axes.

${\text{sum}} = 0$     A1

${\text{product}} = 6$     A1

[2 marks]

$P(1) = 1 - 10 + 15 - 6 = 0$     M1A1

$\Rightarrow (z - 1)$ is a factor of $P(z)$     AG

Note:     Accept use of division to show remainder is zero.

[2 marks]

METHOD 1

${(z - 1)^3}\left( {{z^2} + bz + c} \right) = {z^5} - 10{z^2} + 15z - 6$     (M1)

by inspection $c = 6$     A1

$({z^3} - 3{z^2} + 3z - 1)({z^2} + bz + 6) = {z^5} - 10{z^2} + 15z - 6$     (M1)(A1)

$b = 3$     A1

METHOD 2

$\alpha$, $\beta$ are two roots of the quadratic

$b = - (\alpha + \beta ),{\text{ }}c = \alpha \beta$     (A1)

from part (a) $1 + 1 + 1 + \alpha + \beta = 0$     (M1)

$\Rightarrow b = 3$     A1

$1 \times 1 \times 1 \times \alpha \beta = 6$     (M1)

$\Rightarrow c = 6$     A1

Note:     Award FT if $b = - 7$ following through from their sum $= 10$.

METHOD 3

$({z^5} - 10{z^2} + 15z - 6) \div (z - 1) = {z^4} + {z^3} + {z^2} - 9z + 6$     (M1)A1

Note:     This may have been seen in part (b).

${z^4} + {z^3} + {z^2} - 9z + 6 \div (z - 1) = {z^3} + 2{z^2} + 3z - 6$     (M1)

${z^3} + 2{z^2} + 3z - 6 \div (z - 1) = {z^2} + 3z + 6$     A1A1

[5 marks]

${z^2} + 3z + 6 = 0$     M1

$z = \frac{{ - 3 \pm \sqrt {9 - 4 \bullet 6} }}{2}$     M1

$= \frac{{ - 3 \pm \sqrt { - 15} }}{2}$

$z = - \frac{3}{2} \pm \frac{{{\text{i}}\sqrt {15} }}{2}$     A1

(or $z = 1$)

Notes:     Award the second M1 for an attempt to use the quadratic formula or to complete the square.

Do not award FT from (c).

[3 marks]

$\frac{{{{\text{d}}^2}y}}{{{\text{d}}{x^2}}} = 20{x^3} - 20$     M1A1

for $x > 1,{\text{ }}20{x^3} - 20 > 0 \Rightarrow$ concave up     R1AG

[3 marks]

$x$-intercept at $(1,{\text{ }}0)$     A1

$y$-intercept at $(0,{\text{ }} - 6)$     A1

stationary point of inflexion at $(1,{\text{ }}0)$ with correct curvature either side     A1

[3 marks]