Edexcel IAL Mathematics FP1.1.6 quartic complex roots

(z=) 5+i

Correct complex number

B1

(1)

Mark (b) and (c) together - ignore any labelling seen.
Award marks in the order given for their choice of method

/(c)

With (b) first

$$z=5 \pm i \Rightarrow(z-(5+i))(z-(5-i))=\ldots$$

Or e.g.

Sum of roots =10

Product of roots =26

Correct strategy to find the quadratic
factor using the conjugate pair

M1

$z^{2}-10 z+26$

Correct quadratic

A1

$f(z)=\left(z^{2}-10 z+26\right)\left(2 z^{2}+\ldots z+k\right)$

Attempts to find the other quadratic. May
use inspection (apply rules for quadratic
factorisation ie " 26 " |k|=156 ) or e.g.
long division with quotient $2 z^{2}+\ldots z+\ldots$

M1

NB long division gives quotient $2 z^{2}+z+(A-42)$ and remainder
(10 A+B-446) z+936-26 A}

$2 z^{2}+z-6$

Correct quadratic

A1

$\Rightarrow z=\frac{3}{2},-2(, 5 \pm \mathrm{i})$

Correct real roots. The complex roots do
not have to be stated.

A1

(5)

$$f(z)=(z^2-10z+26)(2z^2+z-6)$$
Multiplying out or extracting coefficients gives
$$A=36,\qquad B=86$$

M1 A1