21M.1.AHL.TZ1.7

METHOD 1

other two roots are $a-b\text{i}$ and $b-a\text{i}$        A1

sum of roots $=-4$ and product of roots $=400$       A1

attempt to set sum of four roots equal to $-4$ or $4$ OR
attempt to set product of four roots equal to $400$        M1

$a+b\text{i}+a-b\text{i}+b+a\text{i}+b-ai=-4$

$2a+2b=-4(\Rightarrow a+b=-2)$         A1

$(a+b\text{i})(a-b\text{i}) (b+a\text{i})(b-a\text{i})=400$

${\left(a^2+b^2\right)}^2=400$         A1

$a^2+b^2=20$

attempt to solve simultaneous equations             (M1)

$a=2$ or $a=-4$           A1A1

 

METHOD 2

other two roots are $a-b\text{i}$ and $b-a\text{i}$        A1

$\left(z-\left(a+b\text{i}\right)\right)\left(z-\left(a-b\text{i}\right)\right)\left(z-\left(b+a\text{i}\right)\right)\left(z-\left(b-a\text{i}\right)\right)\left(=0\right)$        A1

$\left({\left(z-a\right)}^2+b^2\right)\left({\left(z-b\right)}^2+a^2\right)\left(=0\right)$

$\left(z^2-2az+a^2+b^2\right)\left(z^2-2bz+b^2+a^2\right)\left(=0\right)$        A1

Attempt to equate coefficient of $z^3$ and constant with the given quartic equation        M1

$-2a-2b=4$ and ${\left(a^2+b^2\right)}^2=400$        A1

attempt to solve simultaneous equations             (M1)

$a=2$ or $a=-4$           A1A1

 

[8 marks]

[N/A]