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Date	May 2022	Marks available	1	Reference code	22M.3.AHL.TZ2.2
Level	Additional Higher Level	Paper	Paper 3	Time zone	Time zone 2
Command term	State	Question number	2	Adapted from	N/A

Question

This question asks you to investigate conditions for the existence of complex roots of polynomial equations of degree $3$ and $4$ .

The cubic equation $x^{3} + p x^{2} + q x + r = 0$ , where $p, q, r \in ℝ$ , has roots $α, β$ and $γ$ .

Consider the equation $x^{3} - 7 x^{2} + q x + 1 = 0$ , where $q \in ℝ$ .

Noah believes that if $p^{2} \geq 3 q$ then $α, β$ and $γ$ are all real.

Now consider polynomial equations of degree $4$ .

The equation $x^{4} + p x^{3} + q x^{2} + r x + s = 0$ , where $p, q, r, s \in ℝ$ , has roots $α, β, γ$ and $δ$ .

In a similar way to the cubic equation, it can be shown that:

$p = - (α + β + γ + δ)$

$q = α β + α γ + α δ + β γ + β δ + γ δ$

$r = - (α β γ + α β δ + α γ δ + β γ δ)$

$s = α β γ δ$ .

The equation $x^{4} - 9 x^{3} + 24 x^{2} + 22 x - 12 = 0$ , has one integer root.

By expanding $(x - α) (x - β) (x - γ)$ show that:

$p = - (α + β + γ)$

$q = α β + β γ + γ α$

$r = - α β γ$ .

[3]

Show that $p^{2} - 2 q = α^{2} + β^{2} + γ^{2}$ .

[3]

b.i.

Hence show that ${(α - β)}^{2} + {(β - γ)}^{2} + {(γ - α)}^{2} = 2 p^{2} - 6 q$ .

[3]

b.ii.

Given that $p^{2} < 3 q$ , deduce that $α, β$ and $γ$ cannot all be real.

[2]

Using the result from part (c), show that when $q = 17$ , this equation has at least one complex root.

[2]

By varying the value of $q$ in the equation $x^{3} - 7 x^{2} + q x + 1 = 0$ , determine the smallest positive integer value of $q$ required to show that Noah is incorrect.

[2]

e.i.

Explain why the equation will have at least one real root for all values of $q$ .

[1]

e.ii.

Find an expression for $α^{2} + β^{2} + γ^{2} + δ^{2}$ in terms of $p$ and $q$ .

[3]

f.i.

Hence state a condition in terms of $p$ and $q$ that would imply $x^{4} + p x^{3} + q x^{2} + r x + s = 0$ has at least one complex root.

[1]

f.ii.

Use your result from part (f)(ii) to show that the equation $x^{4} - 2 x^{3} + 3 x^{2} - 4 x + 5 = 0$ has at least one complex root.

[1]

State what the result in part (f)(ii) tells us when considering this equation $x^{4} - 9 x^{3} + 24 x^{2} + 22 x - 12 = 0$ .

[1]

h.i.

Write down the integer root of this equation.

[1]

h.ii.

By writing $x^{4} - 9 x^{3} + 24 x^{2} + 22 x - 12$ as a product of one linear and one cubic factor, prove that the equation has at least one complex root.

[4]

h.iii.

Markscheme

attempt to expand $(x - α) (x - β) (x - γ)$ M1

$= (x^{2} - (α + β) x + α β) (x - γ)$ OR $= (x - α) (x^{2} - (β + γ) x + β γ)$ A1

$(x^{3} + p x^{2} + q x + r) = x^{3} - (α + β + γ) x^{2} + (α β + β γ + γ α) x - α β γ$ A1

comparing coefficients:

$p = - (α + β + γ)$ AG

$q = (α β + β γ + γ α)$ AG

$r = - α β γ$ AG

Note: For candidates who do not include the AG lines award full marks.

[3 marks]

$p^{2} - 2 q = {(α + β + γ)}^{2} - 2 (α β + β γ + γ α)$ (A1)

attempt to expand ${(α + β + γ)}^{2}$ (M1)

$= α^{2} + β^{2} + γ^{2} + 2 (α β + β γ + γ α) - 2 (α β + β γ + γ α)$ or equivalent A1

$= α^{2} + β^{2} + γ^{2}$ AG

Note: Accept equivalent working from RHS to LHS.

[3 marks]

b.i.

EITHER

attempt to expand ${(α - β)}^{2} + {(β - γ)}^{2} + {(γ - α)}^{2}$ (M1)

$= (α^{2} + β^{2} - 2 α β) + (β^{2} + γ^{2} - 2 β γ) + (γ^{2} + α^{2} - 2 γ α)$ A1

$= 2 (α^{2} + β^{2} + γ^{2}) - 2 (α β + β γ + γ α)$

$= 2 (p^{2} - 2 q) - 2 q$ or equivalent A1

$= 2 p^{2} - 6 q$ AG

attempt to write $2 p^{2} - 6 q$ in terms of $α, β, γ$ (M1)

$= 2 (p^{2} - 2 q) - 2 q$

$= 2 (α^{2} + β^{2} + γ^{2}) - 2 (α β + β γ + γ α)$ A1

$= (α^{2} + β^{2} - 2 α β) + (β^{2} + γ^{2} - 2 β γ) + (γ^{2} + α^{2} - 2 γ α)$ A1

$= {(α - β)}^{2} + {(β - γ)}^{2} + {(γ - α)}^{2}$ AG

Note: Accept equivalent working where LHS and RHS are expanded to identical expressions.

[3 marks]

b.ii.

$p^{2} < 3 q \Rightarrow 2 p^{2} - 6 q < 0$

$\Rightarrow {(α - β)}^{2} + {(β - γ)}^{2} + {(γ - α)}^{2} < 0$ A1

if all roots were real ${(α - β)}^{2} + {(β - γ)}^{2} + {(γ - α)}^{2} \geq 0$ R1

Note: Condone strict inequality in the R1 line.
Note: Do not award A0R1.

$\Rightarrow$ roots cannot all be real AG

[2 marks]

$p^{2} = {(- 7)}^{2} = 49$ and $3 q = 51$ A1

so $p^{2} < 3 q \Rightarrow$ the equation has at least one complex root R1

Note: Allow equivalent comparisons; e.g. checking $p^{2} < 6 q$

[2 marks]

use of GDC (eg graphs or tables) (M1)

$q = 12$ A1

[2 marks]

e.i.

complex roots appear in conjugate pairs (so if complex roots occur the other root will be real OR all 3 roots will be real).

a cubic curve always crosses the $x$ -axis at at least one point. R1

[1 mark]

e.ii.

attempt to expand ${(α + β + γ + δ)}^{2}$ (M1)

${(α + β + γ + δ)}^{2} = α^{2} + β^{2} + γ^{2} + δ^{2} + 2 (α β + α γ + α δ + β γ + β δ + γ δ)$ (A1)

$\Rightarrow α^{2} + β^{2} + γ^{2} + δ^{2} = {(α + β + γ + δ)}^{2} - 2 (α β + α γ + α δ + β γ + β δ + γ δ)$

$(\Rightarrow α^{2} + β^{2} + γ^{2} + δ^{2} =) p^{2} - 2 q$ A1

[3 marks]

f.i.

$p^{2} < 2 q$ OR $p^{2} - 2 q < 0$ A1

Note: Allow FT on their result from part (f)(i).

[1 mark]

f.ii.

$4 < 6$ OR $2^{2} - 2 \times 3 < 0$ R1

hence there is at least one complex root. AG

Note: Allow FT from part (f)(ii) for the R mark provided numerical reasoning is seen.

[1 mark]

$(p^{2} > 2 q) (81 > 2 \times 24)$ (so) nothing can be deduced R1

Note: Do not allow FT for the R mark.

[1 mark]

h.i.

$- 1$ A1

[1 mark]

h.ii.

attempt to express as a product of a linear and cubic factor M1

$(x + 1) (x^{3} - 10 x^{2} + 34 x - 12)$ A1A1

Note: Award A1 for each factor. Award at most A1A0 if not written as a product.

since for the cubic, $p^{2} < 3 q (100 < 102)$ R1

there is at least one complex root AG

[4 marks]

h.iii.

Examiners report

The first part of this question proved to be very accessible, with the majority of candidates expanding their brackets as required, to find the coefficients $p, q$ and $r$ .

The first part of this question was usually answered well, though presentation in the second part sometimes left a lot to be desired. The expression $2 (p^{2} - 2 q) - 2 q$ was expected to be seen more often, as a 'pivot' to reaching the required result. Algebra was often lengthy, but untidily so, sometimes leaving examiners to do some mental tidying up on behalf of the candidate.

b.i.

[N/A]

b.ii.

A good number of candidates recognised the reasoning required in this part of the question and were able to score both marks.

Most candidates found applying this specific case to be very straightforward.

Most candidates offered incorrect answers in the first part; despite their working suggested utilisation of the GDC, it was clear that many did not appreciate what the question was asking. The second part was usually answered well, with the idea of complex roots occurring in conjugate pairs being put to good use.

e.i.

[N/A]

e.ii.

Some very dubious algebra was seen here, and often no algebra at all. Despite this, a good number of candidates seemed to make the 'leap' to the correct expression $p^{2} - 2 q$ , perhaps fortuitously so in a number of cases.

f.i.

[N/A]

f.ii.

Of those finding $p^{2} - 2 q$ in part f, a surprising number of answers seen employed the test of checking whether $p^{2} < 3 q$ .

Part i was usually not answered successfully, which may have been due to shortage of time. However, it was pleasing to see a number of candidates reach the end of the paper and successfully factorise the given quartic using a variety of methods. The final part required the $p^{2} < 3 q$ test. Though correct reasoning was sometimes seen, it was rare for this final mark to be gained.

h.i.

[N/A]

h.ii.

[N/A]

h.iii.

Syllabus sections

Topic 1—Number and algebra » AHL 1.14—Complex roots of polynomials, conjugate roots, De Moivre’s, powers & roots of complex numbers

Show 55 related questions

22M.3.AHL.TZ1.2a.i:
Given that $1$ and $4 + i$ are roots of the equation, write down the third root.
22M.3.AHL.TZ1.2a.ii:
Verify that the mean of the two complex roots is $4$ .
22M.3.AHL.TZ2.2e.ii:
Explain why the equation will have at least one real root for all values of $q$ .
22M.1.AHL.TZ2.12c.i:
Express $z_{1}$ in terms of $z_{2}$ .
22M.1.AHL.TZ2.12d:
Use the result from part (c)(ii) to show that $a^{2} - 3 b = 0$ .
19M.2.AHL.TZ2.H_9b:
Sketch the graph of $y = {x^4} - 6{x^3} - 2{x^2} + 58x - 51$ , stating clearly the coordinates of any maximum and minimum points and intersections with axes.
16N.1.AHL.TZ0.H_12c:
Find the values of $x$ that satisfy the equation $\left| p \right| = \left| q \right|$ .
19M.2.AHL.TZ2.H_8b:
One of the roots ${w_1}$ satisfies the condition ${\text{Re}}\left( {{w_1}} \right) = 0$ .

Given that ${w_1} = \frac{z}{{z - {\text{i}}}}$ , express $z$ in the form $a + b{\text{i}}$ , where $a$ , $b \in \mathbb{Q}$ .
19N.2.AHL.TZ0.H_10b:
Sketch the graph of $f$ . State the coordinates of the end points and any local maximum or minimum points, giving your answers in terms of $a$ .
19N.2.AHL.TZ0.H_8c:
the girls do not sit on either end and do not sit together.
19N.2.AHL.TZ0.H_10c.iii:
the median of $X$ .
19N.2.AHL.TZ0.H_8a:
the girls do not sit together.
16N.1.AHL.TZ0.H_12b:
Show that $(\omega - 3{\omega ^2})({\omega ^2} - 3\omega ) = 13$ .
19M.2.AHL.TZ1.H_11a:
Show that $abc = 8$ .
19M.2.AHL.TZ2.H_8a:
Find the roots of the equation ${w^3} = 8{\text{i}}$ , $w \in \mathbb{C}$ . Give your answers in Cartesian form.
19M.2.AHL.TZ1.H_2b:
$a + {\text{i}}b$ where $a$ , $b \in \mathbb{R}$ .
22M.3.AHL.TZ2.2h.iii:
By writing $x^{4} - 9 x^{3} + 24 x^{2} + 22 x - 12$ as a product of one linear and one cubic factor, prove that the equation has at least one complex root.
19N.2.AHL.TZ0.H_10c.ii:
${\text{E}}\left( X \right)$ .
22M.1.AHL.TZ2.12b:
Given that $Re (z_{1} {z_{2}}^{*}) = 0$ , show that $Z_{1} {OZ}_{2}$ is a right-angled triangle.
22M.1.AHL.TZ2.12c.ii:
Hence show that ${z_{1}}^{2} + {z_{2}}^{2} = z_{1} z_{2}$ .
22M.1.AHL.TZ2.12e:
Consider the equation $z^{2} + a z + 12 = 0$ , where $z \in ℂ$ and $a \in ℝ$ .

Given that $0 < α - θ < π$ , deduce that only one equilateral triangle $Z_{1} {OZ}_{2}$ can be formed from the point $O$ and the roots of this equation.
19N.2.AHL.TZ0.H_10a:
Find, in terms of $a$ , the probability that $X$ lies between 1 and 3.
20N.2.AHL.TZ0.H_4:
Find the term independent of $x$ in the expansion of $\frac{1}{x^{3}} {(\frac{1}{3 x^{2}} - \frac{x}{2})}^{9}$ .
19M.2.AHL.TZ1.H_2a:
$r{{\text{e}}^{{\text{i}}\theta }}$ where $r$ , $\theta \in \mathbb{R}$ , $r > 0$ .
19N.2.AHL.TZ0.H_5:
Consider the expansion of ${\left( {2 + x} \right)^n}$ , where $n \geqslant 3$ and $n \in \mathbb{Z}$ .

The coefficient of ${x^3}$ is four times the coefficient of ${x^2}$ . Find the value of $n$ .
17M.1.AHL.TZ1.H_2a.i:
By expressing ${z_1}$ and ${z_2}$ in modulus-argument form write down the modulus of $w$ ;
EXN.1.AHL.TZ0.12c:
Use the identity from part (b) to show that the quadratic equation $x^{2} - 6 x + 1 = 0$ has roots ${cot}^{2} \frac{π}{8}$ and ${cot}^{2} \frac{3 π}{8}$ .
20N.2.AHL.TZ0.H_7a:
There are more males than females in the group.
17N.1.AHL.TZ0.H_8:
Determine the roots of the equation ${(z + 2{\text{i}})^3} = 216{\text{i}}$ , $z \in \mathbb{C}$ , giving the answers in the form $z = a\sqrt 3 + b{\text{i}}$ where $a,{\text{ }}b \in \mathbb{Z}$ .
19N.2.AHL.TZ0.H_8b:
the girls do not sit on either end.
19N.2.AHL.TZ0.H_10c.i:
$a$ .
EXN.1.AHL.TZ0.12b:
Use de Moivre’s theorem and the result from part (a) to show that $\cot 4 θ = \frac{\cot^{4} θ - 6 \cot^{2} θ + 1}{4 \cot^{3} θ - 4 \cot θ}$ .
20N.2.AHL.TZ0.H_7b:
Two of the teachers, Gary and Gerwyn, refuse to go out for a meal together.
21N.1.AHL.TZ0.12b:
Plot the points $A$ , $B$ and $C$ on an Argand diagram.
21N.1.AHL.TZ0.12d:
By using de Moivre’s theorem, show that $α = \frac{1}{1 - e^{i \frac{π}{6}}}$ is a root of this equation.
22M.1.AHL.TZ2.12a:
Show that $z_{1} {z_{2}}^{*} = r_{1} r_{2} e^{i (α - θ)}$ where ${z_{2}}^{*}$ is the complex conjugate of $z_{2}$ .
22M.3.AHL.TZ2.2e.i:
By varying the value of $q$ in the equation $x^{3} - 7 x^{2} + q x + 1 = 0$ , determine the smallest positive integer value of $q$ required to show that Noah is incorrect.
22M.3.AHL.TZ2.2d:
Using the result from part (c), show that when $q = 17$ , this equation has at least one complex root.
22M.3.AHL.TZ2.2g:
Use your result from part (f)(ii) to show that the equation $x^{4} - 2 x^{3} + 3 x^{2} - 4 x + 5 = 0$ has at least one complex root.
21N.1.AHL.TZ0.12e:
Determine the value of $Re (α)$ .
17M.1.AHL.TZ2.H_11c.i:
Find the modulus and argument of $z$ in terms of $\theta$ . Express each answer in its simplest form.
17M.1.AHL.TZ1.H_2a.ii:
By expressing ${z_1}$ and ${z_2}$ in modulus-argument form write down the argument of $w$ .
17M.1.AHL.TZ2.H_11b:
Show that $\sin 105^\circ + \cos 105^\circ = \frac{1}{{\sqrt 2 }}$ .
18N.1.AHL.TZ0.H_8:
Consider the equation ${z^4} + a{z^3} + b{z^2} + cz + d = 0$ , where $a$ , $b$ , $c$ , $d \in \mathbb{R}$ and $z \in \mathbb{C}$ .

Two of the roots of the equation are log₂6 and $i\sqrt 3$ and the sum of all the roots is 3 + log₂3.

Show that 6 $a$ + $d$ + 12 = 0.
16N.1.AHL.TZ0.H_12d:
Solve the inequality $\operatorname{Re} (pq) + 8 < {\left( {\operatorname{Im} (pq)} \right)^2}$ .
19N.2.AHL.TZ0.H_6:
Let $P\left( z \right) = a{z^3} - 37{z^2} + 66z - 10$ , where $z \in \mathbb{C}$ and $a \in \mathbb{Z}$ .

One of the roots of $P\left( z \right) = 0$ is $3 + {\text{i}}$ . Find the value of $a$ .
16N.1.AHL.TZ0.H_12a:
Determine the value of

(i) $1 + \omega + {\omega ^2}$ ;

(ii) $1 + \omega {\text{*}} + {(\omega {\text{*}})^2}$ .
17M.1.AHL.TZ1.H_2b:
Find the smallest positive integer value of $n$ , such that ${w^n}$ is a real number.
17M.1.AHL.TZ2.H_11c.ii:
Hence find the cube roots of $z$ in modulus-argument form.
19N.1.AHL.TZ0.H_5a:
Solve the equation, giving the solutions in the form $a + {\text{i}}b$ , where $a{\text{, }}b \in \mathbb{R}$ .
17M.1.AHL.TZ2.H_11a:
Solve $2\sin (x + 60^\circ ) = \cos (x + 30^\circ ),{\text{ }}0^\circ \leqslant x \leqslant 180^\circ$ .
21M.1.AHL.TZ1.7:
Consider the quartic equation $z^{4} + 4 z^{3} + 8 z^{2} + 80 z + 400 = 0, z \in ℂ$ .

Two of the roots of this equation are $a + b i$ and $b + a i$ , where $a, b \in ℤ$ .

Find the possible values of $a$ .
21N.1.AHL.TZ0.12a.i:
Verify that $ω_{1} = 1 + e^{i \frac{π}{6}}$ is a root of this equation.
21N.1.AHL.TZ0.12a.ii:
Find $ω_{2}$ and $ω_{3}$ , expressing these in the form $a + e^{i θ}$ , where $a \in ℝ$ and $θ > 0$ .
21N.1.AHL.TZ0.12c:
Find $AC$ .

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