Roots of a cubic real polynomial

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Paul McFadden	said	Has some problems	2 years, 4 months ago
Bill Foster	said	Ready to use	6 years, 2 months ago

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Paul McFadden 2 years, 4 months ago

Gave some feedback: Has some problems

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Bill Foster 6 years, 2 months ago

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Newcastle University Mathematics and Statistics 9 years, 2 months ago

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Name	Status	Author	Last Modified
Roots of a cubic real polynomial	Has some problems	Newcastle University Mathematics and Statistics	18/01/2023 08:23
Luis's copy of Roots of a cubic real polynomial	draft	Luis Hernandez	30/11/2018 17:44
Maria's copy of Roots of a cubic real polynomial	draft	Maria Aneiros	25/05/2019 08:24
Roots of a cubic real polynomial	draft	Xiaodan Leng	10/07/2019 22:40
Roots of a cubic real polynomial	draft	JPO AddMath	16/09/2019 21:58
Harder practice Q - roots of a cubic	Ready to use	Shaheen Charlwood	27/10/2022 18:59

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Question statement

From the list of complex numbers, choose the one which is a root of the given equation $f(z)=0$ , and hence find all roots.

Give any introductory information the student needs.

			Name	Type	Generated Value

tx3a1	integer	-2
f1	integer	8
tx3a2	integer	-1
b2	integer	3
tx2a2	integer	-7
x1b2	integer	-6
x1b1	integer	5
b1	integer	8
tx1a1	integer	-2
tx2a1	integer	-8
xs2	integer	-1
xs3	integer	-1
xs1	integer	1
x3a1	integer	-2
x3a2	integer	-1
s3	integer	-1
s2	integer	1
s1	integer	-1
s6	integer	-1
s5	integer	1
s4	integer	-1
x2a1	integer	-8
x2a2	integer	-7
x1a2	integer	-7
x1a1	integer	-2
a1	integer	-6
a2	integer	-2
z4	number	-2 - 6i
c2	integer	-2
c1	integer	-5
z1	number	-6 + 8i
z2	number	-2 + 5i
z3	number	-8 - 9i
x3b1	integer	-6
x3b2	integer	6
x2b1	integer	-9
x2b2	integer	-9
tx1a2	integer	-7

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
s4*random(1..9)

JME function reference

Description

Describe what this variable represents, and list any assumptions made about its value.

Can an exam override the value of this variable?

Generated value: `integer`

-2

← Depends on:

→ Used by:

x3a1

This variable doesn't seem to be used anywhere.

Parts

Part a) Gap-fill
1. Gap Choose one from a list
  Add
Part b) Gap-fill
1. Gap Gap 0. Mathematical expression
  Add
2. Gap Gap 1. Mathematical expression
  Add

Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

Given $\displaystyle f(z) = \simplify[std]{z ^ 3 + {( -2) * a1 -c1}*z ^ 2 + {2 * a1 * c1 + a1 ^ 2 + b1 ^ 2} * z -{c1 * (a1 ^ 2 + b1 ^ 2)}}$ , one of the following complex numbers is a root $z_1$ of the equation $f(z)=0$ .

Choose the correct value for $z_1$ :Unnamed gap

Advice

a) Finding a root.

In order to find which one of the four choices is a root you have to evaluate $f(z)$ for each choice.If you find for a choice of $z$ that $f(z)=0$ then that choice of $z$ is a root of the equation.

Note that

$\begin{eqnarray*} \simplify{f({z1})} &=&\simplify[std]{{z1}^3+{-2*a1 -c1}*{z1} ^ 2 + {2 * a1 * c1 + a1 ^ 2 + b1 ^ 2} * {z1} -{c1 * (a1 ^ 2 + b1 ^ 2)}}\\ &=&\simplify[std]{{z1^3}+{-2*a1 -c1}{z1 ^ 2} + {2 * a1 * c1 + a1 ^ 2 + b1 ^ 2} * {z1} -{c1 * (a1 ^ 2 + b1 ^ 2)}}\\ &=&\simplify[std]{{z1^3}+{(( -2) * a1 -c1)*z1^2}+ {(2 * a1 * c1 + a1 ^ 2 + b1 ^ 2)*z1}-{c1 * (a1 ^ 2 + b1 ^ 2)}}\\ &=&0 \end{eqnarray*}.$ So of the list of choices

$z_1=\var{z1}$ is a root.

b) The other roots

Now that you have found a complex root it is very simple to find another complex root.

Since $f(z)$ is a polynomial with real coefficients then if $z=z_0$ is a root we have that the conjugate $z=\overline{z_0}$ is also a root.

Hence the complex number $z_2=\overline{\var{z1}}=\var{conj(z1)}$ is a root.

To find the real root $z_3=c$ we note that the constant term of

$f(z) =(z-z_1)(z-z_2)(z-c)$ is

$-z_1z_2c = -(\var{z1})(\var{conj(z1)})c=\var{-z1*conj(z1)}c$ .

But we know that the constant term of $f(z)$ is $\simplify{-{c1 * (a1 ^ 2 + b1 ^ 2)}}$ .

Hence

$\begin{eqnarray*} \var{-z1*conj(z1)}c &=&\simplify{-{c1 * (a1 ^ 2 + b1 ^ 2)}}\\ \Rightarrow c &=& \simplify[]{{c1 * (a1 ^ 2 + b1 ^ 2)}/{abs(z1^2)}}\\ &=&\var{c1} \end{eqnarray*}$

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Paul McFadden 2 years, 4 months ago

Bill Foster 6 years, 2 months ago

Newcastle University Mathematics and Statistics 9 years, 2 months ago

Error in variable testing condition

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Generated value: `integer`

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Roots of a cubic real polynomial

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Paul McFadden 2 years, 4 months ago

Bill Foster 6 years, 2 months ago

Newcastle University Mathematics and Statistics 9 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`