Finding unknown coefficients of a polynomial, using the remainder theorem

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Christian Lawson-Perfect	said	Ready to use	7 years, 8 months ago
Elliott Fletcher	said	Needs to be tested	7 years, 8 months ago
Bradley Bush	said	Has some problems	7 years, 9 months ago

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Elliott Fletcher 7 years, 8 months ago

Published this.

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Christian Lawson-Perfect 7 years, 8 months ago

Gave some feedback: Ready to use

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Christian Lawson-Perfect 7 years, 8 months ago

Saved a checkpoint:

Good stuff!

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Elliott Fletcher commented 7 years, 8 months ago

Thanks for your feedback Brad,

I did try to make it so that s and t were always integers but i think there are so many numbers involved in the calculation that it would be very difficult to do. Instead i have restricted the values that the variables can take to make the fractions in the answers slightly nicer.

I see what you mean about the error message, i'm not sure how to fix this but i have said that you can leave your answer as a fraction in the question.

I have also split the question up into parts to make it more accessible to students with less algebraic skill.

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Elliott Fletcher 7 years, 8 months ago

Gave some feedback: Needs to be tested

○

Bradley Bush 7 years, 9 months ago

Gave some feedback: Has some problems

○

Bradley Bush commented 7 years, 9 months ago

Amazing question, most of my feedback revolves around the idea that I think it might be too hard for the student so I only have a few comments on this one:

I didn't find an example of when s and t weren't both fractions, I think this adds to the complexity and the challenge of the question but is this what we want to be testing? Could this be imporved through making the variables 'nicer' so the answers are whole numbers more often than not?

If you are leaving the answers as their more complex fraction form then you need to have a go at changing the error message that comes up. The answer gap still accepts a fractional answer if you type the fraction in full, however, if you're slow like me and you type "x/" and then pause for a second, you'll get an error message "your answer is invalid. You must enter an integer or a decimal. This is weird because when you then finish the expresion, the error message goes away. I am just thinking that this might be misdirective to see for anyone typing in their answer slowly. If you can't fix the error message, and you are keeping the fractional answers then it might be worth specifying that answers can be left as fractions in the question.

On that note, I might write "reduce your answer to simplist [fractional] form" instead of "reduce your answer to it's lowest terms"; or something along those lines.

I am actually thinking that the question still might be too hard for some students, making it inaccessible to the students who have less algebraic skill; even with the changes I suggested above. Maybe a solution of this is to create a copy of the question (labelling it 'easy version') and have the same question, just break it down into parts by asking students to give an expression for p(factor 1) then p(factor 2) then with the third part asking them to use simultaneous equations to find t and s.

Your parts and advice seem free of typos and formatting errors (I'm very jealous).

Feel free to ask if you don't understand any of what I've been getting at. High quality question.

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Elliott Fletcher 7 years, 9 months ago

Gave some feedback: Needs to be tested

○

Elliott Fletcher 7 years, 9 months ago

Created this.

Name	Status	Author	Last Modified
Finding unknown coefficients of a polynomial, using the remainder theorem	Ready to use	Elliott Fletcher	20/11/2019 14:39
Considere el polinomio $p(x)$ con parametros $t$ y $s$ . $p(x) = ax^3+sx^2+t$ , halle los valores de $s$ y $t$ .	Ready to use	Luis Hernandez	19/06/2023 13:40
Simon's copy of Finding unknown coefficients of a polynomial, using the remainder theorem	draft	Simon Thomas	28/02/2019 12:15
Christian's copy of Finding unknown coefficients of a polynomial, using the remainder theorem	draft	Christian Lawson-Perfect	20/05/2019 13:55
Finding unknown coefficients of a polynomial, using the remainder theorem	draft	Xiaodan Leng	10/07/2019 21:44
Finding unknown coefficients of a polynomial by the remainder theorem	Ready to use	Wan Mekwi	11/10/2020 21:25

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Question statement

Consider the polynomial

$p(x) = \simplify{{coef2_x3}x^3+s*x^2+{coef2_x}x+t}\text{.}$

The polynomial:

has remainder $\var{rem1}$ when divided by $(\simplify{x+{c}})$
has remainder $\var{rem2}$ when divided by $(\simplify{x+{d}})$

Give any introductory information the student needs.

			Name	Type	Generated Value

c	integer	-3
rem1	integer	3
d	integer	2
rem2	integer	3
coef2_x3	integer	1
coef2_x	integer	-2
numerator	number	-100
denominator	number	5
s	number	-20
x	number	9
y	number	4
t	number	27

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(-3..3 except [0,1])

JME function reference

Description

Dividing term 1.

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`

-3

→ Used by:

coef2_x
d
denominator
numerator
t
x

Statement
Advice
"Part a)" - prompt
"Part a)" → "Coefficient of s" - Minimum accepted value
"Part a)" → "Coefficient of s" - Maximum accepted value
"Part a)" → "Right-hand side" - Minimum accepted value
"Part a)" → "Right-hand side" - Maximum accepted value
"Part d)" → "Unnamed gap" - Minimum accepted value
"Part d)" → "Unnamed gap" - Maximum accepted value

Parts

Part a) Gap-fill
1. Gap Coefficient of s Number entry
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2. Gap Right-hand side Number entry
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1. Step Information only
Part b) Gap-fill
1. Gap Gap 0. Number entry
  Add
2. Gap Gap 1. Number entry
  Add
Part c) Gap-fill
1. Gap Number entry
  Add
1. Step Information only
Part d) Gap-fill
1. Gap Number entry
  Add
1. Step Information only

Gap-fill

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

Using the remainder theorem for the remainder when $p(x)$ is divided by $(\simplify{x+{c}})$ , create an equation involving $s$ and $t$ .

Coefficient of s $s + t$ = Right-hand side.

Advice

We are told that the polynomial:

has a remainder $\var{rem1}$ when divided by $(\simplify{x+{c}})$
has a remainder $\var{rem2}$ when divided by $(\simplify{x+{d}})$

a)

Firstly, substituting $x = \simplify{-{c}}$ into $p(x)$ gives us

$\begin{align} p(\simplify{-{c}}) &= \simplify[all,!collectNumbers, fractionnumbers]{{coef2_x3*(-c)^3}+{(-c)^2}s+{coef2_x*(-c)}+t},\\ &= \simplify[all,fractionnumbers]{{coef2_x3*(-c)^3}+{(-c)^2}s+{coef2_x*(-c)}+t}. \end{align}$

But, by the remainder theorem $p(\simplify{-{c}}) = \var{rem1}$ (using the first bullet point), so this becomes

$\begin{align} \simplify[all,fractionnumbers]{{coef2_x3*(-{c})^3}+s*{(-{c})^2}+{coef2_x*(-{c})}+t} &= \var{rem1},\\ \simplify[all,fractionnumbers]{s*{x}+t} &= \simplify[all,fractionnumbers]{{rem1}-{coef2_x3*(-{c})^3}-{coef2_x*(-{c})}}. \end{align}$

b)

Similarly, substituting $x = \simplify{-{d}}$ into $p(x)$ , gives us

$\begin{align} p(\simplify{-{d}}) &= \simplify[all,!collectNumbers, fractionnumbers]{{coef2_x3*(-{d})^3}+{(-{d})^2}s+{coef2_x*(-{d})}+t},\\ &= \simplify[all,fractionnumbers]{{coef2_x3*(-{d})^3}+{(-{d})^2}s+{coef2_x*(-{d})}+t}. \end{align}$

But, by the remainder theorem $p(\simplify{-{d}}) = \var{rem2}$ (using the second bullet point), so this becomes

$\begin{align} \simplify[all,fractionnumbers]{{coef2_x3*(-{d})^3}+s*{(-{d})^2}+{coef2_x*(-{d})}+t} &= \var{rem2},\\ \simplify[all,fractionnumbers]{s*{y}+t} &= \simplify[all,fractionnumbers]{{rem2}-{coef2_x3*(-{d})^3}-{coef2_x*(-{d})}}. \end{align}$

c)

We now have two simultaneous equations for $s$ and $t$ :

$\begin{align} \simplify[all,fractionnumbers]{s*{x}+t} = \simplify[all,fractionnumbers]{{rem1}-{coef2_x3*(-{c})^3}-{coef2_x*(-{c})}} \\ \simplify[all,fractionnumbers]{s*{y}+t} = \simplify[all,fractionnumbers]{{rem2}-{coef2_x3*(-{d})^3}-{coef2_x*(-{d})}} \end{align}$

Next, we subtract the second equation from the first equation.

This allows us to cancel out the terms involving $t$ and gives us an equation only in terms of $s$ , which we can then rearrange to find the value of $s$ .

Subtracting the two equations gives

$\simplify{s*{(-{c})^2-(-{d})^2}} = \simplify[all,fractionnumbers]{{rem1 - coef2_x3*(-c)^3-coef2_x*(-c)-rem2+coef2_x3*(-d)^3+coef2_x*(-d)}}.$

Then, we can rearrange this equation so that

$s = \simplify[all,fractionnumbers]{{rem1 - coef2_x3*(-c)^3-coef2_x*(-c)-rem2+coef2_x3*(-d)^3+coef2_x*(-d)}/{{(-c)^2-(-d)^2}}}.$

d)

We can calculate $t$ by substituting our value of $s$ into one of our original simultaneous equations. For example, let's use the equation

$\simplify[all,fractionnumbers]{s*{(-{d})^2}+t} = \simplify[all,fractionnumbers]{{rem2}-{coef2_x3*(-{d})^3}-{coef2_x*(-{d})}}.$

Substituting our value of $s$ into this equation gives us

$\begin{align} \simplify[all,fractionnumbers,!noleadingMinus]{{numerator/denominator}+t} &= \simplify[all,fractionnumbers]{{rem2-coef2_x3*(-d)^3-coef2_x*(-d)}},\\ t &= \simplify[all,fractionnumbers]{{rem2-coef2_x3*(-d)^3-coef2_x*(-d) - numerator/denominator}}. \end{align}$

This same answer would've also been obtained if we had substituted our value of $s$ into the other equation instead.

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Finding unknown coefficients of a polynomial, using the remainder theorem

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Elliott Fletcher 7 years, 8 months ago

Christian Lawson-Perfect 7 years, 8 months ago

Christian Lawson-Perfect 7 years, 8 months ago

Elliott Fletcher commented 7 years, 8 months ago

Elliott Fletcher 7 years, 8 months ago

Bradley Bush 7 years, 9 months ago

Bradley Bush commented 7 years, 9 months ago

Elliott Fletcher 7 years, 9 months ago

Elliott Fletcher 7 years, 9 months ago

Error in variable testing condition

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Warning

Generated value: `integer`

Parts

Gap-fill

a)

b)

c)

d)

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Finding unknown coefficients of a polynomial, using the remainder theorem

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Elliott Fletcher 7 years, 8 months ago

Christian Lawson-Perfect 7 years, 8 months ago

Christian Lawson-Perfect 7 years, 8 months ago

Elliott Fletcher commented 7 years, 8 months ago

Elliott Fletcher 7 years, 8 months ago

Bradley Bush 7 years, 9 months ago

Bradley Bush commented 7 years, 9 months ago

Elliott Fletcher 7 years, 9 months ago

Elliott Fletcher 7 years, 9 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

a)

b)

c)

d)

Generated value: `integer`