Identifying common errors in polynomials of eighth grade students

International Journal of Evaluation and Research in Education (IJERE)
Vol. 13, No. 1, February 2024, pp. 57~68
ISSN: 2252-8822, DOI: 10.11591/ijere.v13i1.25131  57
Journal homepage: http://ijere.iaescore.com
Identifying common errors in polynomials of eighth grade
students
Thayarat Ekamornaroon1
, Parinya Sa Ngiamsunthorn2
, Mingkhuan Phaksunchai1
,
Ratchanikorn Chonchaiya2
1
Science Education Program, Faculty of Science, King Mongkut’s University of Technology Thonburi, Bangkok, Thailand
2
Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi, Bangkok, Thailand
Article Info ABSTRACT
Article history:
Received Aug 7, 2022
Revised Mar 18, 2023
Accepted May 1, 2023
This research aims to study and classify errors in polynomials made by
secondary school students. The data for error identification was collected
from exercise books of 72 eighth grade students. Three types of errors were
examined: careless, computational, and conceptual errors. The errors were
considered according to four topics in polynomials: similar terms of
monomials; addition of polynomials; subtraction of polynomials; and
multiplication of polynomials. It is found that students made the highest
computational errors in identifying monomials’ similarity, which accounts
for 17.86%. They have the highest percentage of making computational
errors in the addition and subtraction of polynomials, which account for
10.88% and 12.04%, respectively. Lastly, they have the highest percentage
of making careless errors in the multiplication of polynomials, which
accounts for 14.44%. Furthermore, it can be seen that the source of errors is
learners’ carelessness when writing the question and its answer. In addition,
the basic knowledge of computing addition, subtraction, and multiplication
of integers is the most crucial factor that leads to incorrect answers.
Nevertheless, most students understand the principle of polynomials, but
frequently make errors on other issues.
Keywords:
Careless error
Computational error
Conceptual error
Polynomials
Secondary students
This is an open access article under the CC BY-SA license.
Corresponding Author:
Parinya Sa Ngiamsunthorn
Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi
126 Pracha Uthit Rd, Bang Mot, Thung Khru, Bangkok 10140, Thailand
Email: parinya.san@kmutt.ac.th
1. INTRODUCTION
Mathematics plays an essential role in constructing 21st-century skills [1], and it is intimately
related to the details of everyday human life and activities [2]. Nowadays, mathematics is a necessary subject
and a crucial requirement in every field [3], developing human thinking to be more creative, logical, and able
to analyze problems or conditions extensively. Furthermore, mathematics can assist people in anticipating,
planning, deciding, and appropriately solving any problems encountered in daily life [4], [5]. Because
mathematics is both abstract and concrete, most students find it challenging to comprehend when solving
mathematics problems since it contains several rules, formulas, and definitions that are unrelated to life.
These are causes that make students lack understanding and may generate difficulties that might give rise to
errors [6]. Therefore, it is crucial for mathematics teachers to assist learners to learn from their errors and
mistakes [7] to improve their understanding of the higher levels of mathematics.
Learning mathematics can be defined as a learning process in which students actively engage in
constructing mathematical knowledge [8] and combine various abilities to master a variety of mathematical
concepts to apply them to solve problems in everyday life [9]. Every mathematical knowledge is essential for

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Int J Eval & Res Educ, Vol. 13, No. 1, February 2024: 57-68
58
students’ understanding of the contents of higher concepts in mathematics [10] or other subjects. Many
factors can influence students’ mathematical learning success [11]. Capuno et al. [12] discovered that
students have a positive attitude toward mathematics, which is one of the elements that indicate the students’
learning achievement in mathematics, and that students have confidence in focusing more attention on one
subject than on others [13]. In contrast, there are factors that cause students to struggle with learning, which
is an impediment to mathematical achievement. Both internal factors such as problem-solving, intelligence,
learning motivation, subject mindset, and personality, and external factors like a teacher's performance,
family support, campus environment, and learning methods all have an impact [14].
For all levels of mathematics education, algebra is necessary for achieving success in all branches of
mathematics [15]. It is one of the most essential topics in mathematics because its application is very
fundamental and integrated into other mathematical concepts [16]. In learning abstract concepts, students’
acquisition of abstract thinking ability is essential [17], and they would like to make those concepts more
concrete to be simple to understand. Furthermore, algebra concepts help students to form the relationship
between numbers and real-life applications [18]. On the other hand, algebra is one of the primary topics in
mathematics that students commonly make errors [19] and is difficult to understand which has an impact on
their ability to apply concepts in other topics and subjects [20].
Polynomials are also one of the topics in algebra, they are material that requires reasoning and is not
enough to just memorize it. Students must understand the concepts of the formulas presented so that they can
solve the problem well and, in the end, can be seen as a picture of student understanding [21]. If students do
not understand the fundamental concept, it will have an impact on their ability to understand higher-level
content, such as factoring polynomials, which will require them to apply addition and multiplication of
polynomials to solve problems. Consequently, polynomials are essential for pupils to comprehend at a
fundamental level to learn at a higher level with ease.
When students want to solve problems in mathematics, they need to have two types of knowledge,
namely, procedural and conceptual knowledge. Procedural knowledge involves the ability to carry out
memory of definitions, rules, principles and procedures in mathematics, and to utilize them when solving
problems without substantially understanding of them, while conceptual knowledge refers to mathematical
concepts and interconnected components of mathematical knowledge which contribute to an understanding
of mathematical concepts, rules, and propositions [22]. It is important for students to be able to relate
conceptual understanding to procedural skills. In particular, students are expected to explain the concept and
choose steps that will be applied for solving mathematical problems [23].
Many researchers recognize that these two types of mathematical knowledge are useful in learning
and aid student comprehension. On the other hand, school teachers tend to be more concerned with acquiring
accurate answers using rules or procedures than with the concept about why and how procedures work [24].
In other words, they emphasize procedural rather than conceptual knowledge when teaching mathematics.
For instance, some teachers prepare several exercises as a repetitive process to demonstrate whether students
understand the content. Unfortunately, students with only procedural knowledge cannot solve real-world
situations as a result of a lack of conceptual knowledge. They cannot make the connections between the
concepts and the problem-solving situations [25], which corresponds to the findings in previous studies [26],
[27] that the process of describing and justifying solutions for accurate and inaccurate examples is more
valuable for attaining learning outcomes than describing and justifying solutions for the accurate solutions
only. Teaching with a focus only on procedural skills will diminish learning in the classroom. It is not
adequate to provide students with mathematical skills for the future [24]. Therefore, they can easily forget
and may not be aware when they make a mistake.
If there is a problem with some knowledge, problem-solving errors may occur. According to
Pomalato et al. [28], mistakes are a word used in science and mathematics to describe systematic, consistent,
or unintentional deviations from an accurate value, which corresponds to Riantini et al. [29], in which an
error is described as an alteration from the real solution of a problem. As defined in mathematics, an error is a
deviation from the correct solution to a problem. Errors can also be observed in incorrectly answered
problems from students’ learning styles when solving a mathematical problem [30]. It impedes mathematics
learning as it prevents students from achieving their learning goals. These errors can occur for some students.
According to Luneta and Makonye [31], errors in mathematics could be caused by various factors, including
carelessness, a lack of concentration, or a pattern of mistakes. Errors do not occur regularly, but they can
occur through the existing basic knowledge. The answer may be wrong, but students can improve, or
correctness can be achieved more readily.
Many researchers have divided types of errors from different perspectives. Baidoo [32] studied
types of errors in algebraic fractions and identified students’ errors in four types. These four types are:
i) Mathematical language errors which result from a learner’s lack of comprehension of mathematical
technical jargon, a misunderstanding of how to use operation symbols, and a misunderstanding of how to use

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Identifying common errors in polynomials of eighth grade students (Thayarat Ekamornaroon)
59
letters; ii) Procedural errors which occur when rules and formulas are mixed up when solving mathematical
problems; iii) Concept errors which occur when learners connect their understanding to a topic that they do
not comprehend its principles or qualities; and iv) Application errors which occur when learners understand
the concept but are unable to utilize it correctly to solve problems. According to Agustyaningrum et al. [33],
there are three sorts of errors: i) Careless errors which occur when students are not paying attention or
working too quickly in mathematics; ii) Computational errors which arise when students are unable to
identify a sign, digit, or place value, or when they utilize the wrong formula; and iii) Conceptual errors which
occur from misconceptions about the fundamental principles and concepts associated with the mathematical
problem.
In another study [34], errors in mathematics made by students are categorized into three categories,
including: i) Factual mistakes which are errors that occur when students are not mastering in a fact required
to solve the problem; ii) Procedural mistakes which are errors occurs when students inaccurately apply
mathematical operations; iii) Conceptual mistakes which are errors caused by misunderstandings or
misconceptions about the theories and concepts related to the problem. Oktaviani [35] identified four types of
errors as: i) Conceptual errors which are made when students have a limited understanding of mathematical
concepts and misconceptions; ii) Procedural errors which occur when students work in the wrong order;
iii) Factual errors, also called, computational errors, which happen when students are unable to identify a
sign, a digit, and a place value, or when they employ the wrong formula; and iv) Careless errors occurring
when students are not paying concentration or working too quickly in mathematics. In study by Makhubele,
Nkhoma, and Luneta [36], three types of errors are identified which are: i) Slips, little blunders made by
students who are in a hurry; ii) Conceptual errors which occur when learners lack conceptual understanding
as a result of a lack of comprehension of basic concepts, facts, and skills; and iii) Procedural errors occurring
when student understand a concept but are unable to utilize it to solve a problem. They carry out the
calculation without fully comprehending what they are doing.
According to Herholdt and Sapire [37], error analysis is performed to find interpretations for the
reasoning of errors and mistakes formed in learners’ work. Besides, it can assist teachers in identifying
students' weaknesses, allowing them to identify problems with any topic and solve them to get on point,
allowing students to learn mathematics in a simple and more effective way. Moreover, error analysis is the
process of reviewing errors in order to provide feedback and remedial instruction to enhance learning and
performance [38]. According to Lee [39], the analysis of students’ work from worksheets and exercise books
would help teachers to understand the students’ process of understanding and problems with conceptual
understanding in mathematics. It is critical to provide opportunities for students to practice, review, or
reinforce the material already covered in the class, and determine whether they have comprehended the
materials and have achieved the expected learning outcomes [40]. Furthermore, it allows students to direct
their own learning and select how and where to apply assigned tasks [41]. These errors will occur when
students complete the task, and teachers should check virtually all of them for causes of errors.
As polynomials is a basic concept for consequent topics in mathematics and is one of the topics that
students frequently make mistakes, it is crucial to identify common errors made by students. This research
aims to answer the following questions: i) What are common errors made by students in polynomials
according to similar terms of monomials, addition of polynomials, subtraction of polynomials, and
multiplication of polynomials?; ii) Which type of errors among careless error, computational error, and
conceptual error is the most common errors made by students in polynomials? The main contribution in this
paper is that the errors identification and classification were considered in detail for each topic, which is in
sequential, including similar terms of monomials, addition of polynomials, subtraction of polynomials, and
multiplication of polynomials. Therefore, it could help instructors to be aware of difficulties that would affect
students’ understanding of polynomials. For that reason, the researchers are interested in analyzing and
classifying errors on polynomial topics with a focus on mistakes made by students while working in their
exercise books.
2. RESEARCH METHOD
2.1. Research design
This research employs quantitative approaches to examine student errors in polynomials. Students’
comprehension provides the required conditions for learning a higher level of mathematics. Therefore, an
error assessment is required to effectively enhance the teaching at the next chance and help students to be
able to apply the knowledge at a higher level of education. The concept of quantitative approaches used in
this research can be summarized in Figure 1.

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2.2. Sample
A total of 97 eighth grade students from a school in Bangkok, Thailand was used to create a sample.
A purposive sampling strategy was used to select the sample, which included 72 students in eighth grade of
the English and Mini-English programs. The sample consisted of students who had already studied
polynomials and had an understanding of the mathematical terms used in the test.
Figure 1. Quantitative approach for identifying errors
2.3. Data collection
Students’ work from an exercise book was collected for error analysis on polynomials. There were
several patterns in each exercise, such as true or false questions for similar terms, and open-ended question
requiring solution details for addition, subtraction, and multiplication of polynomials as indicated in Table 1.
Students' mistakes for each topic were collected and categorized into three types of errors, after which the
percentage of students with an incorrect answer was calculated. For students whose mistakes differed from
others, an interview was used to acquire the reason for mistakes found in each topic, which is also used for
the discussion to support the data analysis.
To analyze mistakes found in students’ exercise books, the researchers categorized mistakes into
three types of errors as careless error, computational error, and conceptual error [34]. Their definitions and
examples are given in Table 2. This process is carried out for each topic in polynomials.
Table 1. The topic for error analysis and type of questions in polynomials
No Topic Type of question
1 Similar terms of monomials True-false
2 Addition of polynomials Open-ended question
3 Subtraction of polynomials Open-ended question
4 Multiplication of polynomials Open-ended question
Table 2. Type of errors
Types of errors Definition Example
Careless error Students lack concentration and
are careless from working too
fast when doing mathematics.
Find the sum of 4a3
– 8a2
+ 5a + 3 and 6a3
+ 7a2
– 5.
4a3
– 8a2
+ 5a + 3
6a3
+ 7a2
– 5
10 – a2
+ 5a – 2
It shows that students do not write the term a3
because they are careless in
writing the answer.
Computational
error
Students know the concept, but
they make mistakes when
adding, subtracting, multiplying,
or dividing.
Find the difference of 9 + 3x – 7x2
and 5x2
– 13x.
9 + 3x – 7x2
– 13x + 5x2
9 + 16x – 2x2
The example shows that students make mistakes in the calculation of the
coefficient of x2
.
Conceptual error Students have a poor
understanding of the principles
and ideas connected to
mathematical concepts or cannot
apply their knowledge correctly
to mathematical concepts.
(6x + 2)(3x2
– 5)
= 18x3
– 30x + 6x2
– 10
= 18x3
+ 6x2
– 40
The example shows that students add -30x and -10 together to get the result
is -40 but these two terms are not similar terms so they cannot be added
together which is not consistent with the principle of adding two monomials.
Group
Addition of polynomials
Students’ work from
exercise book
Multiplication of polynomials
Mistakes
Percentage
Three types of errors
Subtraction of polynomials
Similar terms of monomials
+
–

61
2.4. Data analysis
In this research, a quantitative approach is used to analyze and classify mistakes and errors. The
mistakes made by students are collected for each topic in polynomials. For the topic of similar terms, there
are seven common mistakes, called ST1-ST7, which are classified as careless error, computational error or
conceptual error shown in Table 3. There were two careless errors, two computational errors and three
conceptual errors. For each item, the percentage of students with an incorrect answer is subsequently
calculated.
As shown in Table 3, the mistake ST1 shows that students thought that two monomials 𝑥𝑦 and 𝑦𝑥
have different variables, so they answered that the two monomials are not similar. Secondly, the mistake ST2
indicates that students thought that when two monomials have different coefficients, it means the monomials
are not similar terms. Thirdly, the mistake ST3 shows that students thought that two monomials are not
similar because they are in different forms. Fourthly, the mistake ST4 found that students thought that two
monomials have the same variables: 𝑥 and 𝑦, and they concluded that two monomials are similar terms.
Next, the mistake ST5 shows that students thought that two constants which are different types of
numbers, such as 15 is a positive number and –10 is a negative number, are not equal numbers, so they
answered that these two constants are not similar terms of monomials. Sixthly, the mistake ST6 shows that
students answered that these two monomials are similar terms because they have the exponent of the
variables more than 1 even though the exponent of the same variable is not equal. It makes students confused
as to whether the two monomials are similar or not. Lastly, for the mistake ST7, it is found from interviews
that students looked at the exponent of variables, so they concluded that two monomials are similar terms.
Table 3. Type of mistakes and errors in similar terms
Topic in
polynomials
Type of errors Mistakes Code Example of mistakes
Similar terms Careless Two monomials with the same variable and exponent
but the different positions are similar terms.
ST1 Students answer 9𝑥𝑦 and
−15𝑦𝑥 are not similar
terms.
Two monomials that have the same variable but have
different coefficients that are similar terms of
monomials.
ST2 Students answer −𝑥3
𝑦2
and −3𝑥3
𝑦2
are not similar
terms
Computational Two monomials that have different forms but once
simplified are similar terms of monomials.
ST3 Students answer −3𝑥𝑦𝑧
and
−5𝑥2𝑦2𝑧2
𝑥𝑦𝑧
are not similar
terms.
Two monomials that have different forms but once
simplified are not similar terms of monomials.
ST4 Students answer
6𝑥3𝑦4
𝑥𝑦
and
7𝑥𝑦3
are similar terms.
Conceptual Two monomials are constants. ST5 Students answer 15 and
−10 are not similar terms.
Two polynomials that have the same variables, but the
exponent of the same variable is not equal are not
similar terms of a monomial.
ST6 Students answer 16𝑎2
𝑏
and 61𝑎𝑏2
are similar
terms.
Two monomials that have the same exponent, but
different variables are not similar terms of a monomial.
ST7 Students answer 4𝑧2
and
5𝑦2
are similar terms.
For the topic of addition of polynomials, there are five common mistakes, called A1-A5, which are
classified as careless error, computational error or conceptual error shown in Table 4. There were three
careless errors, one computational error and one conceptual error. For each item, the percentage of students
with an incorrect answer is subsequently calculated.
As presented in Table 4, in the topic of addition of polynomials, students made several mistakes.
The mistake A1 shows that students subtracted two polynomials instead of adding two polynomials because
they did not know whether two polynomials should be added or subtracted. The question uses the comma
sign between two polynomials without providing the mathematical operator, so students got confused about
the operation of two polynomials and gave an incorrect answer. The mistake A2 shows that students wrote
the term in the dividend polynomial by writing 5𝑥3
instead of −5𝑥3
because students were careless in doing
their work. They incorrectly wrote the coefficient of 𝑥3
so the result came out incorrectly.
The mistake A3 shows that students made an incorrect addition of the term 4𝑎3
+ 6𝑎3
. The correct
answer is 10𝑎3
but students wrote only 10 because they were careless in writing their answers. The answer
after simplifying is thus not correct. The mistake A4 shows that students were wrong in the calculation for
adding the coefficients of 𝑎2
. The correct answer of −8𝑎2
+ 7𝑎2
is −𝑎2
, but students got the answer 𝑎2
so
that the answer given is not correct. The mistake A5 shows that students did not add two similar terms which

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led to the wrong answer as −14𝑎𝑏3
. This deviates from the polynomial addition principle where two similar
terms are added into a simplified form.
Table 4. Type of mistakes and errors in addition of polynomials
Topic in polynomials Type of errors Mistakes Code Example of mistakes
Addition Careless Students misunderstood that the
two polynomials were subtracted
despite being added.
A1 Find the sum of 3xy – 4xz + 6xyz and
-7xyz – 5xy – 10xz !
Answer: 3xy – 4xz + 6xyz
5xy + 10xz + 7xyz
8xy + 6xz + 13xyz
Students used wrong polynomials
in calculation.
A2 Find the sum of 3 – 2x2
– 5x3
+ 6x and
7x2
+ 4x !
Answer: 5x3
– 2x2
+ 6x + 3
+ 7x2
+ 4x
5x3
+ 5x2
+ 10x + 3
Students wrote incorrect answers. A3 Find the sum of 4a3
– 8a2
+ 5a + 3 and
6a3
+ 7a2
– 5
Answer: 4a3
– 8a2
+ 5a + 3
6a3
+ 7a2
– 5
10 – a2
+ 5a – 2
Computational Students were unable to determine
the sum of coefficients of two
similar terms when combining
two polynomials.
A4 Find the sum of 4a3
– 8a2
+ 5a + 3 and
6a3
+ 7a2
– 5
Answer: 4a3
– 8a2
+ 5a + 3
6a3
+ 7a2
– 5
6a3
+ 1a2
– 2
Conceptual Students made a mistake when
calculating two not similar terms.
A5 Find the sum of a3
b – 10ab3
– a4
7b4
and
-5ab3
+ 3a3
b – 4b4
Answer: a3
b – 10ab3
– a4
7b4
3a3
b – 5ab3
– 4b4
8a3
b – 14ab3
– a4
7b4
For the topic of subtraction of polynomials, there are five common mistakes, called S1-S5, which
are classified as careless error, computational error or conceptual error shown in Table 5. There were four
careless errors and one computational error. For each item, the percentage of students with an incorrect
answer is subsequently calculated.
Table 5. Type of mistakes and errors in subtraction of polynomials
Subtraction Careless Students misunderstood that the
two polynomials should be added
even when they should be
subtracted.
S1 Find the difference of 6a2
+ 8a – 5 and
8 + 9a – 7a2
Answer: 6a2
+ 8a – 5
– 7a2
+ 9a + 8
–a2
+ 17a + 3
Students used wrong polynomials
in calculation.
+ 3 and
9a2
– 5a + 4
Answer: 8a2
+ 3
– 9a2
– 4
– a2
– 1
Students wrote incorrect answers. S3 (4x2
+ 6y + 9) – (7y + 3) – (5x2
– 9y + 1)
Answer: 4x2
+ 6y + 9
7y + 3
5x2
– 9y + 1
–x2
– y + 5 + 9x
Students wrote the opposing
polynomials incorrectly.
+ 8a – 5 and
8 + 9a – 7a2
Answer: 6a2
+ 8a – 5
– 7a2
– 9a – 8
– a2
– a – 13
Computational Students were unable to calculate
the difference in coefficients of two
similar terms when subtracting two
polynomials.
S5 Find the difference of 9 + 3x – 7x2
and
5x2
– 13x
Answer: 9 + 3x – 7x2
– 13x + 5x2
9 + 16x – 2x2
+
+
+
+
+
+
+
-
+
-

63
As shown in Table 5, in the topic of subtraction of polynomials, the mistake S1 shows that students
added two polynomials instead of subtracting because students thought that the previous question was adding
polynomials and they were careless in doing their work. Thus, they considered adding two polynomials when
they should subtract them, so the answer is not correct. The mistake S2 shows that students wrote the
opposite polynomials incompletely as −9𝑎2
− 4 instead of −9𝑎2
+ 5𝑎 − 4, because they were careless in
doing their work. This resulted in a mistake after subtracting incomplete terms.
The mistake S3 shows that students wrote the wrong answer because they were careless in writing,
making the wrong variable in their answer from 𝑦 to 𝑥. By doing so, they were unable to add the term to get
the correct answer that is −𝑥2
+ 8𝑦 + 5. The mistake S4 shows that students wrote the sign of the opposite
term incorrectly because they were careless when writing the sign of the subtrahend, making a mistake for
subtracting two polynomials. Indeed, students made an error in writing the opposite sign for the coefficient of
𝑎2
as −8 − 9𝑎 − 7𝑎2
instead of the correct term −8 − 9𝑎 + 7𝑎2
, which contributes to mistake in
subtraction. The mistake S5 shows that students subtracted the coefficient of 𝑥2
incorrectly. The correct
answer is −7𝑥2
− 5𝑥2
= −12𝑥2
, but the students’ answer is −7𝑥2
− 5𝑥2
= −2𝑥2
which is the incorrect
answer.
For the topic of multiplication of polynomials, there are five common mistakes, called M1-M5
which are classified as careless error, computational error or conceptual error shown in Table 6. There was
one careless error, three computational errors and one conceptual error. For each item, the percentage of
students with an incorrect answer is subsequently calculated.
Table 6. Type of mistakes and errors in multiplication of polynomials
Multiplication Careless Students wrote incorrect
answers.
M1 (6x + 2)(3x2
– 5)
= 18x3
– 30x + 6x2
– 10
= 18x3
+ 6x2
– 30 – 10
Computational Students could not multiply
two monomial coefficients
accurately.
M2 (5x2
– 3x)(2x2
+ 6x – 9)
= 10x4
+ 30x3
– 45x2
+ 6x3
+ 18x2
– 27x
= 10x4
+ (30x3
+ 6x3
) + (–45x2
+ 18x2
) + (– 27x)
= 10x4
+ 36x3
– 27x2
– 27x
Computational Students multiplied two
polynomials incorrectly using
the wrong indices properties.
M3 (6x + 2)(3x2
– 5)
= 18x2
– 30x + 6x2
– 10
= 24x2
– 30x – 10
Students were unable to add
two similar terms after
multiplying two polynomials.
M4 (5x2
– 3x)(2x2
+ 6x – 9)
= 10x4
+ 30x3
– 45x2
– 6x3
– 18x2
+ 27x
= 10x4
+ 24x3
– 57x2
+ 27x
Conceptual Students did add two
unsimilar terms.
M5 (6x + 2)(3x2
– 5)
= 18x3
– 30x + 6x2
– 10
= 18x3
+ 6x2
– 40
From Table 6, the mistake M1 shows that students wrote some terms in the answer incorrectly
because they were careless in writing the term −30𝑥 to −30. So, the incorrect answer was given. The
mistake M2 shows that students multiplied the term −3𝑥 ⋅ 2𝑥2
incorrectly. The correct answer is −6𝑥3
, but
they got 6𝑥3
and gave an incorrect answer. The mistake M3 shows that students multiplied the term 6𝑥 ⋅ 3𝑥2
incorrectly. The correct answer is 18𝑥3
; nevertheless, they got an incorrect answer as 18𝑥2
which came from
an error in applying the property of indices 𝑎𝑚
⋅ 𝑎𝑛
= 𝑎𝑚+𝑛
. The mistake M4 shows that students made a
mistake in subtracting the coefficient of 𝑥2
. The correct answer is −45𝑥2
− 18𝑥2
= −63𝑥2
but they got
−57𝑥2
which is an incorrect answer. The mistake M5 shows that students added two non-similar terms
which were 30x
− and 10
− , and gave the answer −40. They made mistakes in adding −30𝑥 and −10 which
are not similar terms. Indeed, the two terms cannot be combined since they are not similar and do not
correspond with the principle of adding monomials, so the answer is not correct.
3. RESULTS AND DISCUSSION
3.1. Results
Based on students’ work collected from the exercise books, the number of students who made
mistakes is recorded for each incorrect item listed in Table 3 to Table 6. Then the percentage of incorrect
answers from the total number of students is calculated for each topic of polynomials according to the three
categories of errors, namely, careless error, computational error, and conceptual error. The average
percentages of incorrect answers from the total number of students for each topic according to three types of
errors are presented in Table 7.

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Table 7. Average of percentage of errors in learners’ exercise book in polynomials
No Topic in polynomials Mistakes Type of errors Percentage of incorrect
1 Similar terms ST1, ST2 Careless 5.98
ST3, ST4 Computational 17.86
ST5, ST6, ST7 Conceptual 4.44
2 Addition A1, A2, A3 Careless 4.78
A4 Computational 10.88
A5 Conceptual 0.46
3 Subtraction S1, S2, S3, S4 Careless 4.80
S5 Computational 12.04
4 Multiplication M1 Careless 14.44
M2, M3, M4 Computational 6.11
M5 Conceptual 0.83
3.2. Discussion
It can be seen from Table 7 that computational error is the main source for mistakes in the topics of
similar terms of monomials, addition, and subtraction of polynomials. However, for multiplication of
polynomials, students tend to have careless errors more frequently. Further comments and reasons for errors
made by students can be discussed.
3.2.1. Topic 1: similar terms
In similar terms of monomials, it can be seen that 17.86% of all students made computational errors
as they used the properties of indices to simplify terms incorrectly. It caused them to believe that two
monomials are not similar terms, even though they are. Conceptual errors were found in 4.44% of the total
students. This can be divided into three categories. The first group is the students who misunderstand that
two constants are not similar terms because they are different types of numbers, which corresponds to Seng
[42] that discussed about two constants for positive and negative integers. After all, they were confused
between the concept of negative integers and similar terms. The second group consists of students who
incorrectly assume that two monomials with the same number of variables but different exponents are similar
terms of a monomial. The result corresponds to [42], which found that students did not perceive the concept
of like terms, and they made a common blunder of similar terms by comparing their coefficients rather than
their variables. Two monomials are indeed similar if and only if the exponent of the same variables is equal.
The last group is the students who believe that two monomials with the same exponent but different
variables are similar terms of monomials. Because they think that if any two monomials have the same
exponent, the two monomials are similar terms. Then they immediately give answers which correspond to
Ancheta and Subia [43], students did not recognize that the variables must be the same and that the
corresponding exponent must also be the same for the terms to be similar. However, two monomials must
have the same variables before considering the exponent of the same variables. This error may hinder
specifying that two monomials are not similar terms. Finally, 5.98% of all students made a careless error
when comparing any two monomials that are similar.
3.2.2. Topic 2: addition of polynomials
For the error analysis from students’ work in the topic of addition of polynomials, it is discovered
that 10.88% of all learners made computational errors because they added two integers by adding the
coefficient of two similar terms incorrectly. This result corresponds to Makonye and Hantibi [44] that
students made errors in addition between negative and positive numbers. For example, they added 65 45
+
by assigning a subtraction sign. It may occur when they do not correctly understand the principles and rules
about the operation between positive and negative integers [45]. Second, on average, 4.78% of students made
careless errors because they hurried through their assignments. In other words, they already knew the answer
but were negligent in obtaining it, resulting in an incorrect answer. Finally, it is learned that 0.46% of
learners made conceptual errors by combining two non-similar terms, failing to follow the proper principle.
This result agrees with Ferrer [46] that the students added 100𝑥3
– 5𝑥 + 93 for a total of 98𝑥3
. Since the
terms are not similar, this expression cannot be combined. Students frequently mistakenly believe 𝑥3
and 𝑥 to
be similar terms, although they include different exponents, making them different terms. To summarize,
most students comprehend the principle of adding two polynomials, but they make a mistake that is not
involved in polynomials.
3.2.3. Topic 3: subtraction of polynomials
In the topic of subtraction of polynomials, it is found that 12.04% of students made computational
errors because they subtracted the coefficients of two similar terms incorrectly. This result corresponds to

65
Seng [42] that this error happens more often when solving integer and simplifying algebraic calculations.
Students had problems in subtracting with negative integers, indicating that they should review the
subtraction of any two integers in every case. It contributes to the reduction of errors. Second, 4.80% of
students made careless errors caused by three issues: writing the wrong problem, writing the wrong answer,
and misunderstanding the operation of two polynomials. Students were careless in their work and became
confused about the operation of two polynomials, which caused them to be careless with their work, resulting
in mistakes and giving incorrect answers. In addition, students miswrote the opposite polynomials, and most
students changed the sign in front of the term in the polynomial, causing the answer to be incorrect. This
result corresponds to Marpa [47], which found that most students forgot to change the sign of the subtrahend
before they proceeded to the addition. They directly proceed to the process without considering the operation.
Finally, there are no students who made conceptual errors due to most students comprehend the principle of
polynomial subtraction, yet they frequently make calculation errors and are hurry in writing the solution
when subtracting two polynomials so that the answer is not correct.
3.2.4. Topic 4: multiplication of polynomials
For the error analysis from students’ work in the topic of multiplication of polynomials, it is
discovered that 14.44% of students had careless errors at the highest percentage because they made mistakes
such as writing incorrect problems or answers. These errors occurred when they rushed to complete their
tasks, lacked concentration, and failed to verify their solution to obtain the correct answer. Secondly, around
6.11% of students made computational errors due to incorrectly multiplying the coefficients for two integers
in the case of the multiplication of any integers with negative integers. According to Daud and Ayub [48],
when students multiply −3𝑥(2𝑦 − 𝑧), they fail to deal with the negative sign when performing algebraic
multiplications and give the answer as −6𝑥𝑦 − 3𝑥𝑧. This error resulted in an incorrect answer. In addition,
they make errors in using the properties of indices to multiply two polynomials that correspond to Ulusoy
[49]; this appears to corroborate the idea that students’ knowledge of exponents is still procedural and it is
not sufficient to accurately compute exponential expressions without understanding about number systems
and the logic behind the computation. Indeed, these students are unable to comprehend the laws of
exponents. The last is a conceptual error, for which 0.83% of students have made in this category. Students
were wrong in adding two unsimilar terms after multiplying two polynomials which do not correspond to the
principle of combining two monomials. The results show that most students already understand the
multiplication principle of polynomials.
It should be noted that a clear understanding of both procedural and conceptual knowledge is
required for students to successfully work on subsequent topics in polynomials. Students need to recognize
similar terms of monomials in order to correctly compute addition, subtraction and multiplication of
polynomials. They also require a previous understanding of principles and rules about the operation between
positive and negative integers. These factors impact teachers, who must be aware of the need for precision in
their answers at each step of the problem-solving process in order to reduce errors that have little bearing on
other topics. Therefore, teachers should emphasize on correcting errors or instructing about errors before
students could develop inaccurate computational procedures and concepts [50].
4. CONCLUSION
The topic of polynomials contains abstract contents which can be difficult for eighth grade students
to visualize. This is a significant issue when solving polynomials. Students who have insufficient knowledge
on polynomials tend to make errors and are not able to relate their knowledge to other topics in mathematics
and other subjects. Error analysis aids teachers in identifying misunderstandings and providing additional
information to increase understanding. This research identified mistakes that occur from students in four
topics in polynomials according to three categories including careless error, computational error, and
conceptual error. It is found that most students made mistakes in computational errors in similar terms,
addition, and subtraction of polynomials accounted for 17.86%, 10.88%, and 12.04% respectively. However,
in multiplication of polynomials, careless errors were the highest mistake which accounted for 14.44%. From
the result, it shows that most students have computational errors but not conceptual errors.
From the findings and conclusion, it is recommended that teachers should design learning activities
and strategies to improve understanding and visualizing of the operations with integers, which are the core
foundation for students to perform the algebra of polynomials. This would help students to reduce
computational error which is the major mistake in addition and subtraction of polynomials. Further
investigation should be studied on the causes of mistakes and strategies to reduce them focusing on common
mistakes found in this research.

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66
ACKNOWLEDGEMENTS
The paper is successful due to the great support from Mr. Ketum Saraburin, who coordinated with
the school to allow the researchers to collect the data. In addition, the researchers are very thankful to Miss
Arisa Serawong for allowing the data collection from her students. In addition, this research was funded by
the Petchra Pra Jom Klao Master’s Degree Research Scholarship from King Mongkut’s University of
Technology Thonburi (grant number 15/2561). This contributed much to the completion of the paper, making
it available for people who are interested.
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BIOGRAPHIES OF AUTHORS
Thayarat Ekamornaroon is a master’s degree student in science education
program, Faculty of Science at King Mongkut’s University of Technology Thonburi
(KMUTT). He graduated from KMUTT in 2018 and started with a master’s degree in 2019.
He was a pre-service teacher at Debsirin School, Thailand. His research focuses on
mathematics education. He can be contacted at email: thayarat.eka@mail.kmutt.ac.th.

 ISSN: 2252-8822
68
Parinya Sa Ngiamsunthorn is an assistant professor at the Department of
Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi,
Thailand. He obtained B.Sc. and Ph.D. degrees in Mathematics from the University of Sydney,
Australia. His research interests are nonlinear analysis, partial differential equations,
dynamical systems and mathematics education. He can be contacted at email:
parinya.san@kmutt.ac.th.
Mingkhuan Phaksunchai is an assistant professor at Faculty of Science, King
Mongkut’s University of Technology Thonburi, Thailand. She obtained Ph.D. degrees in
Educational Research Methodology from Chulalongkorn University, Thailand. Research
interests are science education, STEM education, and educational research. She can be
contacted at email: mingkhuan.pha@mail.kmutt.ac.th.
Ratchanikorn Chonchaiya received a Ph.D. degree in Mathematics from the
University of Reading, the U.K., and a Master's Degree in Mathematics Education from
Ramkhamhaeng University, Thailand. He has over 15 years of experience in the Mathematics
Educational area. He is currently working at the Department of Mathematics, Faculty of
Science, King Mongkut’s University of Technology Thonburi (KMUTT). His current research
interest includes students’ learning and development at various levels and areas of education.
His publication topic includes promoting students’ mathematical resilience, self-efficacy, and
vocational mathematics. He can be contacted at email: ratchanikorn.cho@kmutt.ac.th.

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