Category: Maths Notes PPT

The process of finding factors is known as factoring. This process can also be called factorization. It can also be defined as, factoring consists of a number or any other mathematical object as the product of two or more factors. For example, 3 and 5  are the factors of integer 15.

()

Factoring Algebra

Factoring algebra is the process of factoring algebraic terms. To understand it in a simple way, it is like splitting an expression into a multiplication of simpler expressions known as factoring expression example: 2y + 6 = 2(y + 3). Factoring can be understood as the opposite to the expanding. Different types of factoring algebra are given below so that you can learn about factoring in brief.

Types of Factoring Algebra

Different types of factoring algebra are discussed below:

Let us discuss the basic two methods of factoring which is used frequently to factorise the polynomial. The most popular formula used to find the factors of a polynomial in the Quadratic equation is Shridhar’s formula.

[x = frac{-b pm sqrt{b^{2} – 4ac}}{2a}]

Greatest Common Factor

In this, we have to find the greatest common factor of the given polynomial to factorise it. This process is a type of reverse procedure of distributive law.

Distributive law p(q + r) = pq + pr

In the case of factorisation, it is opposite of distributive law

pq + pr = p(q + r)

Where ‘p’ is the greatest common factor of the given polynomial.

Factorisation Problems:

1) Factorise 6x² + 3x

Now, take the common multiple out from the above polynomial

3 is the common multiple for the given problem.

= 3x (2x + 1)

Therefore, 6x² + 3x = 3x(2x + 1)

2) Factorise 2x² + 8x

Here, 2 is the common multiple for the given problem 

= 2x(x + 4)

Therefor, 2x² + 8x = 2x(x + 4)

Note: This method is applicable if each term in the polynomial shares a common factor.

Factoring Polynomial by Grouping

In this method, the given polynomial is grouped in the pairs to find the zeros. This method is also called factoring by pairs.

Factorisation examples: Factorise x² – 15x + 50

To solve the problem by grouping, find the two numbers which when added gives -15 and multiplication gives 50.

So, -5 and -10 are two numbers

Like, (-5) × (-10) = 50

(-5) + (-10) = -15

Therefore, The given polynomial can be written as:

X² – 5x – 10x + 50 = 0

x(x – 5) – 10(x – 5) = 0

In this taking (x – 5) as common factor;

We get, (x – 5)(x – 10)

Hence, The factors are (x – 5) and (x – 10).

Note: This method is applicable if the polynomial of the form x² + bx + c and there are factors of ac that add up to b.

Factoring Rules:

Some of the basic rules of the factoring are:

• If the third co-efficient(c) is “plus”, then the factors will be either both “plus” or else both “minus”.

• If the second coefficient(b) is “plus”, then the factors are both “plus”.

• If the second coefficient(b) is “minus”, then the factors are both “minus”.

• In either case, look for factors that add to b.

• If the third coefficient(c) is “minus”, then the factors will be of alternating signs; that is, one will be “plus” and one will be “minus”.

• If the second coefficient(b) is “plus”, then the larger of the two factors is “plus”.

• If the second coefficient(b) is “minus”, then the larger of the two factors is “minus”.

Factor in mathematics is a number that is able to divide another number exactly leaving no remainder. In simple words, factors are whole numbers that can divide a number greater than it exactly and evenly without a remainder. Factors are integers and are never fractions or decimals. They can be both positive and negative integers. Only two factors are possible in the case of prime numbers while for composite numbers there are more than two factors. 

The factor is an easy but very useful topic. It is such an important mathematical tool that is used in all level maths starting from elementary school level to higher advanced levels. It is used in all other science and many arts subjects also. An important tool for measurements involved in day-to-day life. Similarly, by the term factors of 18, we mean all those integers that can divide the number 18 evenly with a remainder of zero.

Factors of 18

Factors of 18 are the products of such numbers which completely divide the given number 18. Factors of a given number have two values; they can be either positive or negative numbers. By multiplying the factors of a number we get the original number. For example 1, 2, 3, 4, 6, 12 are the factors of 12. Hence we have 4 x 3 = 12 or 6 x 2 = 12 as the pair factors of 12. In this article, we will study the factors of 18, what are the factors of 18, what is the prime factorization of 18, the factor tree of 18, all factors of 18, and examples. Factor pairs of the number 18 are the pairs of the whole numbers which could be either positive or negative but not a fraction or decimal number. Factorisation is the common method to find the factors of 18. 

()

Definition

The factors of a number are defined as the numbers which give the original number on multiplying the two factors. The factors can be either positive or negative integers. Factors of 18 are all the integers that can evenly divide the given number 18. Now let us study how to calculate all factors of 18.

What are the Factors of 18?

According to the definition of factors of 18, we know that factors of 18 are all the positive or negative integers that divide the number 18 completely. So let us simply divide the number 18 by every number which completely divides 18 in ascending order till 18.

18 ÷ 1 = 18

18 ÷ 2 = 9

18 ÷ 3 = 6

18 ÷ 4 = not divides completely

18 ÷ 5 = not divides completely

18 ÷ 6 = 3

18 ÷ 7 = not divides completely

18 ÷ 8 =  not divides completely

18 ÷ 9 = 2

18 ÷18 = 1

So all factors of 18: 1, 2, 3, 6, 9, and 18.

We know that factors also include negative integers hence we can also have, 

list of negative factors of 18: -1, -2, -3, -6, -9 and -18.

All Factors of 18 Can be Listed as Follows:

 

Hence 18 has a total of 6 positive factors and 6 negative factors.

All Factor Pairs of 18

All Factor Pairs of 18 are combinations of two factors that when multiplied together give 18.

List of all the positive pair factors of 18

1 x 18 = 18; pair factors are(1, 18)

3 x 6 = 18; pair factors are(3, 6)

2 x 9 = 18; pair factors are(2, 9)

So (1, 18), (3, 6), and ( 2, 9), are the positive pair factors of 18

As we know that Factors of 18 include negative integers too. 

List of all the negative pair factors of 18:

-1 x -18 = 18

-3 x -6 = 18

-2 x -9 = 18

So (-1, -18), (-3, -6) and ( -2, -9) are the negative pair factors of 18

Now we will study what is the prime factorization of 18.

What is the Prime Factorization of 18?

According to the prime factor definition, we know that the prime factor of a number is the product of all the factors that are prime, which is a number that divides by itself and only one. Hence we can list the prime factors from the list of factors of 18.

Or the other way to find the prime factorization of 18 is by prime factorization or by factor tree of 18.

Now let us study prime factors of 18 by division method.

Prime Factors of 18 by Division Method

To calculate the prime factors of 18 by the division method, first, take the least prime number that is 2. Divide it by 2 until it is completely divisible by 2. If at a point it is not divisible by 2 take the next least prime number that is 3. Perform the same steps and move forward, till we get 1, as the quotient. Here is the stepwise method to calculate the prime factors of 18

Step 1: Divide 18 with 2

18 ÷ 2 = 9

Step 2: Divide 9 with 2

9 ÷ 2 = not divisible

Step 3: So take another prime number 3 divide with 3

9 ÷ 3 = 3

Step 4: Now again divide 3 by 3

3 ÷ 3 = 1 

We get the quotient 1.

From the above steps, we get a prime factor of 18 as 2 x 3 x 3 = 2 x 32

Here is the factor tree of 18.

()

Solved Examples

Example 1: Find the Prime Factors of 180.

Solution: 

  180 = 2 x 90

         = 2 x 2 x 45

         = 2 x 2 x 5 x 9

         = 2 x 2 x 5 x 3 x 3

This is all about the factors of 18 and how to calculate them. Learn how the factors of a number can be calculated step by step and utilize the process for determining the factors of other numbers on your own.

Factors are the pair of numbers which on multiplication give us the original number. Factors are also defined as the number which on dividing the original number gives us the quotient as a whole number. Factors can be positive numbers or negative numbers. Decimal numbers and fractions cannot be the factor. Factors and multiples are not the same. Multiples are the extended terms of the given number. There are different methods by which we can find the factors. They are the factorization method, prime factorization method, division method, divisibility method. 

We will discuss all these methods in detail further.

Factors of 40 by Division Method

Following the multiplication tables up to 20, we can easily find the factors of 40. 

So, the factors of 40  are 40,  20, 10, 8, 5, 4, and 2.

In order to find more factors of 40, we can start dividing 40 with the factors that we have obtained until now.

The common factors for 40 are 2, 4, 5, 8, 10, 20.

Divisibility Method

In this method, we check the divisibility of 40 by every number. This is the simplest method to find the factors of 40.

Steps to find Factors of 40 by Divisibility Method

1. We will check whether the number is divisible by 2 since 40 is the even number is divisible by 2.

2. Now we will check the divisibility by 3. 40 is not divisible by 3.

3. Now by 4, it is divisible because multiplying 4 by 10 gives us 40.

4. It is also divisible by 5 because it has 0 in the unit’s place.

5. Similarly, we will check the divisibility till 10 and we will get all the factors of 40.

Tips and Tricks

This article must have given you an idea about the factors of 40 and also made you clear about the concept. If you want to learn more about factors,  you may visit ‘s website which will definitely clear all your doubts. Get into learning but don’t drown in it. 

The only boat which can save you is the below-mentioned strategies, especially for subjects like maths who ask for loads of skills. You should add these tricks to your schedule and you will see the changes super soon. 

Let’s dive into these tricks. 

It is not a trick but advice, if you get things learned easily without any confusion, then, you should just try to memorize all the tables and that too till 20. Learning these tables on your tips is the best thing that you can do and is also a very helpful technique that can help you solve the questions faster. If you feel that your memory is good and you can get the tables stored in it, you should try and that is how you can make your memory your best friend. 

The foremost and the most important trick which is completely dependable is that all multiplication has a twin. For example, if you don’t remember 7×3 you might remember 3×7, this way you can save a lot of time, which you generally waste thinking about the same. This way you need to remember only half the table. It will help you save time. Hence, students shall not be missing on this one. 

Number 2: 2 has the simplest trick, in this, you just need to double the number to itself. 

For example 2×5=5+5=10

Number 5: For this, you need to cut the number in half and then multiply the number by 10

For example 5×4= 4, half of 4 is 2, then 2 ×10 is 20 which is the answer for 5×4. 

Number 8: Double, double, and double! 

You just need to double the number thrice and the answer is ready. 

For example, 8×6, double of 6 is 12, 12 doubled is 24 and 24 doubled is 48 which is the answer. 

Number 10: Again a simple one in the list of number 10. You just need to put a zero at the end and your job is over. 

For example, 8×10 put a 0 at the end. The answer is 80. 

Number 9: This is also quite easy, you need to multiply the number by 10 and then subtract the number by itself. 

For example 9×6. Multiply 6 and 10, and subtract 9 from it. The answer is 51. 

These number tricks will help you get all the answers very quickly and easily. 

Remembering the squares can also help you. But, learning too much is quite impossible. So, to crack it, there is a quick trick. You need to multiply the number by itself and that gets you the square of the number. And this can get you another trick, in when you multiply the number by itself it gets you an answer in which if you subtract 1 it is the first number and when you add 1, it is the second number. Make a combination of these numbers.  

For better clarity, refer to the example given below. 

For example, 

5×5=25

25-1=24, and then we will do 

5+1 and 5-1 (both the 5s of 5×5) which gets us 

6×4 which is equal to 24

This trick goes with all the squares. 

Factorization is the study of finding the whole numbers which give you the product as a given number when you multiply each other. Maths could be tough for some students, as some are good with numbers, and some are not. But one thing is sure, and when it comes to finding out the factors, it’s an easy score for every student. The questions of finding out the factors are quite easy than other questions that might come in your exam. But you need to make sure that you don’t make any silly mistakes and lose marks in it. 

Today we are going to show you the factors of 80 and find out the prime factorization of 80. We will be discussing the importance of factors and will be dropping some cool facts here and there, which makes learning a bit more fun. 

What are the Factors of 80?

Let’s start with the very basics, and here we are going to show the factors of 80. Before we move any further, let’s talk about 80 in the first place. Have you seen a puzzle that has moving parts? Each puzzle that has 15 moving parts can be solved in just 80 single tile moves. In addition to this, 80 is an even, composite, and abundant number. On the other hand, 80 is not your perfect square; neither is your perfect cube. 

Now let’s find out the factors of 80; as you can see, 80 is a big number so that you will find plenty of factors for this particular number. A factor is a number that, when multiplied by another whole number, gives you the product that is your given number, and in this case, we have a product i.e., 80. 

So, starting our factorization.

Factors of 80

These are the only numbers that can make a product of 80, but numbers bigger than 80 will always have a product greater than 80, so there’s no use in going any further. As a result, we have our list of factors of 80, and it looks like this. 

These are all factors of 80 =1, 2, 4, 5, 8, 10, 16, 20, 40, 80. 

One more interesting fact about 80, in some, find the factor puzzles, 80 is a clue, which can be used to find other numbers in the puzzle, you need to use the 8X10 fact only.

Prime Factorization of 80 

Now let’s look at the question, what is the prime factorization of 80, and how can we find it. See prime factors are those numbers, which are prime in nature, meaning they are only divisible by 1 and the number themselves. 

We have found out the factors of 80 now. We need to take out the prime factors and multiply them to get the answer. 

From the above solution, we have 1,2,4,5,8,10,16,20,40,and 80 as factors. 

Now, we take out prime numbers. First, these two numbers are 2 and 5. 

After that, we multiply 2×5, but it only comes out to be 10. 

So what we do is, multiply 2×5 from 3 times 2. 

Thus, (2X2X2) X (2X5) = 8 X 10 = 80. 

You can also write this as (2)4 *5, and this is your only prime factorization of 80.

Solved Example

So now we have found out the factors of 80 and have understood the concept of prime factorization of 80. We think it’s time we try our hands-on solving some problems related to factors of 80.

Question: Find the common factors of 80 and 100. 

Solution: from the question we can see, we have to find the numbers that are present in the factorization of both 80 and 100. 

So first let’s see the factors of 80 = 1,2,4,5,8,10,16,20,40,and 80.

In the same way, let’s look at the factors of 100 = 1,2,4,5,10,20,25,50, and 100.

After comparing both, we take out the numbers that are appearing in both factorizations. 

Common factors = 1,2,4,5,10 and 20. 

We have given you a solved example, and we hope a lot of things have been cleared regarding the factors of 80. Now it’s the right time for you to take out your pen and notebook and try to solve some questions yourself. If you are stuck somewhere, comment down the problem, and we will be happy to help you solve that problem. 

A finite set in mathematics is a set that has a finite number of elements. In simple words, it is a set that you can finish counting. For example, {1,3,5,7} is a finite set with four elements. The element in the finite set is a natural number, i.e. non-negative integer. A set S is called finite if there exists a bijection f:S = {1,……,n} for natural number n. The empty set {} is also considered finite. So, S is a finite set, if S admits a bijection to some set of natural numbers of the form {|x| < n}.

An infinite set is a set with an uncountable number of elements. We use dots to represent the infinite elements in a roaster. For example, a set of infinite natural numbers. {1,2,3,4,…}.

Join to read about these topics in detail from the expert maths faculties. 

The Cardinality of Finite Set

The cardinality of a finite set is n(A) = a, here a represents the number of elements of set A.

Whereas, the cardinality of the set A of all English Alphabets is 26, as the number of elements (alphabets) is 26. So, n(A) = 26.

It shows that you can list all the elements of a finite set and write them in curly braces or the form of Roster. Sometimes, the number of factors may be too big, but somehow it is countable or has a starting and ending point. Then this type of set is called a Non-Empty Finite Set. The number of elements is denoted with n(A) and if n(A) is a natural number then only it is a finite set.

 

Is an Empty Set a Finite Set?

An empty set is a set which has no elements in it. It is represented as { }, which shows that there is no element in the given set. The cardinality of an empty set is 0 (zero) as the number of elements is zero.

A={ } or n(A)=0.

The finite set is a set with countable elements. As the empty set has zero elements in it, so it has a definite number of elements.

Therefore, an empty set is a finite set with cardinality zero.

 

What is the Infinite Set?

A set which is not a finite set is infinite. If the number of elements is uncountable, then also it is called an infinite set. Unlike finite sets, we cannot represent an infinite set in roster form easily as its elements are not limited. So, dots are used to describe the infinity of the set.

 

What will you learn in the Finite and Infinite Sets Chapter?

In this particular chapter, you come to know about the basic definitions of the finite set and infinite sets. Further in the topic, maths experts have explained the properties of the sets. 

Following are the conditions of the finite sets.

1. Two subsets always form a subset.

2. The power set of finite sets is finite.

3. A subset of finite sets is finite.

Following are the conditions of the infinite sets.

1. Union of two infinite sets is an infinite set.

2. The power set of infinite sets is infinite.

3. The superset of the infinite set is an infinite set.

• Graphical Representation of Finite and Infinite Sets

Also, maths teachers have provided you with a graphical representation of the sets that clear the unwanted confusion about finite sets. 

In addition to that, has brought a good amount of solved examples to help you understand these topics thoroughly and expose you to higher-order thinking skills (HOTS) questions.

Solved Example

Q1. Which of the following sets are finite or infinite?

1. The set of months of a year.

2. {1,3,5,…..}.

Answer: 1. The set of months can be represented as A= {Jan, Feb, Mar, Apr, May, Jul, Aug, Sep, Oct, Nov, Dec}. It forms a set of countable elements with the number of elements =12. Hence, it is a finite set.

Answer: 2. The set {1,3,5,…} has all the natural numbers but does not consist of any ending point. This makes it an uncountable set, and so it is an infinite set.

 

Q2. What is the Cardinality of Infinite Sets?

Answer: Cardinality of a set is expressed as n(A) = x, where x is the number of elements in the set A.

The number of elements in an infinite set is unlimited, so the cardinality of the infinite set n(A) = infinity.

Benefits of Using  

Learning becomes fun and easy with . is an online learning platform that provides various study materials for all classes. We provide an online tuition class to help students understand things with 3-dimensional visuals. Also, you get a chance to clear your doubts from different subject matter experts. All these things are available under one roof called . We are also available to download on your phone so that you can access indefinite study materials to boost your studies. 

In our everyday lives, collecting, recording, and maintaining information is indeed a crucial task. When it comes to the field of mathematics, statistics has an integral part as it refers to the organization, distribution, collection, and interpretation of the set of observations or data (representation of facts or a piece of information that can be further processed). With the help of statistics, we can have a better understanding of what a dataset or a set of observations reveals about a specific phenomenon. We can also predict the nature of data using statistics, by studying large amounts of data and trends and also by interpreting the results. We have various methods for representing the statistical data, including pie charts, bar graphs, tables, histograms, frequency polygons, amongst many others. So, let us now discuss the concepts of collecting and recording data using a frequency distribution table.

 

For having a better understanding of the frequency distribution, let us consider an example. Suppose we have the marks obtained by ten students out of 25 in a unit test conducted at their school as follows:

 

12, 23, 22, 8, 19, 15, 24, 25, 17, and 9

 

The data given above is in the raw form and is referred to as the raw data. We can calculate its range, which is the difference between the largest value and the smallest value in a set of observations or data set. In this particular scenario, the range is 25-8 = 17.

 

Understanding the Frequency Distribution Table

Without any second thoughts, the process or representation of data explained in the example above would become way more difficult and cumbersome if there were a larger number of observations in a set. In such cases, the concept of a frequency distribution table proves to be exceedingly beneficial as it organizes a larger data set into a table and makes the interpretation and analysis a lot easier and convenient.

 

Ungrouped Data

Let us consider a situation in which the scores of 20 students out of 25 in their mathematics unit test are given as follows:

 

15, 18, 21, 24, 18, 11, 15, 21, 13, 9, 23, 14, 6, 18, 20, 11, 10, 20, 25, and 17

 

Please keep in mind that the term ‘frequency’ denotes the number of times an observation appears or occurs in a dataset. Hence, it is quite evident that the frequency shall increase in the case of repetitions. Now, let us draw the frequency distribution table for the given dataset.

 

Student Marks

 

In this example, the frequency refers to the number of students scoring the same marks in the mathematics unit test. It is imperative to note that the sum of the frequencies should be equal to the total number of observations in the dataset. The frequency distribution table of this example is known as an ungrouped frequency distribution table as it takes into consideration the ungrouped data and calculates the frequency of every observation one by one.

 

Grouped Data

Let us consider another situation in which we have the scores of 200 students instead of 20 students out of 25 in their mathematics unit test. It will prove to be a hectic task of tallying the scores of all the 200 students. Moreover, the length of the table will increase as well, and it will not at all be understandable. In such cases, the concept of a grouped frequency distribution table becomes handy as it considers groups or data in the form of class intervals for tallying the frequency of observations like which observation belongs to a specific class interval.

 

For having a better understanding of a grouped frequency distribution table, take a look at the table given below based on the dataset of the previous section.

 

Marks Obtained by Students in Unit Test

 

The first column of the grouped frequency distribution tables denotes the scores of students represented in the form of class intervals. “Lower Limit” is the lowest number in the class interval, and “Highest Limit” is the highest number in the class interval. The example explained above falls under the case of continuous class intervals as the upper limit of a specific class is the lower limit of the next class.

 

In the case of continuous class intervals, the extreme values are included or counted in that class interval where they are the lower limit, for instance – if there is a student who has scored ten marks in the mathematics unit test, then his marks would be counted or included in the class interval 10-15 and not 5-10.

 

Disjoint class intervals are analogous to continuous class intervals, in which the class intervals will be of the form 0-3, 4-7, 8-11, and so on, and their frequency distribution table is constructed in the same manner as explained above.

 

Solved Example

Consider the frequency distribution table given below corresponding to the marks scored by students in their science unit test out of 20 and answer the questions that follow.

 

Student Marks

 

Question 1

Find out the lower limit of the second-class interval?

Answer 1

The lower limit of the second-class interval, that is, 5-10, is 5.

 

Question 2

What is the class size?

Answer 2

The class size refers to the difference between the upper- and lower-class limits, which is 5-0 = 5 or 10-15 = 5 (the answer is 5 in all the cases).

 

Question 3

What is the class mark for the interval 10-15?

Answer 3

The average of the upper and the lower limit is the class mark. So, for the interval 10-15, the class mark is [frac{(10+15)}{2} = frac{25}{2} = 12.5].

 

Question 4

What are the class limits of the second interval?

Answer 4

The class limits of the second interval, that is, 5-10 are 5 (lower limit) and 10 (upper limit).

Key Takeaways

In statistics, a frequency distribution is a visual depiction of the number of observations within a certain interval.

Normal distributions, which display the observations of probabilities divided across standard deviations, benefit from frequency distributions in particular.

A frequency distribution can be represented graphically or tabulated to make it easier to understand.

Grouped Frequency Distribution and Ungrouped Frequency Distribution are the two forms of frequency tables.

Traders utilize frequency distributions to track market activity and spot patterns in finance.