Factoring polynomials

When factoring out polynomials,  we find   the  polynomial  that divide out evenly from the original polynomial  .   How to factor  Polynomials:  For this we have to find all the terms that if multiplied together we get the  original polynomial.  This is continued to all the terms until this cannot be simplified any more.  If the polynomial cannot be factored any more then the polynomial is said to be completely factored.


A factor of a polynomial is any polynomial which divides evenly into   the given polynomial   For example, x + 2 is a factor of the polynomial x^2 – 4.
The factorization of a polynomial is its representation as a product its factors. For example, the factorization of x^2 – 4 is (x – 2)(x + 2).

In mathematics, factorization or factoring is the breaking apart of a polynomial into a product of other smaller polynomials. If you choose, you could then multiply these factors together, and you should get the original polynomial.
This example shows how to factoring polynomials.    Take 3x^2 – 12x + 9.   The common term in these is 3.  So take 3 out and divide each term by 3.  We get   3 (x 2-4x + 3).

Step by step explanation as how to Factor out Polynomials is given below
Factor 4a^2 + 20a – 3a - 15


The first two terms have a common factor in 4a.  The last two terms have a   common factor in 3.
We need to factor those terms out.

4a ( a + 5)  -3(a+5)

Now you have a binomial.  Each term     has a factor of (a + 5).

(4a -3) (a+5) is the factored terms.


Another example is given below on factoring of polynomials
X2 -8x + 15

start by looking at the factor pairs of 15.  We are looking for a pair of factors which add up to equal -8.  Look for the factor pairs of  +15 so that they add up to -8    The negative factor pairs of 15 are:
-15  and  -1
 -5 and  -3
Since -5 + -3 = -8 this is the pair we are looking for and we can factor the
original expression into:  (x-5)(x-3)

Matrix types: Diagonal matrix

The term diagonal matrx refers to the topic of linear algebra which is a branch of mathematics. In general a matrx would look like as follows:
[a11 a12 a13 … … a1m]
[a21 a22 a23 … … a2m]
[a31 a32 a33 … … a3m]
[… … … … … …    ]
[… … … … … …    ]
[an1 an2 an3 … … anm]

The above matrx has n rows and m columns. The diagonal of such a matrx consists of all the entries where in the row number = column number. Therefore if r = row number and j = column number. Then the entries of the type a(ij) where i=j are the diagonal entries. Therefore in the above matrx the diagonal would be the highlighted entries as shown below:

[a11 a12 a13 … … a1m]
[a21 a22 a23 … … a2m]
[a31 a32 a33 … … a3m]
[… … … … … …    ]
[… … … … … …    ]
[an1 an2 an3 … … anm]

Now if we have a matrx where in all the entries except the diagonal entries are zero, then such a matrx would be called a diagonal matrix. In general a diagonal-matrx would look like this:

[a11 0 0 … … 0]
[0 a22 0 … … 0]
[0 0 a33 … … 0]
[0 0 0 … … 0]
[… … … … .. 0]
[0 0 0 … … ann]

Note that a diagonal-matrx has to essentially be a square matrx. An example of a 3x3 diagonal-matrx is shown below:

[2 0 0]
[0 -1 0]
[0 0 5]

Scalar matrix:

A diagonal-matrx in which all the entries of the diagonal are equal is called a scalar-matrx. The general form of a scalar matrx would be like this:
[a 0 0 … … 0]
[0 a 0 … … 0]
[0 0 a … … 0]
[0 0 0 … … 0]
[… … … … .. 0]
[0 0 0 … … a]

Determinant of diagonal matrix:
Let us try to understand how to calculate the determinant of a diagonal matrix using an example.

Example: Calculate the determinant of the following diagonal-matrx:
[2 0 0]
[0 -1 0]
[0 0 5]
Solution:
D = 2*(-1*5 – 0*0) – 0*(5*0-0*0) + 0*(0*0-(-1)*0))
= 2*(-1)*(5) – 0 + 0
= 2*(-1)*5 = -10
So we see that the value of determinant of a diagonal-matrx is the product of the terms on the diagonal (also called the principal diagonal)

Discount and Discount Percentage

Discount and market price are two of the most important concepts in mathematics. These two concepts are also studied in finance and economics and play an important role in the real life market scenario. Let’s try to understand both the concepts in this post.

Market price is the price that is written in the price tag of a product. It is the amount that one needs to pay to buy a product. At times, customers are allowed to buy the product at a lesser price than the market price. For example: The market price of cot mobile for kids was Rs. 50. Maria was allowed to buy the same in Rs.45. Therefore, Rs. 5 is the discount offered to Maria while buying cot mobile for kids. The mathematical formula of finding discount price is Discount = Market Price – Selling Price (MP – SP).

Example 1: The market price of action figure toy is Rs.199. It is available in online shops for Rs.190. What is the discount offered?
Discount = Market Price – Selling Price
Discount = 199 – 190
= Rs. 9 is the discount offered on action figure toy in online shops.

Example 2: Mary bought kids’ toy guns at Rs. 50 and sold it for Rs. 40. What is the discount offered?
Discount = Market Price – Selling Price
Discount = 50 – 40
= Rs. 10 is the discount offered on kids toy guns by Mary.
Discount Percentage
The discount percentage is calculated on market price. The mathematical formula to find out discount percentage is [Discount / MP] X 100.

Example: Arun bought mobile phone for Rs.10000 and sold it out at Rs.9500. Find the discount and discount percentage:
Discount = Market Price – Selling Price
Discount = 10000 – 9500 = 500
Discount Percentage = [Discount / MP] X 100
= [500 / 10000] x 100
= 5%
The discount offered by Arun is 5%.
These are the basics on discount and discount percentages.

Subset

So what is the basic difference between a set and a sub-set? The answer is simple. A set has all the elements that are present in  a subset but a subset. does not contain all the elements of a set, it contains only a few elements. But it crucial to note that a subset. of a particular set will have no element that is other then what is present in that set.

So a subset. is a smaller or condensed form of a set. Let us consider a set of integers less than 5 and greater than 1. A = {  2,3,4 } now B which is a subset. of A can be any one of these: B = { 2,3} or B = {2} or B = { 3,4} etc. but not that B has no element other than that of A.

Even a null set is a sub-set of all the set may it be any set. A set of numbers can be either a set of real numbers or a set of integers or a set of fractions or a set of floating point number etc.  A set of numbers hence can be considered as a superset of all the other sets. A super set is which that contains all the sets elements of the given sub-sets. In the above example A can be called a superset of B and B can be called a S. S of A or just Subsets of Numbers.

The sum of subsets of a particular set will make up that set. It should be noted that null set belongs to every set.
To explain the Subsets of integers we can take up another example. The subsets can be A (sub-set of I) : { -1, -2 , 0 ,1, 2, 3} etc. All subsets are the part of a universal set which contains all the numbers. For a S. s of a integer we can have as many numbers as there are integers and then we can add a null set to it to get the original set.

The Set and Subset are the essentials for analyzing particular type of data and hence should be understood very clearly. The proper set and the improper set are yet another types of set, in a proper set we also have null element or entity. Similarly we can define s. s. for the same set.
R denotes the set of real numbers. Say ‘r’ is a s. s. of real numbers we write it as r = {0}

Percentage Increase

When we get a discount at a shop, we say that we got this much % off.  Now if the same shop offers 5 % more off, then the price of a product will be decreased by a certain amount. The increase and decrease in anything is usually calculated in terms of %. This is actually the measure of perc change which helps us to know the extent to which a thing has gained or lost its value over time.

For example if Sam works at shop with the pay of $ 10 per hour and if his pay gets increased to 4 15 per hour then we can Find Percentage Increase in his pay by making use of Formula for Percentage Increase which is difference of old value and new value divided by old value multiplied by hundred which can be written as (new value – old value) / old value X 100. Some more example of this would be change in temperature in two days, change in height or weight gain in children, share prices of a company or stock market or the prices of any product in a specific year compared to the last year.

How do you Calculate Percentage Increase- Finding Percentage Increase is done by following the series of steps given below: -

1. In the first step, we find out the difference between the two numbers.
2. In the second step, we take the difference and divide it by the original number.
3. In the third step, we multiply the number obtained in second step by 100.
4. In the last step, we put a % sign with the number or figure obtained in the third step.

Let us take an example to understand this concept more clearly. If the staff of a company is increased from 20 to 30 in one year then what is the Increase Percentage? To solve this, we follow the same steps given above. First we find the difference between two numbers which is 30 - 20 = 10. Then in the next step, we divide 10 by 20, which is the original number. That gives 10/20. Now in the next step, we multiply it by 100 which give 50. And in the last step, we put a 5 sign with 50 which makes the solution as 50 %.

Less than symbol

Less than symbol
A less than symbol looks like this “ <”  which is commonly used in mathematics. A less than symbol shows the relationship between in two values. The value that is on the left hand side is less than the value that is on the right hand is a less than symbol.

What is less than Symbol? In a way we can think about this is symbol with, two lines when they meet it becomes a point and that is where we put smaller value. But the Right hand side it is open wide and that side we put larger value. So for example if we are putting number 3 and number 5 like this, 3 less than(<) 5 Here we will put 3 at the left side and number 5 at right side, so we can say 3 is less than 5, this is less than sign example.

Symbols less than can be useful when we do not know the value of something, but we can compare to a value, we know that it is less than a certain value. So for instance I can never remember how old is my cousin sister is. But I know that she is younger to me. Let us say I am 20 years old, but I do know my cousin sister’s age is less than 20 years old. So we can write like this, age of my cousin < 20 years. Here we can say by looking this less than symbol, that age of my cousin sister is less than 20 years.

Another example in math symbols less than we shall understand it by, let us say that I have the expression 12 times 6 less than 8000,that is 12*6 < 8000 Maybe I cannot do that very quickly in my head but I am pretty confident that whatever I get 12 times 6 is going to be less than 8000. So there are times, in which we know rules of relationship between two numbers and which we can demonstrate it with the less than symbol.

The less than symbol does not tell us, how much less one value than the other is. It just puts them in that order. Which indicates one portion is smaller than the other. So let us use few more examples, where we can use less than symbols. Let us say, 7 and 56 can we use less than symbol here? Yes, very much as we know on the left side 7 and on the right side 56, so as per the symbol indicates <   56 is larger than 7. So we can use less than symbol here. Thus we can write 7 < 56.

Word Problems on Subtraction

Word problem is a very important concept of basic mathematics. A Word problem is nothing but a textual representation of a mathematical operation. Word problems are usually used to solve a problem easily. Word problems can be on all four operations of mathematics including addition, subtraction, multiplication and division. Let’s have a look at word problems on subtraction.
Word problem on subtraction is nothing but a text representation of subtraction. For example: Rohan bought 6 toys from Melissa & Doug brand. He gave 2 Melissa & Doug toys to Sam’s child and kept the others for his son. How many toys is he left with? This word problem is a text representation of mathematical problem (6 toys – 2 toys =?).

Solving word problems on subtraction:

Solving word problems on subtraction includes certain steps.  Firstly, both the numbers to be subtracted needs to be identified. Secondly, the word problem needs to be converted to a mathematical expression and finally, the number needs to be subtracted. For example: Mohan gifts three maternity wear from Mother Care India brand to his wife. His wife gives one of the Mother Care India maternity wear to her friend. How many maternity wear does Mohan’s wife now? Identifying the numbers, we get 3 maternity wear and 1 maternity wear. Converting the word problem to a mathematical expression and then subtracting, (3-1 = 2). Thus, the answer is 2 maternity wear.

Examples of Word Problems on Subtraction:

1. Philip has bought five fun toys from baby store India online collection. He gave two toys to Rohan’s son and kept one for his own son. How many toys from baby store India online does his son get?
Answer: 5 toys – 2 toys
(5-2) = 3 toys.
2. There are 18 Angry Bird toys in a basket. 7 Angry Bird toys were given to Mohan and 2 toys to Divya. How many Angry Bird toys were there in the basket after giving toys to Mohan and Divya?
Answer: 18 Angry Bird toys – 7 Angry Bird toys – 2 toys
(18-7-2) = 9 toys.
These are some basic examples on word problems on subtraction.