Steps to Factor Trinomials

In the previous post we have discussed about Polynomial Long Division and In today's session we are going to discuss about Steps to Factor Trinomials and How To Factor Trinomials Step By Step,
1. Firstly we are going to compare the given trinomial with the standard form of the trinomial i.e. ax>2 + bx + c and recognize the values of a, b, and c.
2. Now we are going to first look for the factor which is common in all the three terms. Once the common factor is recognized, we will bring it out of the three terms.
3. In the next step, we will split the middle term in such a way that the product of the two splitted terms will be equal to the product of the first and the third term and the sum will be equal to the middle term.
4. Further we will take out the common terms and make the factors.
Let us take the following trinomial :
18x>2 + 48*x*y + 32y>2
Here we will first take out the factor 2, from all the three terms and we get :
= 2 * (9x>2 + 24 * x * y + 16y>2)
= 2 * (3x)>2 + 2 * 3 * 4 * x * y + (4y )>2
= 2 * 3x + 4y>2
Thus we come to the observation that the step by step procedure must be followed in order to get the factorization of the trinomial.
If we are able to recognize the factors directly, relating it to some of the identity, then it becomes more easy for us to factorize. In case the trinomial is
4x>2 + 12x + 9
= (2x)>2 + 2 * 2 * 3x + 3>2
= (2x + 3 )>2
= (2x + 3 ) *(2x + 3 )
homework help online is available in math tutorials. Online cbse class 8 books is also available.

Polynomial Long Division

In the previous post we have discussed about Degree of Polynomial and In today's session we are going to discuss about Polynomial Long Division. In this blog we are going to discuss the Polynomial Long Division. An operation that is used to dividing a polynomial value with the polynomial value is called as polynomial long division. The process in which a value is dividing by the same value or lower degree is also known as polynomial Long Division. Polynomial long division is denoted the term that is added, subtract and multiplied. This equation 7xy² + 3x – 11 is the representation of the polynomial long division. The polynomial word came from the two words the first one is ‘poly’ and second one is ‘nomial’. Poly means 'many' and nomial means 'term'. By combining both the meaning we get the word I. e. many terms. Polynomial may be denotes constant, variables, and exponent values and we can combine them with addition, subtraction and multiplication operation. (know more about Polynomial long division, here)
Lets consider a polynomial p (n), D (n) where degree (D) < degree (p), then the quotient polynomial Q(n) and remainder polynomial R(n) with degree(R) < degree(D),
P(n) = Q(n) + R(n) ⇒ P(n) = D(n) Q(n) + R(n),
D(n) D(n)

By following some steps we can easily find the polynomial long division.
Step1: Firstly we need to focus on the higher coefficient term which is present in the equation.
Step2: We need to multiply the divisor with the leading term by doing that we can get coefficient term that will be exact.
Step3: After getting the coefficient term we only have to change the sign of the variable. If negative sign is present, then we change it into positive sign and vice-versa.
Step4 : At last cancel the term of same coefficient or variable.

In differential equation solver we use some methods and somewhere it is related to the concept of how compound interest works. Indian Certificate of Secondary Education is a type of exam that is comes under the Council for the Indian School Certificate Examinations.

Degree of Polynomial

In the previous post we have discussed about How to deal with Polynomials and In today's session we are going to discuss about Degree of Polynomial. We know that the polynomial is the combination of the terms joined together with the sign of addition or subtraction. (know more about Polynomial, here)
 If the Polynomial has only one term we call it a monomial. If there are two terms in the polynomial, then we say that the polynomial is called the binomial and the polynomial with three terms is called trinomial. The polynomial with more than three terms is simply called the polynomial. By the term degree of polynomial, we mean the highest power of the term among all the terms in the given expression. If we have the polynomial 2x + 3x>2 + 5 x>4, then we say that the term 5x>4 has the highest power. SO we say that the degree of the polynomial  2x + 3x>2 + 5 x>4, is 4. On the other hand if we have the polynomial 4x + 3, here the degree of thee polynomial is 1 as the power of x in 4x is maximum, which is equal to 1.
 We must remember that if the degree of the polynomial is 1, then we call the polynomial as the linear polynomial. In case the degree is 2, then the polynomial is quadratic polynomial and if the degree of the polynomial is 3, then the polynomial is called the conical polynomial. Here we write 2x + 5 is a linear polynomial,  5x>2 + 2x  +5 is a quadratic polynomial; and 2x>3 + 5x>2 + 4x + 8 is a cubical polynomial.
    To learn more about the Green s Theorem, we can take the help from the online math tutor and understand the concept of the lesson clearly. The sample papers of Andhra Pradesh board of secondary education are available online to learn about the patterns of the Question paper which will be very useful to prepare about the exams.

How to deal with Polynomials

In the previous post we have discussed about Factoring Trinomials and In today's session we are going to discuss about How to deal with Polynomials.  In mathematics, there is a term algebra in which we studied about the statements that express the relationship between the things. In the algebraic notation, relationship between the things can be described as a relationship between the variables and operators that are vary over time. In the same aspect polynomials also consider as a part of algebra that deals with the real numbers and variables. Through the polynomials we can simply perform the basic operation like addition, subtraction and multiplication with the variables. In a more appropriate way we can say that polynomial is a combination of different types of terms with different mathematical operators.   (know more about Polynomial , here)
 In the standard definition we can say that polynomial is an expression that contains the combination of number and variable into it with basic operations and positive integer exponents. In the below we show how to we represent the polynomials into algebraic expression:
3ab² – 4a + 6
In the above given algebraic notation 3ab², 4a, 6 can be consider as a terms through which we perform the basic operation that is addition and subtraction. In the above 3, 4 and 6 can be consider as a constants and ‘ab, a’ can be consider as variables. The power of two with the variable ab can be consider as positive exponents. In the other aspect exponents value describe the degree of the term. It means in above notation the term 3ab2 has a degree of 2. In the absence of exponent value we can take the degree of term as a one. Basically in mathematics polynomials are used for describing relationship between the numbers, variables and operations that generate some values. The output of polynomial expression helps the students to get the value of unknown variables. In the study of algebraic expression the concept of polynomial can be categorized into three categories that is monomial, binomial and trinomial. In mathematics Definite Integral can be consider as part of calculus which is used to integrate the function's values between upper limit to lower limit. The ICSE board books help the students to make their study according to their syllabus that are conducted by the Indian certificate of secondary education.

Factoring Trinomials

We know that the trinomial is the polynomial formed by three terms. To learn about Factoring Trinomials, we say that the standard form of the trinomial is ax>2 + bx + c.

 To find the factors of the above given polynomial , we say that we will either write it in form of some or the other identity or we will try to factorize it by the splitting method. In the splitting method, we say that the middle term of the trinomial is split in such a way that the sum of the two terms is equal to bx (i.e. the second term) and the product of the two split term is equal to the product of the first and the third term of the trinomial.

 Let us look at the following examples:

If we have the polynomial:  4x>2  + 12x + 9

 Here we can   write the above given polynomial as  ( 2x )>2 + 2 * 2x * 3 + 3>2

We observe that the above given polynomial is in the form of the identity (a + b) >2 = a>2 + 2 * a * b + b>2

 So it can be written as (2x + 3) >2

 If we solve the polynomial by splitting the second term we say it can be written as :

4x>2 + 6x + 6x + 9

= 2x * ( 2x + 3 ) + 3 * ( 2x + 3 )

= ( 2x + 3 ) * ( 2x + 3 ) = ( 2x + 3 ) >2 Ans

 We will learn about Function Notation, by visiting the online math tutors and understand the concept related to this topic.  We can also download Previous Year Question Papers Of CBSE for the particular subject and it will help us to  have a look on the  pattern of the  previous year question papers.

 

Factoring Trinomials

We know that the trinomial is the polynomial formed by three terms. To learn about Factoring Trinomials, we say that the standard form of the trinomial is ax>2 + bx + c.

 To find the factors of the above given polynomial , we say that we will either write it in form of some or the other identity or we will try to factorize it by the splitting method. In the splitting method, we say that the middle term of the trinomial is split in such a way that the sum of the two terms is equal to bx (i.e. the second term) and the product of the two split term is equal to the product of the first and the third term of the trinomial.

 Let us look at the following examples:

If we have the polynomial:  4x>2  + 12x + 9

 Here we can   write the above given polynomial as  ( 2x )>2 + 2 * 2x * 3 + 3>2

We observe that the above given polynomial is in the form of the identity (a + b) >2 = a>2 + 2 * a * b + b>2

 So it can be written as (2x + 3) >2

 If we solve the polynomial by splitting the second term we say it can be written as :

4x>2 + 6x + 6x + 9

= 2x * ( 2x + 3 ) + 3 * ( 2x + 3 )

= ( 2x + 3 ) * ( 2x + 3 ) = ( 2x + 3 ) >2 Ans

 We will learn about Function Notation, by visiting the online math tutors and understand the concept related to this topic.  We can also download Previous Year Question Papers Of CBSE for the particular subject and it will help us to  have a look on the  pattern of the  previous year question papers.

 

How to use Gcd Calculator

In the previous post we have discussed about How to use LCM Calculator and In today's session we are going to discuss about How to use Gcd Calculator, By GCD, we mean greatest common divisor. If we want to learn about the divisor of the given numbers, we mean that   the greatest divisor, which is divisible by all the given numbers.  To learn about gcd, we will first find the factors of the given numbers. We must ensure that the factors of the given number should be all prime factors. Now we will pick the factors of the given number such that  they are common to both the numbers whose gcf is to be calculated.  In case the gcf is one, the two numbers are not having any of the common factors. We use gcf to find the lowest form of the fraction numbers, rational numbers or the ratios. It simply signifies that the two numbers whose gcf is 1, cannot be divided by the same number. We must remember that the  gcf of two prime numbers is always 1
 Suppose we want to find the gcf of 9 and 12
Here we will first find the prime factors of 9 and 12. So we say that the prime factors of 9 = 3 * 3 * 1 and the prime factors of  12 = 2 * 2 * 3 * 1
 In both the prime factors we find that the numbers 1 and 3 are the prime factors of both the numbers. So  1 * 3 is the gcf  of  9 and 12.
 We can also use gcd calculator to practice the problems based on gcd and understand its logics clearly.  If we want to learn about the Central Tendency, we can take online help from the math tutor which is available online every time. To know more about the contents of CBSE syllabus we will visit the site of CBSE and collect the recent information to update our self.

How to use LCM Calculator

Lcm stands for the least common multiple. By the term Lcm, we mean the process of finding the number, which is the least common multiple of all the given numbers.  Let’s first see what is a multiple.  By the word multiple, we mean that the number which exactly divides the given number. Thus if we say that 12 is the multiple of 3, it means that the number 3 completely divides the number 12.
 Now if we  have to find the Lcm  of the two numbers say 3 and 6, then we will first write the  multiples of 3 and the multiples of 6, which are as follows :

Multiples of 3 = 3 , 6 , 9, 12, 15, 18,  21,  24,  . . . .  . .
Similarly, we have the multiples of  6 = 6,  12,  18, 24,  . . . . . .
 Now we will observe the common multiples of the two numbers 3 and 6 as 6, 12, 18, 24, 30 . . . . which means that these numbers  divides both the numbers completely.  Out of these numbers we say that the number 6 is the smallest number. So we come to the conclusion that  6 is the L.C. M of  the numbers 3 and 6.
In the same way we can find the LCM of 3 or more numbers also. We must remember that the LCM of 1 and any number n is n itself.

  We can also download Lcm Calculator or use it online to understand the concepts of Lcm. To understand the concept of the Circle Graph, we can study and learn from the books of CBSE or take the help of online math tutor.  We also have CBSE Sample Paper online which the students can download and  use as a guidance tool for the  students preparing for the exams and in the next session we will discuss about How to use Gcd Calculator.

Least Common Denominator

 In today's session we are going to discuss about Least Common Denominator, By Least Common Denominator, we mean the LCM of the denominators of given fractions. Before we learn about Least Common Denominator, let us quickly recall the terms LCM, fractions & denominator; we are already familiar with. By LCM, we mean lowest common denominator. It can be calculated for 2 or more numbers by long division, listing of multiples of the given numbers or in the best way, by prime factorization. Now, fractions are numbers in the form a/b . A fraction, as we can see in the expression, has two parts, a & b. Here, a is the numerator & b is the denominator of the fraction. Thus, we have recalled denominator also that it is the lower part in the fraction.
Now coming back to the topic of discussion, i.e. , Least Common Denominator or LCD ; as stated above is the LCM of the denominators of two or more fractions . But why do we need to find such LCD. As we know that the fractions may be like or unlike depending on whether their denominator is same or not & also that to add or subtract fractions; the fractions must be like fractions. If the fractions to be added or subtracted are like, we can add or subtract them easily. But for unlike fractions, we need to make them like by changing their denominators to Least Common Denominator. This is done by finding the LCM of the denominators of all unlike fractions. As for example; if we have to add 3/7+2/5+4/3. Here the fractions are unlike. So we’ll find the LCM of their denominators which comes out 105. Now we can change the fractions & make their denominator 105 by finding equivalent fractions. Thus, finally we have 3/7+2/5+4/3 = (45+42+140)/105=227/105. You can find the Independent Variable Definition at different places online . Also look for ICSE class 10 syllabus online and in next session we will discuss about How to use LCM Calculator.

Adding Fractions Calculator

Fractions are the numbers which can be expressed in the form a/b , where a is called the numerator & b is called the denominator of the fraction . Numbers, in Mathematical terms can be added, subtracted, multiplied & even divided. Thus, we can do any of these on fractions as well; they being numbers. Here, we will learn about Adding Fractions Calculator.
As we are already familiar with the term addition of numbers, we will extend the same to addition of fractions now. Addition means more or something increased by. So, when we add fractions, we increase them actually of find a fraction one more than the other.
Now let us recall that fractions may be like or unlike depending whether the denominator of the given fractions is same or not. If the fractions have same denominator, we call them like fractions; else they are unlike fractions.
Adding like fractions is quite simple. We just have to add the numerators of the fractions to be added & give the denominator of the fractions given to the sum of numerators so obtained. Thus, if we have to add 5/9 & 2/9 , we observe that the fractions are like . So, we will just add the numerators and we get 5/9 + 2/9 = (5+2)/9 =7/9.
But, when we have to add unlike fractions, we first need to find the LCM of the denominators of the fractions to be added; thereafter making the given fractions like by taking their equivalent fractions with LCM as the denominator of all the fractions to be added. Next, we proceed like addition of like fractions. e.g., 1/6 + 2/5; these are unlike fractions & the LCM of denominators 6,5 is 30. Thus, changing the fractions to those with denominator 30, we get 1/6=5/30 & 2/5=12/30. Now, adding them we get 1/6+2/5=5/30+12/30 = (5+12)/30=17/30
You can get more on simplifying fractions online. Also class 12 icse sample papers are available online.