polynomial factoring

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Polynomials can be factored using either the synthetic division technique or mid – term splitting (in case of quadratic polynomials). If the quadratic polynomials are possible to be factored, their middle term is split and we get the roots. The other way is to use the formula directly to get two roots. Let us consider some examples to know how polynomial factoring is done:
Example 1: Factorize the polynomial 5 x² + 8 x + 3 = 0?
Solution: The given polynomial is of degree two and so number of roots it would possess will be 2. As it is possible to split the middle term of this polynomial, we do it as follows:
5x² + 8 x + 3 = 0,
Or 5 x² + 5 x + 3 x + 3 = 0,
Or 5 x (x + 1) + 3 (x + 1) = 0,
Or x = -1, -3 /5.
Example 2: Suppose we need to find the highest common factor between x³ + 2x² + 8 and x² + x + 4?
Solution: In such a case we use the synthetic division technique as follows:
x³ + 2 x² + 8 / x² + x + 4; Remainder = x² + 4,
x² + x + 4 / x² + 4; Remainder = x,
x² + 4 /x; Remainder = 4,
We see 4 is the common factor between two polynomials x³ + 2 x² + 8 and x² + x + 4.

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factor polynomials

In the previous post we have discussed about  How do you Determine if a Polynomial is the Difference of Two Squares and In today's session we are going to discuss about factor polynomials. We define polynomial as the expression which has the combination of different terms. If a polynomial has only one term, then we say that it is a linear polynomial. On the other hand, we say that the expression with two terms is a binomial and the expression with three terms is called a trinomial.

We can factor polynomials and write them as the product of different expressions. There are different methods to find the factor of the given polynomials.

First method is by finding the common factors and taking them common. Suppose we have the polynomial say 4x>2 + 2x, so we find that 2x is common factor of both the terms in the given polynomial. So we will write the given polynomial as follows :

2x ( 2x + 1 ) .

Thus the given polynomial can be written as the product of 2x * ( 2x + 1 ).

Another method of finding the factors of the given polynomial is by breaking the middle term.

Let's us take the polynomial x>2 + 7x – 18:

We will break the middle term of the given polynomial such that the sum is 7x and the product is - 18x>2, which is the product of the first and the third term. (know more about factor polynomials, here)

Thus the given polynomial can be written as follows:

=2x>2 + 9x – 2x -18 ,

=2x>2 -2x + 9x -18,

= 2x (x – 1) + 9 (x – 1),

= (2x + 9) (x – 1) .

Some times the polynomial are similar to the standard identities, which can be directly written in their form.

For example if we have 9x>2 - 4y>2,

= (3x)>2 – (2y) >2 ,

=(3x – 2y ) (3x + 2y).

This solution is based on the identity a>2 – b>2 = ( a + b ) * ( a – b ) .

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How do you Determine if a Polynomial is the Difference of Two Squares

Hi friends, we will discuss How do you Determine if a Polynomial is the Difference of Two Squares. Polynomials are expressions in such a way that it consist of variables with exponents and constants values. Exponent values present in a polynomial expression is of any degree. Generally these Polynomials expression are used in Trigonometry, calculus, algebra and so on. There is a rule defined in polynomials so that polynomial contain constant, variables, exponents and operations but they cannot have any type of division operator in expression. Polynomial expression don’t have Radicals, infinite number or any type of negative exponent. Now now we will understand that How do you Determine if a Polynomial is the Difference of Two Squares.

Now we will use some step to solve polynomial:
Step 1: To find polynomial first we need to solve the given expression. For example: suppose that we have given a polynomial expression 2p² + 2p² - 10 – 6. Now we have to solve it as 4p² – 16.
Step 2: Then test the Integer value present in equation. The integer value present in equation is a perfect Square. In the equation integer value is 16 that is a perfect square. If we want to write it in terms of exponent then we can also write. It can be written in exponent form as 4².
Step 3: Now we will see again the equation and also check that if it is make a difference of two perfect square number that this equation is denotes a subtraction of two perfect square terms. Now we have to set above equation in format of subtraction of two square terms that is p² – q².

Step 4: Now we have to find the factor of this equation by using the difference of two square formula that is (p + q) (p - q). Then we get the equation 4p² – 16 that is written in factorized form as (2p + 4) (2p - 4).

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How do you Determine if a Polynomial is the Difference of Two Squares

In the previous post we have discussed about Factoring Polynomials Calculator and In today's session we are going to discuss about How do you Determine if a Polynomial is the Difference of Two Squares. Hi friends, in mathematics, we will see different methods to solve a polynomial expression. Before learning How do you Determine if a Polynomial is the Difference of Two Squares. First it is necessary to learn about definition of polynomial. Polynomial can be defined as any types of expression that can be written using constant, variable and exponent values in it. For example: 4ab² + 7xy² – 4x – 25. Given example is a polynomial expression. Now we will understand that How do you Determine if a Polynomial is the Difference of Two Squares. Steps to follow to determine polynomial differences are shown below: (know more about Polynomial, here)

Step 1: The word difference means subtraction. It means subtract one value to other value. For example: The difference of 9 and 3 is given as 3, in mathematical it can be written as: 9 – 6 = 3.

Step 2: If we want to calculate if a polynomial is the difference we need to subtract one polynomial value other polynomial value.

Step 3: Then we have to check the answer that it matches a given polynomial or not.

Step 4: To satisfy the above statement, the given polynomial can be ready to be factorized into two different factors. For example we have an polynomial expression: p² – q². As we see this, it is an difference of polynomial two squares. If we find the factor of given example then it can be written as:

= p² – q², on finding it factor we get:

= (p + q) (p – q), here we get the difference of two squares.

In this square polynomial case power value of square should be even, if power of polynomial expression are odd then it is not square polynomial. In this way we can easily solve the square polynomial expression.

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Factoring Polynomials Calculator

In the previous post we have discussed about polynomial factoring calculator and In today's session we are going to discuss about Factoring Polynomials Calculator. Hello friends, in this blog we will understand that how to Factoring Polynomials Calculator. If in any equation constant value, variables and exponent values are present then it is polynomial expression. For example: 3xy² – 6x + 2y³– 20. As we see in given expression that polynomial expression is joined with mathematical operators. In mathematics, Negative and fraction values are also present in case of polynomial expression. It never joined by division operator.

Polynomial factoring calculator is a type of machine that help us to calculate tough problem very easily. Let’s discuss some steps to calculate the factor of polynomial expression.

Step 1: Put polynomial expression in first text box.

Step 2: Or enter coefficient of square, cube in one text box and coefficient of ‘x’ in next text box and constant in last text box.

Step 3: Then press solve button to get result.

By using the given steps we can calculate the factor of polynomial expression.

Now we will discuss how to find factor of polynomials expression. Here we will discuss quadratic method to find polynomial expressions.

Let’s have a polynomial expression u²+ 4u – 10, we can factorize this polynomial as shown below:

We can find its factor by quadratic formula. Formula to find factors is given as:

U = -b + √ (b² - 4ac) / 2a, here value of 'a' is 1, value of 'b' is ‘4’ and value of 'c' is ‘-10’. So put these values in formula. (know more about Polynomials , here)

U = - 4 + √ [(4)² - 4(1) (-10)] / 2(1),

U = - 4 + √ (16 + 40) / 2,

U = - 4 + √ (56) / 2. So, we get two factor of this polynomial expression, one positive and other negative.

U = -2 + √ 28 and U = -2 - √ 28.

This is how we can find the factor.

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polynomial factoring calculator

Polynomial expression can be defined as any equation contain constant value, variables and exponent values joined by mathematical operators. Exponents values can be 0, 1, 2, 3, 4 and 5 ….etc. For example: 9xy² – 3x + 7y³ – 20. In mathematics, Polynomial expression can also have negative and fraction values. It cannot be joined by division operator.

Polynomial factoring calculator is a mathematical tool that help us to solve hard problem very easily. Those are unknown about polynomial factor can also find the factor of polynomial. Let’s understand some steps to find the factor of polynomial expression.

Step 1: First enter the polynomial expression in the text box.

Step 2: In other word put the coefficient of square, cube in one text box and coefficient of ‘x’ in other text box and the constant in last text box.

Step 3: Then enter the solve button to get the result.

Using these steps we can find the factor of polynomial expression. (know more about polynomial factoring calculator, here)

Now we will understand how to find Factoring Polynomials. We can find the polynomial expressions factored by two methods i,e. direct method and by quadratic method. Here we will understand quadratic to find polynomial expressions.

Let’s take a polynomial expression u² + u – 4, we can factorize this polynomial as mention below:

We can find its factor by quadratic formula. Formula to find factors is given as:

U = -b + √ (b² - 4ac) / 2a, here value of 'a' is given as 1, value of 'b' is given as ‘1’ and value of 'c' is given as ‘-4’. So put these values in formula.

U = - 1 + √ [(1)² - 4(1) (-4)] / 2(1),

U = - 1 + √ (1 + 16) / 2,

U = - 1 + √ (17) / 2. So, here we get two factor of this expression, one positive and other negative.

U = -1 + √ (17) / 2 and U = -1 - √ (17) / 2.

This is how we can find the factor.

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prime factorization calculator

Prime factorization defines as a process in which a number is expressed in form of its prime factors. First we have to know about the prime factors that are numbers which have only two factors that are 1 and itself means numbers which are only divided by 1 and itself are known as prime numbers. So when we want to do prime factorization , It will be expressed in the multiplication form of prime numbers.

For process of prime factorization there is an on line tool that is known prime factorization calculator. It will help in calculation of prime factorization in easy and effective manner. It gives the perfect result quickly. It internally follow all the rules of prime factorization that is also called as integer factorization. We can calculate prime factorization of a number just by entering it into the text box of calculator and by clicking on the submit button. It provide the appropriate answer without delay.

It will be explained by an example as if there is a number 15 then prime factorization is calculated as divide the number by smallest prime number that will divide it that is 3 means 15 / 3 = 5 so first prime factor is 3 and 5 is itself a prime number so it can not be further divided, so prime factorization of 15 is 3 * 5. So it is a way of finding prime numbers that generate the original number when multiplied together.

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Prime Factorization Calculator

Prime Factorization is the way of finding prime factors and prime factors are the numbers that are only divided by one or itself. These prime factors are whole numbers that are greater than one.

When we talk about the prime factorization it is describe as a process in which find the multiples of a given number that are in form of prime number.

We can explain it as 6 can be prime factorized as 2 * 3 .Both 2 and 3 are prime numbers and these are factors of 6 , so it is called as prime factorization. (want to Learn more about Prime Factorization, click here),

There is an on line tool that is known as Prime Factorization Calculator used for calculation of prime factors of given number. In Prime Factorization Calculator there is a text box in which user enter number of their choice and calculator will find prime factorization easily and accurately. It is a very time efficient tool that provide the answer of given problem related to prime factorization quickly.

Internally it follows all the rules of finding the prime factorization. We can describe all the rules related with the process of prime factorization that are as follows:

In the first step check that by which number given value is divided by using the division rule.

And in next step divide number and check whether quotient is prime or not ,if not then it will further divided by the prime number using the division rule.

At last when the generated quotient is a prime number than show all prime factors in multiplication form.

If we have a number 12 then it prime factors are first divide it with 2 that gives 6 and it is also divide by 2 that gives 3 and 3 is prime number and it will not further divided so the prime factorization of 12 is 2 * 2 * 3.

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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 )
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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.

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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.
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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.

How to deal with factoring trinomials worksheet

To understand about factoring trinomials worksheet, we will first learn about what are trinomials. Trinomials are the polynomials, which has three terms. To find the factors of the trinomials,  say ax>2 + bx + c we will first break the middle term bx such that the  sum of the two terms is equal to bx and the product of the two spitted terms equals to the product of ax>2 and c.  Once the middle term is split into two parts, we observe that the polynomial now has 4 terms. Now we take common terms from the first two terms and similarly we take common from last two terms in such a way that we are left with the common terms in the brackets. Thus taking the common terms common, we get the factors of the given polynomial. (want to Learn more about trinomials , click here),
Sometimes we have the trinomials such that it forms the perfect squares of the given terms, in such situation we will write the terms in the forms of the perfect squares. Let us try some of the examples:  4x>2 +   12x + 9
    = (2x)>2 + 2 * 2 * 3 * x + (3)>2
 It is equivalent to the formula for  (a + b ) >2, so  that a =  2x and b = 3, so the resultant factors of the equation are
  = ( 2x + 3 ) >2
This solution can also be attained by the method of breaking, which is done as follows:
= 4 x >2 +  6x + 6x + 9
= 2x *  (  2x  + 3 ) + 3 ( 2x + 3 )
= ( 2x + 3) * ( 2x + 3 )
To learn more about the Box and Whisker Plot Definition, you need to take the help from online tutors. The detailed curriculum of ap state education board for different grades is available online.

In the next session we will discuss about Binomial Experiments and Read more maths topics of different grades such as subtracting integers worksheet in the upcoming sessions here.

factor algebra calculator

If we talk about algebra then you always need to deal with factors, factors can be done easily you need to have good knowledge of divisibility rule. If you are going to do the factor of any number then the first thing you need to notice is that the number is divisible by any number or not . We can make the factor algebra calculator by factoring any number but as I told you earlier that you also need to have knowledge about divisibility rule. If a number has 0, 1,2,4,6,8 at its end then it is divisible by 2. If you add the digit and the sum is a factor of three then the number will be divisible by 3, if the last two digit of any number is divisible by four then the whole number is divisible by 4. And if a number contain the end digit as 0 and 5 then number is divisible by 5. In this way we can find the factor of any given number , now if we find the factor of 126 then this number contains 6 at the end so it is a factor of two but when we divide it by two the resulting number will be 63 , now it doesn’t contain a even number  at the end so now 2 will not be the factor if we see the sum of the number is 6 + 3 =9 and 9 is divisible by 3 so three will be the factor, if we divide 63 by 3 we have 21 , now 21 is again divisible by 3, so if we divide 21 by 3 we will have 7 and as we  know that seven is a prime number so only 7 will be the factor. So required factors are 2, 3, 3, and 7. Factors play a very important role in solving Algebra Problems. As the tamilnadu board exams are near so u need to buy Tamilnadu Board Sociology Sample Papers from the nearest shop.
In the next session we will discuss about How to deal with factoring trinomials worksheet and Read more maths topics of different grades such as Properties of Irrational Numbers in the upcoming sessions here.

Perfect Square Trinomials

Hello friends. Previously we have discussed about consecutive exterior angles and In today's session we are going to discuss about Perfect Square Trinomials which comes under school secondary board of andhra pradesh,

Perfect Square:  It is a number whose square root is a rational number.
Example:
121 = 11²
121 is a perfect square of 11 which is a rational number.
0, 1, 4, 9, 16, 25, 36, 49, 64, 81, etc all these are the perfect squares.
The perfect squares can also be in the form of p/q
Example:
9/49, 16/81, etc are also perfect squares.
Trinomials: In polynomial expression there can be many terms in an equation but in factoring trinomials there must have exactly three terms connected by a plus or a minus sign.
Example:
4m² – 3m + 2
m² + 7m – 8
Perfect Square Trinomial: It can be represented by the following formula x² ± 2xy ± y²  (in another way we can see this formula as the square of the binomials); such trinomials are called perfect square trinomials.  Perfect square trinomials can be formed when a binomial is multiplied by itself.
(x ±y)² = x² ± 2xy ± y²
X² + 18X + 9is an example of perfect square trinomials, it can be shown in the form of (x + 3)²
Let’s see what the outcome is when we square any binomial, take (x + y)
(x +y)² =(x + y)(x + y) = x² + 2xy + y²
The square of a binomial expression gives rise to the following three terms:
1.       x²  as Square of the first term of binomial
2.       2xy as Twice the product of two terms
3.       y² as Square of the second term of binomial
It going to be the same if there is minus sign in place of plus sign.
But we have to check first whether a trinomial is a perfect square trinomial or not.
In the next session we are going to discuss factor algebra calculator
and if anyone want to know about Binary Numbers then they can refer to Internet and text books for understanding it more precisely.
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Binomial Theorem

Previously we have discussed about subtracting integers worksheet and In today's session we are going to discuss about Binomial theorem which comes under board of secondary education ap, It is defined as the algebraic expansion of the powers of a binomial expression . A binomial expression consists of two terms containing the positive or negative sign between them . For example: ( x + y ) or ( p / q² ) - ( k / q ⁴ ) etc .
We can explain Binomial theorem as when the binomial expression have the power of ' n ' then it would be expand by the help of binomial theorem . It would be describe as ( 1 + a ) ⁿ = ∑ ⁿ_r=0 c _rⁿ x ^r .
The above expansion can understand by an example as
( p + q ) ⁴ = p ⁴ + 4 p ³ q + 6 p ² q ² + 4 p q ³ + q ^{4 .}In the example binomial coefficient in the expansion of ( p + q ) ⁴ are define as the coefficient in expansion of ( x + 1 )ⁿ are c _r ⁿ or _n c _r or ( ⁿ _r ) . for finding the values of the coefficient Pascal's triangle is used .
                      1

                    1 1
                   1 2 1
                 1 3 3 1
                1 4 6 4 1
              1 5 10 10 5 1
            1 6 15 20 15 6 1
But by calculating Binomial Probability Formula for computing the numbers in the pascal triangle so that we can easily expand the formula easily without referring the triangle it is stated as :
( a + 1 ) ⁿ = c ⁿ _n a ⁿ + c ⁿ _{n -1}a ^n-1 + c ⁿ _{n -2} a ^n-2 + ….......+c ⁿ₂ a ² + c ⁿ₁ a + c ⁿ_{0 .}
In the next session we are going to discuss Perfect Square Trinomials 
and Read more maths topics of different grades such as Properties of Numbers
  in the upcoming sessions here.     

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Binomial Experiments

Hi friends,Previously we have discussed about multiplying polynomials worksheet and topic we are going to discuss today is binomial experiments which is a part of ap board of secondary education. The binomial experiments are part of the algebra mathematics. The binomial experiments are experiments in which have four conditions.
1 ) the number of trials are fix.
2 ) each trial is independent to others.
3 ) only two outcomes are possible.
4 ) the probability of each outcomes are constant from trial to trial.
These processes are performed with a fixed number of independent trials, each have two possible outcomes.
The binomial experiment examples: tossing a coin 10 times and see how many heads occur; Asking 100 people and find the result, if they watch xyz news; rolling a dice and see if the number 6 appears. Examples of the experiments that are not binomial experiments: rolling a dice a 6 appears (in this not a fixed number of trials), asking to the 10 people and how old they are (this means at least not two outcomes).
Binomial Probability example:
Two coins are tossed simultaneously 300 times and it is found that two heads appeared 135 times, one head appeared 111 times and no head appeared 54 times. If two coins are tossed at random, what is the probability of getting 1) 2 heads 2) 1 head 3) 0 head?
Solution : total number of trials = 135.
Number of times 1 head appears = 111.
Number of times 0 head appears = 54.
In a random toss of two coins, let e₁, e₂, e₃ be the events of getting 2 heads, 1 head, 0 head respectively. Then, 1) p(getting 2 heads)=p(e₁)= number of times 2 heads appear / total number of trials.
135 / 300 = 0.45
2) p (getting 1 head)= p(e₂)= number of times 1 heads appear / total number of trials.
111 / 300=0.37
3)p(getting 0 head)= p(e ₃ )= number of times no heads appear / total number of trials.
54 / 300=0.18
the possible outcomes are e₁, e₂, e₃ and p(e₁), p(e₂), p(e₃)=(0.45+0.37+0.18)=1
In the next session we are going to discuss Binomial Theorem and if anyone want to know about Properties of Complex Numbers then they can refer to Internet and text books for understanding it more precisely.

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Solving Binomial Expansion

In college algebra is the part of arithmetic, which perform the calculation on the variables. In the algebra, expressions are written in the form polynomial which is a part of secondary board of education andhra pradesh. The binomial polynomial (polynomials worksheet) contains two terms. These two terms can be considered as a collection of or arranging the two monomial with operators and brackets option. Through this session we are discussing about Solving Binomial Expansion. Dividing Polynomials can be defined as two variables which perform some expression or some operation. Like a² – b² = (a + b) (a – b)
The algebra provides the lots of way to solve the binomial expression. In the mathematics to solve the binomial expression we use the concept of expansion of binomial theorem. Binomial theorem can generally be represented by the formula given by Blaise Pascal in the 17th century. The most generic example of binomial expression is given below:
(a + b)² = a² + 2ab + b²
 In the more general term binomial formula can be expressed as a:
     (a + b)ⁿ = ∑∞i=0  (n/i) aⁱ b^n-i
Here (n/i)  is a binomial coefficient and ‘n’ is a real number.
In the simple term binomial expression can be written as:
(a + b)² = a² + 2ab + b²
(a + b)³ = a³ + 3a²b + 3ab²  + b³
As same like binomial theorem formula can be written as:
(a + b)ⁿ = aⁿ + n(a^n-1 b¹) + n(n-1)/2! (a^n-2 b²) + n(n-1) (n-2)/3! (a^n-3 b³)+……….+ bⁿ

Here we will show you binomial expansion examples and how to solve the binomial expression. (Know more about Binomial Expansion in broad manner, here,)
Example1: Solve the binomial given expression (a + 5)³.
Solution:  Given that (a + 5)³, as we can see that it’s a binomial expression. So this can be solved by following the binomial expansion formula:
So, solution become in simplified formula:
      (a + 5)³ = a⁵ + 3x² (5) + 3x (5)² + 5³  
      (a + 5)³ = a⁵ + 15 x² + 75x + 125
As clear from the above solution, we can easily solve the binomial related problems.
In the next session we are going to discuss Binomial Experiment and Read more maths topics of different grades such as  How to do Estimate Quotients in the upcoming sessions here.

Binomial Expansion

Hello students ,Previously we have discussed about column multiplication and in this blog we are going to discuss about the Binomial Expansion which is a part of secondary school board andhra pradesh. Binomial Expansion is related to the algebraic expansion of powers of a Binomial Distributions. We have a case when n is a positive integer then the expansion of ( 1 + a )ⁿ is equal to ∑ⁿ_r=0 c_rⁿ a^r. Coefficients of 'a' that appear in the expansion of ( 1 + a )ⁿ are known as binomial coefficients. (Know more about Binomial Expansion in broad manner, here,)

By using the formula of expansion you can easily expand the series without doing the multiplication. There are mainly two properties of binomial expansion that include :

1. ( n + 1 ) terms contained by an expansion .
2. Binomial coefficients c_rⁿ should be integers .
   We can understand it by an example as ( a + b )² is described in terms of expansion as
   a² + 2ab + b² where ab is the coefficient .
   Binomial Expansion examples :
   ( a + b ) ⁴ : It expands as a⁴ + 4 a³ b + 6 a² b² + 4 a b³ + b⁴ . Here in each term exponents of a and b are non negative integers with sum of the powers of a and b is equal to n. In above example n is equal to the 4 .
   Binomial Expansion can be understood using Pascale's triangle that applies to expand the terms in form of ( a + b ) ⁿ .
                                                                        1

                                                                         1  1
                                                                        1 2 1
                                                                       1 3 3 1
                                                                      1 4 6 4 1
                                                                    1 5 10 10 5 1
                                                                  1 6 15 20 15 6 1
We can take an example to understand the Pascale's triangle as ( a + b )³ is expanded by taking the row of triangles starting with 1 and 3 is 1 3 3 1 therefore expansion of ( a + b )³ is described as
( a + b )³ = a³ + 3 a²b + 3 ab² + b³ .
In the next session we are going to discuss Solving Binomial Expansion
and if anyone want to know about Math Blog on Estimating Quotients then they can refer to Internet and text books for understanding it more precisely.

 

Math Blogs on Solving Trinomial Multiplication

Mathematics is an interesting subject when all the concepts of the problem are very well understood and questions become easier to solve. In the mathematics there are several rules, theorems and method to solve the problem in a mean time. Previously we have discussed about antiderivative of cos2x and In today's session we are going to discuss about Solving Trinomial Multiplication which comes under ap secondary education board, In  general meaning solving the trinomial equation is referred to as factorizing (or factoring) the trinomial. factoring trinomials is very easy to understand and easy to solve the problem.

    Factoring the trinomial equation is the part of algebra , before performing the factorization user must be aware of fluent addition and multiplication practices. Here is the example given below which shows the basic step of Solving Trinomial by Multiplication. (Know more about Trinomial in broad manner, here,)
  example          X² + 10X + 16
In above example, we can see that an equation with three variables are known as trinomial equations. In this equation we deal with 10x. To solve the trinomial we have to factor the 10x in two parts in the manner whose multiplication get equals the multiplication of  X²  and 16. Here we show you how to solve the problem.
              X² + 10X + 16  ( here multiplication of X² and 16 is 16X²)
Here we partitioning the 10x in two parts which satisfying the above condition.
             X² +  8X + 2X +16    here we can see that factoring the 10X into 8X and 2X perfectly satisfy the above condition, which is multiplication equals to 16X.
 Now move forward, we take common value form the first two and last two equations.
     X(X + 8) + 2(X + 8)
     (X + 8) (X + 2) that’s our final answer means solution of Trinomials equation.
In the next session we are going to discuss Binomial Expansion  and if anyone want to know about Math Blog on Estimating Quotients then they can refer to Internet and text books for understanding it more precisely.    

 

Multiplying Trinomials

Hello friends as I believe you have understood the previous topics in algebra, I am going to discuss about what Multiplying Trinomials is, what the examples of multiplying trinomials are and how to multiply trinomials. Firstly we will talk about how to multiply trinomials. For learning trinomial multiplication we need to know what is a trinomial so we will discuss first about trinomial then we will move to find out what is multiplication of trinomial. (To get help on cbse click here)
Trinomial is on the whole a part of polynomials. ‘Tri’ means three so trinomial can be seen as an expression that contains three terms. It has broad applications in quadratic functions. 2a+3b+7 is a trinomial because it contains three terms 2a, 3b and 7. So by this example we can create a general form i.e.-AX+BY+C.
When we multiply one trinomial with other trinomial it is called as multiplication of trinomials. Now we will discuss how to do multiplication operations between two trinomials with some examples -
Example 1 : What is the result of following multiplication -
(4x+5y+7) * (2x+3y-9) = ?
Solution : For solving these type of question we multiply each term with every term -
Step 1 : Firstly we multiply trinomial (4x+5y+7) with 2x -
(4x+5y+7) * 2x = 8x2+10xy+14x …....eq(1)
Step 2 : Now we multiply trinomial (4x+5y+7) with 3y -
(4x+5y+7) * 3y = 12xy+15y2+21y …....eq(2)
Step 3 : After this we multiply trinomial with 9 -
(4x+5y+7) * 9 = 36x+45y+63 ….....eq(3)
Step 3 : Now we perform operations which are giving in question -
so, we add eq(1) and eq(2) because here given that 2x+3y :
8x2+10xy+14x + 12xy+15y2+21y = 8x2+22xy+14x+21y+15y2 ….....eq(4)
Now, we subtract eq(3) from eq(4) :
8x2+22xy+14x+21y+15y2 – (36x+45y+63) = 8x2+22xy-22x-24y+15y2-63
So, the result of (4x+5y+7) * (2x+9) = 8x2+22xy-22x-24y+15y2-63
This is a Multiplying Trinomials example which shows how to multiply trinomials.
Now I think you have understood about the multiplication of polynomials. In the next topic we are going to discuss Solving Trinomial Multiplication and In the next session we will discuss about Math Blogs on Solving Trinomial Multiplication. 

How to Factor Trinomials

Hello students, in this topic we are going to learn about Factoring Trinomials. First we have to understand the term trinomial. A trinomial is basically an equation which is present in form of x² + bx + c, these are types of equations you read in algebra. So first we have to understand the meaning of the factorization. It means that if we have any value then break that value in their elementary numbers we can understand it by an example as 35 is factor in its smallest terms as ( 7 * 5 ) which means “ 7 times of 5 is 35 “ .  (To get help on icse click here)
Now factoring trinomials can be understand by an example as a² + 14 a + 24 than we factor the terms of the equation in form that it's factor have the multiplication 24 and their addition will be 14a. So we write the factor of the 24 as ( 2 * 2 * 2 * 3 ) then we arrange these factors as their total is equal to 14 , so we do the splitting as (a + 2 ) ( a + 12 ) So after factor trinomials solver we get the equation as
=a² + 12 a + 2a + 24 = a ( a + 12 ) +2 (a + 12 )
= ( a + 12 ) (a + 2 )
The factor of trinomial is a² + 14 a + 24 are ( a + 2 ) and ( a + 12 ).
Now we will take another example for factoring the trinomial as an equation
p² + 7p -30
Here the sign of the third term is negative so the factors of it will be in form if ( p + k)( p - k)
So the factor of -30 which have the difference 7 is as follows:-
30 = ( 2 * 3 * 5 * 1 ) so we make the pair as it have the difference 7 so it is 10 - 3
And we write it as p² + 10p -3p -30 = p ( p + 10 ) -3 ( p + 10 )
= ( p – 3 ) ( p + 10 )
So the factor of equation p² +7p – 30 is ( p -3 ) ( p + 10 ).
In the next topic we are going to discuss Multiplying Trinomials and In the next session we will discuss about Multiplying Trinomials.