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.
.

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.     

.

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.

.

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. 

Factorizing a Trinomial by splitting the middle term

Previously we have discussed about is pi a rational number ? and now we are going to learn about factorizing the trinomial which falls under karnataka education board. A trinomial is of the form: ax²+ bx + c. To proceed further with factoring of a trinomial we are going to understand it with the help of an example. The factoring of a trinomial can be done by following certain steps or you can take help of online tutors.
Suppose we have a trinomial x² + 4x +3. We split the middle term such that on multiplying the first and the last term we get 3x² and on addition we get 4x.
We get x² + 3 x + x +3. Now we take the first two terms and the last two terms and take common out of them. We get x(x+ 3) +1(x+3). Note that both the common terms in bracket should be the same. We get (x+3) and (x+1) as factors on factoring trinomials.(Know more about Trinomial in broad manner, here,)
Thus we can see that factoring a trinomial can be easily understood with the help of the above mentioned example. In similar way other types of trinomials can also be solved irrespective of the signs.
Let’s take another example. Suppose we have a trinomial of the form x² -6x +9. To solve this trinomial we again take the product of the first and last terms on multiplication and get 9x². Now we split the middle term such that we get 9x² on multiplication and -6x on addition as follows: x² – 3x -3x + 9. Now we take first two terms and last two terms and take common factors .We get x(x-3) – 3(x-3). So the factors are (x-3) and (x-3). Thus we have learnt about solving a trinomial with the method of factorization and if anyone want to know about Perfect Square Trinomials then they can refer to Internet and text books for understanding it more precisely. Read more maths topics of different grades such as Simplifying rational expressions in the next session here.

Trinomials

Previously we have discussed about how to do trigonometric functions and now we are starting a new topic that is Trinomials which comes under maharashtra board,It is basically a part of polynomial .as you know tri means three so trinomial means an expression(algebraic expression,rational expressions) that contains three terms. It has a wide application in quadratic functions. 2X+3Y+2 is also a trinomial because it contain three terms 2X,3Y and 2. So by this example we can create a general form i.e.-AX+BY+C. This equation is known as trinomial equation.
Now let’s do some more operation on trinomial before going for multiplying trinomial. Trinomials can be factorized which we will see by an example.
X²+5X+4 is a trinomial and we need to factorize them so what we need to do is we need to factorize the middle term with respect to last term. Here the middle term is 5 and last term is 4 so we can factorize 5 with respect to 4 as 5-1=4 and 4*1=4 so we just need to write the above example as
X²+4X-X+4
We just divided the equation in 2 parts and then just taken X common from 1^st part and -1 from second part
X(X+4)-1(X+4)
X+4 is common in both the parts so we take that as well common
(X+4) (X-1)
These are the required factors.
Now we will see how to multiply trinomials(want to Learn more about Trinomials ,click here),
Example: multiply (x² +4x+5) with (x² +3x +6 )
We can multiply both the trinomials, but for that we need to remember following points given below:
We have to multiply the first member with all the members of second trinomial and same thing have to be done with the other variable, and we also need to remember that powers are added up in multiplication.

(x² +4x+5)* (x² +3x +6)
=x²(x² +3x +6) +4x(x² +3x +6)+5(x² +3x +6)
=x⁴+3x³+6x² +4x³+12x²+24x+5x²+15x+30
Now we will arrange them according to their power in descending order.
=x⁴+7x³+23x²+39x+30
This is the required result
Note: in multiplying Trinomials, order is not important,if we multiply (x² +4x+5) with(x² +3x +6 ) or (x² +3x +6 ) with(x² +4x+5) both will give the same result.
Now I think you well understood about the trinomial and Multiplying Trinomials, you just need to practice more all questions are about practicing so just practice hard. This is a brief introduction about trinomials in the next article we will learn about Binomial Distribution by splitting the middle term.