Exponents in on Grade XII

In previous article, one of the important topics of mathematics that is Quadratic functions was explored and in today's article we are going to move further in grade XII to learn Algebra online syllabus with Exponents and exponential functions. This topic has very short introduction of itself but various situations and applications of it, tends it to cover a wide portion of Algebra unit.
Let us first start with the general description of exponent after that we will move to the main topic of Exponential functions with their growth and decay property. Before coming into 12 standard students learn almost all the aspects of logarithm and in respect of those exponents as well. Whenever an exponent is described it is clear that presence of Euler constant 'e' is surely there in the function.  The main reason to use exponent in any function is that the corresponding constant change in any variable in parallel with other independent variable is measured with the help of graph. For more on exponents visit here.
If any function is described as f(x) = y^p then 'y' is called base and 'p' is its relative exponent. There are some pre-defined properties for exponential function as follows:
Suppose two integer variables are m and n, then
   • yⁿ = y × y × · · · × y with n factors
   • y^1/n means the nth root of a. That is, a1/n is that positive number which satisfies
                                 (y^1/n ) × (y^1/n )) × · · · × (y^1/n ) = y
       Where there are n factors on the left hand side.
   • y^m/n = (y^1/n ) × (y^1/n ) × · · · × (y1/n ) (y^1/n )
   • y⁻ⁿ = 1/ yⁿ
    
The above situations are used when x is an integer but if ‘x’ is a real number then:
Suppose ‘p’ is a rational number
                                
                               p=m/n;          here m, n are integers
                               
Now assume ‘x’ as a real number that cannot be written in a form of rational number. Two common examples of this type or irrational numbers are x = √2 and x = π.
For other possible real numbers that can be easily converted into fraction of rational form use decimal to fraction conversion for example:
                                     x = 3.914712334317 . . .
Then, if we are working to 3 decimal points we would write
                                            x ≈ 3.915
and this number can certainly be expressed as a rational number:                                                
                                       x ≈ 3.915 = 3915/1000                                                  
So, in this case                                                          
                                 y^x = y^3.914712... ≈ y^3.915 = y ^3915/1000
                 
and the final term: y ^3915/1000 can be determined in the usual way by using your calculator.
This is all about Exponents; now let us move towards the next part of this session that includes the description of Exponential functions.
Exponential Functions: - suppose a function is as f(x) = y^x
Now for this function the value of ‘x’ clearly varies according to a fixed base value and its appropriate exponent value.
So the functions (y^x) (as different values are chosen for a) are called exponential functions. Let us explain in more practical manner:
                 If y>z> 0 then    y^x > z^x       only if x>0    
And if x<0 then         y^x < z^x                   
Euler constant ’e’ is the most important and widely used exponential function whose particular base value is 2.7182818 . . . ,
                                          e = 2.7182818 . . .
The number ‘e’ has its own importance in mathematics like number π and, like π it is also an irrational. So ‘e’ also cannot be written as the quotient of two integers. The value of ‘e’ is stored in most of the science calculators to sort out exponential function.
There are various ways of calculating the value of ‘e’. For example, ‘e’ represents the end point of sequence of numbers as follows:
(2/1)¹ . (3/2)² . (4/3)³ . ......  (18/17)¹⁷.....  ..  (65/64)⁶⁴.....
Decimal form of these ( up to 6 decimal points) is as:          2.000000,   2.250000,    2.370370,    ...,   2.637929,     . . . , 2.697345,    ...
This slowly converging unit can represent as single unit as ‘e’s expression as:
e= Lim _{m-> infinity} [( m + 1) /m]^m
There is one more way to calculate the value of ‘e’ and that is infinite series:
e = 1 + 1/1! + ½! + 1/3! + ¼! .........+ 1/m! + .....
Here students know that
                         m! = m × (m − 1) × (m − 2) × . . . (3) × (2) × (1)
According to the standard analysis it can be shown that first eleven terms of this series represents the value of ‘e’ with almost negligible error of 3 × 10⁻⁸.
In general all functions of the form of a^x are called exponential functions but still to be more specific we refer to e^x as
The exponential function.
 Exponential growth:-
If s > 1 then it can be easily represented no matter how large K is:                                  
                                      s^x/x^k   → ∞ as x → ∞
It means if K is a fixed value and can be chosen as large as wanted, still if the value of ‘x’ increases then s^x will also increase and being ahead of it always. It will happen until the value of‘s’ is greater than 1. This phenomenon of s^x growth as x increases is called exponential growth.
 Exponential decay
As explained in previous point that the behavior of ‘e^x’ as x → ∞ is known as exponential growth. Just in similar passion,
 if we explain or characterize the expansion of e^–x then that procedure will called as Exponential decay.
This is all about Exponential functions and their relative properties. As the definition of exponential function defines that in it we observe the relative change in any variable in respect of any other variable. This variation may occur in two different styles as Direct and Indirect.
Let us first talk about the direct variations in exponential function. If two variables are said to be direct variant of each other then it means that if one of them get doubles then other one also changes in same ratio, similarly if the second one is reduced in some amount then first one is also decreased in same ratio. For example: assume a function as y = x + 1
If value of ‘x’ grows up then at the same time value of ‘y’ also gets increased.
Now let’s see “what is Inverse variation?”, this type of variation causes opposite change in any value whenever the related value has increased or decreased. For example, if x and y are two values that are related to each other with Inverse relation then if value if x increases as it doubles then at the same time the value of y gets decreased by half of its value. This type of change gets complicated when more than two independent variables are included in the function. Let us take example of Newton’s kepler theorem:
Formula is as F = Gm/r²
So it represents that whenever mass of the body gets doubled then the strength of force also gets doubled and when distance is increased then the strength of the force gets fall in its fourth.
The thing is that both variations: direct and Inverse, are occurring together so determination of the value gets complicated.
This is all about Exponents, Exponential function with their growth and decay functionality, and types of variations in exponents; all of these grade XII Algebra topics comes as a single chapter in your syllabus because there queries solution requires implementation of their common properties, which are explored in detail in this article.
In the successive article of this one, we will discuss the next included algebra topic in grade XII syllabus and that is co-relation of data.
After reading this article still there is any issue or doubt remains in your mind or you want to solve related math queries of it, and then access the online math tutoring websites where students like you can ask their queries and surely will have immediate assistance by online math tutor. These math education websites provides proper interactive session to students so that they can ask their problems on the regular basis to be clear in every math concept. Students can use some available options like online chat, relevant study material, video seminars, unsolved and solved math worksheets, online math calculators for quick solution of the problem, and many more. All these options increase the ease level of the students’ study that causes their better involvement in studies.
Grade XII math syllabus is comparatively vast than any other standard and the main difficulty in it is that all the topics of it usually have the math application problems whose introductory part was studied by you guys in your previous classes. So always learn every math topic with your pure attention to execute it with ease in the final examination because math is a kind of practical subject that can enrich your aggregate percentage when potentially studied otherwise it also can ruin your overall performance in other subjects.

In upcoming posts we will discuss about Patterns and Functions in Grade XII and Standard distributions. Visit our website for information on 12th physics syllabus Maharashtra board

Quadratic Equations in grade XII

In previous article of Algebra I, we had already explained the fundamentals of Rational expressions and polynomials, today in this article we are going to continue with another topic of algebra 1 that is Quadratic equations and quadratic functions. The introductory part of quadratic equation was learned by students in their previous classes but still to make you guys remember it, let us start with the basics concept of Quadratic equations.
As we all know that every algebraic equation is a polynomial function and so on Quadratic equation is also, but the point is that every quadratic equation is a non-linear polynomial function of second order and it is in the form of monomial because quadratic equation has many derivatives but all of them are of a single variable, Suppose x is an unknown variable then its quadratic equation could be of the form:
Ax² + Bx + C = 0, here A, B, and C are integer constants. One compulsion to have this form as quadratic is that “A is not equal to zero”, if it does then the equation is said to be linear not quadratic.
For example:x² + 2x = 4 is said to be quadratic because it can be written in the form of Ax² + Bx + C = 0,
Let us take another example;
Suppose given equation is as 2x² – 1/x + 2 = 0
Here 1/x can be rewritten as x^-1, and quadratic can’t have x raised to the power other then 1 or 2. So the equation is not quadratic.
Now let us come to the main point and that is “ how to solve any quadratic equation?”, in previous article you guys have learned the factoring procedure and in between that session it was nicely explained that factoring can be used to solve any algebraic equation, so quadratic equation is also an algebraic form.  Let’s see how factoring is being applied to evaluate quadratic equation solutions.
There are 3 simple steps need to be executed to sort out the quadratic equation:
1.    First, write the given equation into standard form and for this you can use distributive property and additional property to combine like terms and put them into one side of ‘=’ sign.
2.    Now factor the obtained equation by using simple factoring technique.
3.    Now evaluate the possible values of variable by putting each factor equals to zero according to Zero product law.

Let us take an example and observe the actual execution of above defined steps:
(6x + 8)x = 14
Apply distributive property
6x² + 8x = 14
Imply addition property
6x² + 8x – 14 = 0 ( in standard form)
Now apply the factoring technique as:
6x² – 14x + 6x -14 = 0
Now take the common factor out
6x( x-1) – 14 ( x-1) = 0
(x - 1) (-6x + 14) = 0
Use the zero product law
 (x - 1) = 0 or (-6x + 14) = 0
x = 1 or x = -14/6 = -7/3

This is the specified way to sort out quadratic equation by using factorization. There is one more way to sort out these types of equations and that is Extracting square roots. After extracting square roots equation automatically comes into the correct form, for example:
( 2x-5 )2 + 5 = 3
Use additional; property to add -5 both sides of the ‘=’ sign
( 2x-5 )2 = -2
Now extract square roots
±√(2x-5)² = ±√(-2) (negative and positive sign both are included when square root of any value is taken)
(2x – 5) = ±i√ 2 ( simplify the radical and apply the definition of ‘I’)
Apply addition property of equality
2x = 5 + ±i√ 2
Now use division property of equality
x = (5 + ±i√ 2)/2
There are two more ways to evaluate any quadratic equation, one of them is “completing the square” and other is “ quadratic formula”. We will discuss both of them but let us start with Completing the square method.
Let’s take an example and in parallel of its solution we will explain the implementation of this method,
Given equation is as: 3x² + 4x – 7 = 0
First step: use addition equality property and isolate the variable terms like x² and x, at one side of the equation.
 3x² + 4x = 7
Second step: use division equality property and divide each term on both sides by integer coefficient of term ‘x²’.
3x²/ 3 + 4x/3 = 7/3
x² + (4/3)x = 7/3
Third step: now take coefficient of term x, square it and add in both sides of equation; here X’s coefficient is 4/3 so its ½ is 2/3, now square of (2/3) is added to both sides of the equation:
x² + (4/3)x + (2/3)² = 7/3 + (2/3)²
x²+ (4/3)x + (4/9) = 7/3 + (4/9)
Multiply 7/3 with 3/3 to have the same denominator on RHS
x² + (4/3)x + (4/9) = 21/9 + (4/9)
x² + (4/3)x + (4/9) = 25/9
Next step: factor the left side by using the standard formula:
( a² + 2ab + b²) = (a + b )²
So , x² + (4/3)x + (4/9) = 25/9
(x + 2/3 )² = 25/9
Next step: use; extract the square root technique for finding the relative roots of the equation
√(x + 2/3 )² =±√( 25/9)
x + 2/3 = ± 5/3
Again use addition property of equality
x = -2/3  ± 5/3
Now two equations can be formed as
x = -2/3  + 5/3  OR x = -2/3  - 5/3
By solving these equations
x = 1 or x = -7/3
This method is used to solve complex problems and for some easy problems prefer the previous method, which is much easier than this one.
Now it’s time to talk about the standard quadratic formula and its use to evaluate Quadratic function. If quadratic equation is in the form of
Ax² + Bx + C = 0 then the standard quadratic formula is represented as:
; here (b² – 4ac) is called determinant of the function, which decides the behaviors of the roots of quadratic equation.
While using this formula, the only task for students is to convert the given equation into standard form, for example:
Given equation is: (y + 3)² = ( y – 2)
Rewrite the above equation as its factors:
( y + 3) ( y + 3) = (y - 2)
Now use the standard distributive property to combine like terms:
y² + 6y + 9 = y – 2
Use addition equality property:
y² + 6y + 9 – y + 2 = 0
y² + 5y + 11 = 0
Now the equation is in standard form so we can implement the quadratic formula, first grab the values of a, b and c from the equation.
a = 1, b = 5 and c = 11.
Put this value in the formula


; by using the ‘i’ definition roots can be easily determined.
This is all about Quadratic equations and functions. In this article we have explored all the possible ways of solving any quadratic equation. These four ways of evaluating quadratic equation covers the topic of quadratic function in grade XII math. Students of this class most of the times fixed in troubles because there are several ways in which a similar problem can be asked and every time there is a bit of changes in the solution process, so it is hard to memorize all the ways and forms of quadratic equation. This task only can be accomplished by doing practice as much they can and to help students in it Online math tutor websites do their role by giving immediate assistance and various worksheets to solve.  If any kind of doubt stays in student's mind related to any topic then the whole fundamentals gets messed up, so it must be required that student asks his queries on regular basis and this will happen only when student does not hesitate in front of tutor which most often happen. To troubleshoot this problem, Online services uses online Chatting option via text or video, to build a better conversation between students and tutors, and because most of us are friendly with Internet platform, the access of this service gets easier.

In grade XII math content Algebra is covering most of the problem region so it gets more essential to learn its each and every problem with all the possible ways of query presentation. For having the detailed descriptions of other topics of Grade XII math syllabus, keep following the successive articles that will surely add a wide range of knowledge.

In upcoming posts we will discuss about Exponents in on Grade XII and Methods of data representation. Visit our website for information on Maharashtra state board

Polynomials in XII Grade

In previous article we had introduced the whole scenario of grade XII^th math and now from here we will start the proper demonstration of each and every topic one by one. Today’s topic is polynomials and their solution by using factoring technique. When the term polynomial arises then its application, rational expression also comes into the account. It is expected from students of this class like you to have the knowledge of polynomials, still for making you guys remind about polynomials, let us start a fast flash back of polynomials:
Polynomials are the mathematics expressions that include variables and constants that are related by using arithmetic operators like addition, subtraction, and multiplication.
Now as the general methodology of math expression, every equation can be of two forms either linear or non-linear. You can also play linear equations worksheets to enhance yours skills, But in both cases polynomials functions are needed to be normalized for simplifying them. Simplification of polynomials every time mean that students need to calculate the values of unknown variables of equation.
Let us start with the representation scenario of any polynomial function after that we will discuss the solution process of it;
F(x)= a_nxⁿ + a_n-1 x^n-1 + .……+ a₂x² + a₁x + a₀

Every polynomial includes three terms as:leading term, constant and integer co-efficient. Here x_n is the leading term and (a_n, a_n-1, ……., a₀) are integer co-efficient.
When any polynomial is said to be as linear then it means all the derivatives of the equation are of same order and similarly when the scenario is opposite means: all the derivatives are not of the same order then it is said to be non-linear polynomial function. visit here for more on polynomials.
For example:
X² + Y² = 8 (here both are of same order so it is Linear equation)
X² + y = 7 (here x and y have different orders, so clearly it is an example of non-linear equation)

There are 4 major elementary properties of any polynomial equation, those are as follows:
1.    Sum of two or more polynomials always results as polynomials.
2.    Product of two polynomials is also a polynomial function.
3.    When two polynomials are combined together then the result is obtained by substituting variable of the first polynomial by the second one.
4.    Suppose a polynomial is a_nxⁿ+ a_n-1x^n-1 + ... + a₂x² + a₁x + a_0,
and  na_nx^n-1 + (n-1)a_n-1x^n-2 + ... + 2a₂x + a₁ is the its derivative thenif the set of co-efficient (an, an-1, ……., a0) does not contain the integer value that time Ka is to be evaluated as k times of a.
Now the basic theme to categorize any polynomial into distinct category is the presence of number of unknown variables in the function. As the name suggests Poly means “many”, so any polynomial function may have various unknown variables. Suppose any equation includes all the derivatives of only one unknown variable so that could be called as Monomial and if two variables are taking part then the formed polynomial is Binomial. Let us learn this one with examples:
-2xy + 3x - 5z ( a trinomial, because of three unknown variables)
-10xy (monomial,)
7xy + z (binomial)

That’s all about polynomial presentation, now it’s time to move on into its solution process. There are various ways of solving any polynomial equation but in this article we are going to elaborate factoring procedure.  As we all know that factoring is a normalization technique and most often used for polynomial evaluation because the only problem while solving any polynomial equation is that it includes number of derivatives of various order, which makes its presentation a bit complex.
So let us first talk about factorization, the problems of factors was studied by you and other grade XII students in their early grades but that time you guys were dealing with linear equations and here the given equation is most of the time of non-linear form.
There are mainly two situations according to which factoring also needs to be implemented in different ways, the situations are:
If the polynomial equation is ax² + bx + c
Then when constant c is positive and other when c is negative,
So let us explore these two situations and will take suitable example for better explanation:
First when the constant c is positive in that case:    polynomial only can be factorized when there are 2 factors of product (ac) that can be added into the absolute value of b.
For example:
6x² + 11x + 3
Here a= 6, b = 11 and c = 3

Now we need to know that are there two factors of (ac= 18) whose sum is 11, the answer is yes, the sum of 2 and 9 is 11 and product is 18.
So by rewriting the equation:
6x² + 11x + 3
6x² + 9x + 2x + 3
Put the common factor out
3( 3x + 1) + 2x( 1 + 3x)
(3 + 2x)(1 + 3x)
Now let us talk about the second situation where c or a  is negative,
-800x² -800x + 600
We can rewrite this equation by taking the common factor out as:
200( -4x² – 4x + 3)
Now if here the product of a and c is taken then the the result is negative.
(ac) = -XII, now the factors are 6 and 2 but 6 should be placed as negative to result the sum as -4.
200( -4x² – 6x + 2x + 3)
200( 3( 1 – 2x) + 2x ( 1-2x))
Now here common factor is (1-2x)
200(3 + 2x)(1- 2x)
This is the way to use the factoring normalization procedure to sort out the complex polynomials. While solving any polynomial related query one thing is to be remembered always that whenever there is any common factor present in all derivatives, then make sure to keep it out. This fundamental is the key statement to sort out the complex rational expressions.
Rational expressions are the fraction form that includes complex polynomials in its numerator and denominator. Now when term polynomial arises students must gets sure that he needs to normalize the expression first by using factorization as done above. Let us take an example to make you better understand this:
(x-2)/ (x+4)  +  (x+1)/(x+6) = (11x + 32)/(x² + 10x + 24)
(x-2)/ (x+4)  +  (x+1)/(x+6) = (11x + 32)/ (x+4)(x+6)
By applying cross-multiplication in the above expression:
(x+4)(x+6) [(x-2)/ (x+4)  +  (x+1)/(x+6) ]  = (11x + 32)
(x+4)(x+6) (x-2)/ (x+4)  + (x-2)/ (x+4)  (x+1)/(x+6) = (11x + 32)
Cancel out the common terms, after that remaining term is as :
(x+6) (x-2) + (x+4) (x+1) = (11x + 32)
Multiply the above terms
X² + 4x – 12 + x² + 5x +4 = 11x +32
2x² + 9x – 8 = 11x + 32
Simplify it now by using the normal arithmetic transitions
2x² -2x -40 = 0
2(x² – x – 20) = 0
Now send 2 into the denominator of RHS.
X² – x -20 =0
(x - 5) (x + 4) = 0
So it gives, x = 5 or -4
This is how any complex rational expression can be sorted out; in this article we have explained the polynomials, factorization and rational expression. All these three terms are related to each other because for solving those similar fundamentals are required to implement like Normalization.
It is quite understanding that when students solve math queries then several doubts occur in their minds and because there is nobody to give them immediate assistance the whole concept of topic is messed up in his mind, that’s why the service of online math tutoring is reaching the success bar because they fulfill the requirement of that immediate assistance. Online math tutor is always virtually present with you through internet platform. Student can ask any kind of mathematical query and surely he will get instant answer in much explained manner. Student can review lessons as many times he want to make himself comfortable in that topic. In present time Online math tutoring is better option than other private tuition classes because private institutes are running as secondary school for students where the rush of students does not allow them to interact in a friendly way with tutor and this is most required term to understand the actual difficulty of the students in particular subject.

When student go with Online math learning then tutors provide some friendly and useful options to student for managing a proper learning session and to compare your analytical skills with other students across the globe, These features are video aids, text chatting option, video conference, 24 x 7 hours availability, Online tests, various worksheets to solve and option of choosing the tutor according to own appropriateness. Every single topic of math is very well categorized in these type of online math websites and you not need to search a lot, just type your query and either the direct solution or the instant assistance by online math tutor is given to you in the mean time.

In upcoming posts we will discuss about Quadratic Equations in grade XII and Correlation and causation. Visit our website for information on 12th state board syllabus Tamilnadu