Significance of Math

Do you what to know what is arithmetic progression:

The arithmetic series is nothing but the addition of the arithmetic sequences. The series of the arithmetic numbers which are given as the sequences. The sequences are to be added to get the series form. The geometric series is nothing but the addition of numbers of the geometric sequences to form the series. Now we are going to see about the arithmetic and the geometric series.

About the Arithmetic and Geometric Series:

We know that the arithmetic series are given as the sum of sequences of numbers. The arithmetic series formula is used to find the certain number of sequences which can be given as follows,

S_n = n(a₁+ a_n)/2

The infinite geometric series are the sum of the arithmetic sequence formula of numbers. The geometric series formula can be determined by using the formula as follows,

S_n = a₁ (1 - rⁿ)/1 – r

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Familiarization with math

Let us learn steps to divide and simplify:

How to divide a fraction is the technique to perform, it only requires multiplication. keep the first fraction as it is when two fractions are dividing and then multiply the first fraction with the reciprocal of the second fraction (i.e. invert of the second fraction). Example: How to divide fractions 1/6 / 7/8 Keep the first fraction 1/6 as it is = 1/6 And then reciprocal the second fraction = 8/7 Need help with linear programming examples 1/6 x 8/7 = (1 x 8)/ (6 x 7) = 8/42 = 4/21.this is after multiplying both the fractions.
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Introduction to Solve exponential equations

Online Solving makes the student to interact with the online tutors and let them to clear their doubts in solving exponential equation.

Solving it online makes exponential equation is one of the instant types of learning where student can solve the problem easily with the guidance of online educators.

Equation is which has the variables in the exponent. Same log,logarithmic rules are used to find Exponential equation do you want to learn how to calculate pi.

Solving Exponential Equations Online ...

• To find the exponential equation, take log on either sides and solve the variables.
  
• When the base of the exponential equation is equal on either sides, then the equation can be solved by using the property, if b^x = b^y, then x = y, where b > 0 and b ≠ 0.
• Stay Step forward to math

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Do you work out fractions

* If they add six fraction values, when the fractions have same denominator values means adding six fractions done basically. Denominators are same means, they add the numerator. If there is a necessity, simplify the result fraction values.

* When the fractions have different denominator values means convert the denominator in to common denominator by using the L.C.M and then they perform rename the fraction values and add fractions get help with online graphing calculator with table.

* When the fractions have mixed numbers means they add the mixed numbers in six different ways such as converting mixed numbers in to improper fractions and another way they add the whole numbers and fractional numbers separately.
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Introduction for Linear Programming:

Linear programming solver is the universal method of most favorable part of limited wherewithal such as labor, substance, engine, resources etc., to quite a few competing behavior such as goods, services, jobs, projects, etc, on the fundamentals of known criterion of optimality. The phrase limited at this point is used to describe the availability of scarce resources during planning period. The principle of optimality usually means either performance, return on investments, utility, time, distance etc.

Structure of Linear programming examples:

The LP model contains the following 3 fundamental elements.

• The decision variable that we seek to determine.
• Objective (goal) that we aim to optimize (minimize or maximize )
• Constraints that we need to satisfy.

Formulation of linear programming problem:

The procedure consists of the subsequent chief steps

Step 1: Study the given situation to find the key decision to be made.

Step 2: Identify the variables involved and designate them by symbols x_j (j=1, 2…)

Step 3: Express the feasible alternatives mathematically in terms of variables, which generally are: x_j >=0 for all j for Math Question and Answer.

Step 4: Identify the constraints in the problem and express them as linear inequalities or equation involving the decision variables

Step 5: Identify the objective of the function and express it as a linear function of the decision variables.

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Let us learn about bar graph examples:

In pictograph, we use pictures to represent data. This Process of representing numerical data take more time and it is difficult to compare two items. Here rather than symbols and pictures it is better to use bars. This type of using bars is called bar graph or bar diagram. The bar-graph represents the information by rectangular bars in which a unit square is taken to indicate the value of certain number or quantity. Here, let us know how to draw or how to construct the simple bar graph.

Simple Bar Graph-example Problem :

1. Draw the simple bar graphs for every month expenditures of a family:

items	expenditures
House rent	2000
Food	3000
Education	800
Electricity	400
Transport	600
miscellaneous	600

Solution:
Bar graphs showing the every month expenditure of a family on various items.

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Let us learn about plane trigonometry:

Trigonometry is the division that deals with study of plane and spherical triangles. Plane trigonometry consists of 2 dimensional triangles the ones that we can draw on the paper.or we can say Plane trigonometry is the sub division of trigonometry in which its principles are applied to the plane triangles.If enough sides and angles are known we can calculate the remaining sides and angles as well as area and hence we can solve the triangles. Triangles can also be solved by the law of sine and law of cosines.

Important things to remember

1) Pythagorean theorem

2) Trignometric Functions

3) law of Sines and Cosines

4) Trigonometric identities

5) Half and double angle formula

Do you need to know more about kinds of triangles .

Basic Formulas Used in Plane Trigonometry

Pythagorean theorem

As square of the hypotenuse is = sum of the squares of the other 2 sides in the Right Triangle.The hypotenuse is the longest side.if two sides of triangles are given as a and b and hypotenuse = c so according to the theorem

From the above right triangle we have ( hypotenuse)²= (leg)²+ (height)²

c²= a²+ b²

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Introduction to algebra factorising:

In algebra equations factorisation is one of the important method. Factorisation gives the x values of the given algebraic equations. Factoring the equations gives the desired value of the function. It is very easy and simple. General form of the algebraic equation can be written as, ax² + bx + c. In factorisation, we find two or more factor values. Quadratic equations are mainly used for factorising process.

Problems - Algebra Factorising

Find the factor value of the given quadratic equation x² - 2x - 35 = 0

Solution:

Given quadratic equation is x² - 2x - 35 = 0

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First, we have find the two numbers that add to be give the value of (- 2) and give the product of (- 35).

For in this case, that two numbers are - 7, 5. So we exchange - 2x as (- 7x + 5x)

Therefore, the given equation can be rearranged and written as,

x² - 2x - 35 = 0

x² - 7x + 5x - 35 = 0

Then, group the first two terms and second two terms

(x² - 7x) + (5x - 35) = 0

Take the biggest common number from the group, we get

x (x - 7) + 5 (x - 7) = 0

(x + 5) (x - 7) = 0

Separately equate the two values to zero, we get

x + 5 = 0, x - 7 = 0

x = - 5, 7

Answer:

The factors are x = - 5, 7

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Introduction to histogram definition online tutoring:

Histogram is an easy method to represent database. The histogram graphs are useful to compare the data that which one is high and low range. The histogram graphs have area of rectangles by representing frequencies. One of the most important source for students since students can easily access the materials for their related subjects; this can be achieved by online. Histogram chart is one of the quality tools. Tutoring helps to online relationship between students and staffs. Students get the information from the tutor. Here we are going to see about histogram definition online tutoring.

Histogram Definition Online Tutoring - Steps:

The following steps used to construct the histogram in definition of histogram tutoring:

Need more clear picture on math symbols and definitions

Step 1:

In histogram graph, the information ranges are dividing by equal intervals in a column. For example (0 to 5, 5 to 10, and 10 to 15)

Step 2:

We need to create x-axis and y-axis. X-axis represented by horizontal position and y-axis represented by vertical position.

Step 3:

Given data (class intervals) are plot into x-axis (horizontal position).

Step 4:

Given data (frequency) are plot into y-axis (vertical position).

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Knowledge of Math

Geometry Helps for Kids

Learn three dimensions;
Here we are going to learn the three dimension objects, it has length, width and height it will be come under the category of solid geometry problems free , there are two types of solid geometry those are

• Polyhedra
• Non polyhedra

Polyhedra:
These kinds of shapes made up of the flat surfaces.
Example: triangular prism, tetrahedron
Non Polyhedra:
These kind of shapes have curved surfaces or it is the combination of mixed and flat surfaces so they are not Polyhedra
Example:
Sphere, torus
Example 6-geometry helps for kids
Which objects is used to measure the degrees?
Solution:geometry answers free
We can use the protractor to measure degree, the shape of the protractor is look like a below

from that we can measure the angle 50 degree it will be shown in the below diagram

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Math is a Game

Euclid’s Definitions, Axioms and Postulates

Consider the three steps from solids to points (solids-surfaces-lines-points). In
each step we lose one extension, We can get online geometry tutor also called a dimension. So, a solid has threedimensions, a surface has two, a line has one and a point has none. Euclid summarized solved geometry problems these statements as definitions. He began his exposition by listing 23 definitions in
Book 1 of the ‘Elements’. A few of them are given below :

1. A point is that which has no part.
2. A line is breathless length.
3. The ends of a line are points.
4. A straight line is a line which lies evenly with the points on itself.
5. A surface is that which has length and breadth only.
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6. The edges of a surface are lines.
7. A plane surface is a surface which lies evenly with the straight lines on itself.

Permutation and combination

The permutation and combination takes place on different types of objects. The permutation of a different object is the number of different ways they can be ordered i.e. which is first, second, third, etc. If you desire to choose some objects from a larger number of objects, the way you place the chosen objects is also important. When comes to combination, on the other hand, one does not consider the order in which objects were selected or placed, just which objects were selected.

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Permutation has two types:

• Permutation with Repetition.
• Permutation without Repetition.
  

Combination has two types:

• Combination with Repetition.
• Combination without Repetition.

Hope the content was helpful to understand the topic to learn more on mathematics symbols

Explain Types of functions

Introduction:
            Function is defined as follows, considering two sets A and B. We form the Cartesian Product, we form relations. From all the relations, we can select a few which satisfy the rule that each element of the set A is related to only one element of the set B.
When a relation satisfies this rule, it is called a function.
Any relation on A x B in which (i) no two second elements have a common first element and (ii) every first element has a corresponding second element is called a function.
Understanding Types of Functions:
Linear functions:
 Linear functions are the functions in which the highest degree of x term is 1. There are 2 types of linear functions depending ion the operator between the variables.
Polynomial function:
The function which contains many terms is called as polynomials. Functions which contain 1 term is called as monomial, function which contain 3 terms is called as trinomial etc can be called as polynomial function.


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Understanding Geometry Circles

Introduction:

• A Circle is a simple shape of Euclidean geometry consisting of those points in a plane which are equidistant from a given set point called the center. The common distance of the points of a circle from its center is called its radius.

• Circles are simple closed curves which divide the plane into two regions, an interior and an exterior. In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure (also known as the perimeter) or to the whole figure including its interior.

• In this article of geometry circles formulas of circumference are discussed.

Geometry Circle Formulas for Circumference:

The pictorial representation showing circumference of circle is given below:

The Circumference of a circle in geometry can be calculated using the following formulas using coordinate geometry:
1) Circumference = 2`pi` r
2) Circumference = `pi` d
where r ----> radius of circle
d ----> diameter of circle.

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How to Factor Trinomials Calculator

Introduction:

        The equation or a function is in the structure of  ax²+bx+c =0 (where a≠0, b, c are constants ) called as trinomials. We can also refer it as a quadratic equation formula. An algebraic expressions which  has 3 terms known as trinomials. The trinomials having highest power 2.The trinomials have the  two roots. There are two ways to factor the trinomial according to the co-efficient of x² . In the following section we are going to learn how to factor the trinomials by using trinomials calculator.

Factor Trinomials Calculator:

For factorindg the trinomials we must know the below two methods.

Factor trinomials calculator Method 1:

Example Problems:

The following problem will help you understand the factoring trinomial calculator method 1 and factorizing trinomials.

Example 1:

Find the factors of the following trinomial. x²- 10 x +16

    Solution:

                                                                       16   (product)

                                                                     /    \    

                                                                 - 8      - 2 

                                                                     \    / 

                                                                      -10     (sum)

x²- 10 x +16 = x² - 2x - 8x +16

                     = x ( x-2 ) - 8( x-2 )

                     = ( x - 8 ) ( x - 2 )

So the factors of the given trinomial is (x-8) and (x-2).

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Geometry Coordinate Proofs

Introduction of  Geometry Coordinate:

             A coordinate is a number that determines the location of a point along some line or curve. A list of two, three, or more coordinates can be used to determine the location of a point on a surface, volume, or higher-dimensional domain.

            For example, the longitude is a coordinate, which determines the position of a point along the Earth's equator, and latitude is another coordinate that defines a position along a meridian. The pair of coordinates consisting of latitude and a longitude determines a point on the surface of the Earth.

Geometry Coordinate Proofs Example:

In geometry, prove that the segment created by joining the midpoints of two sides of a triangle is parallel to the third side and has a distance end to end equal to half that of the third side.

                     

Solution:

            There may be a definite amount of doubt that the shape above is completely common. If we show that theorem for the above figure, have we actually proven it for all similar triangles? It would appear that this is a particular triangle, indeed, that has one side on the x-axis and one vertex at the origin in geometry

            The common formula for distance between two spots will allow us to determine the required length. The formula tells us that for any two spots (x₁,y₁) and (x₂,y₂) the length of the circle of segment between then.

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Trigonometry Radian Measure

Introduction:

       An angle is determined by rotating a ray about its endpoint. One way to measure angles is in radians. To signify that a given angle is in radians, a superscript c, or the abbreviation rad might be used. If no unit is given on an angle measure, the angle is assumed to be in radians. `(3pi^c)/2-=(3pi)/2 rad.-=(3pi)/2`                                                                                                                       

Trigonometry Radian Measure:

        Middle angles of a circle contain an angle measure of 1° if it subtends an arc to be 1/360 of the boundary of the circle. These appearances of angle determine is reasonably general. Another appearance of angle determine namely in utilize is radian measure. If a middle angle subtends an arc i.e. identical toward the radius of the circle after that the middle angle contain a computing of one radian.

If a middle angle θ of a circle through radius r subtends an arc of length s, subsequently their radians establish is describing since  `theta= s/r`

Given, radius is 4 cm, and length of arc is 60 cm.

We know that the formula for radian measure of `theta=s/r` .

As a result,`theta = 60/4`

θ=15

The angle of the arc is 15 radians.

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Explain Rotational Symmetry

Introduction:

           The rotational symmetry is objects that look the same after a certain amount of rotation. A object may have more than 1 rotational symmetry for instance, if reflections or turning it over are not counted, the trickling appearing on the Isle of Man's flag (see opposite) has three rotational symmetries (or "a threefold rotational symmetry").

Key Factors of Rotational Symmetry:-

• The rotational symmetry is proportion with value to some or all rotations in m-dimensional Euclidean space.

• Rotations are through isometrics, i.e, isometrics preserve orientation.

• herefore a proportion group of revolving symmetry is a subgroup of E+ (m)

• Rotational symmetry by way of admiration to all rotations about all points implies translational regularity with respect to all translations, so space is all the same, and the rotational symmetry group is the whole E (m).

• The rotational symmetry through deference to any angle is, in two extents, circular symmetry. The basic domain is a half-line.

• That is, no dependence on the angle use cylindrical coordinate and no dependence on either angle using spherical coordinates.

• The basic sphere of influence is a half-plane from side to side the axis, and a radial half-line, correspondingly.

• Ax symmetric or ax regular are adjectives which refer to an object have cylindrical symmetry, or ax rotational symmetry.

• In mains, continue or separate rotational regularity about a plane correspond to correspond mains rotational symmetry in every perpendicular plane, about the point of intersection.

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Common Data Representation

Introduction to Common Data Representation:

Representing a Numerical Data:
             A numerical data is a set of mathematical values which is used for measuring and counting and it can be represented by graphs(Pie-chart, bar graph). A numerical value is generally used for scaling purposes i.e. to find out the range, value ,mean and count of a particular object. We can even represent the data on number line based on the values. Depending on values we can represent data in statistical tables and can give ranks to the data, we can even find frequency distribution.
We can represent the Data by:
Pie Chart:

     Pie chart is a representation on a circle. Therefore its also known as "circle graph". The circle is divided into different partition depending on Category. It is one of the most common graphs for describing a set of measurements. It is the best graphical display for displaying data arranged in categories. Each category is represented by a wedge of the pie and the size of each wedge is in proportion to the percentage of each category.

Bar Graph:

     A bar graph consist two axes and a series of horizontal bars or vertical bars which are placed in the x ,y plane,depending on the value. It is another way of displaying qualitative data.The frequencies along the vertical axis (ordinate) of the chart and the categories on the horizontal axis of the chart(abscissa). Bar chart is used to display frequency and percentages.
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How to multiply decimals

Introduction to Multiplying decimals:

         A number that contains the decimal point is called as decimal. The decimal numbers are always has ten as its base value. In Hindu-Arabic numeric system, decimal notation refers the positional notation of the number. It can also used to refer the non-positional system in Roman and Chinese numerals. Decimals are always can be represented as another form of fractions.

Place Value:

       To know the decimals, we need to understand the place value. The value of a digit can be determined by its position in a number is called as place value. Each place in a decimal number has a different place value.

For Example, 25.64

Here 2-tens

         5-ones

         6-tenths

         4-Hundredths      

Multiplying Decimals:

Rules for multiplying Decimals:

• Multiply the numbers normally, ignoring the decimal points.

•  After multiplying normally, put the decimal point in the answer – The answer will have as many decimal places as the two given numbers combined.

 Consider the following Example,

              13.7* 12.5

Step1: Multiply normally by ignoring the decimal points

              137*

              125

             685

           274

         137 

        17125

How to find the Mean and Median of a data set

INTRODUCTION:

           A group of variables or a set of information is referred as Data. In general, sets are the collections of numbers and set can contains any kind of data. Data set is nothing but a collection of data. Data set can be represented as tables and graphs. Data collection is the method of collecting or preparing the data. For a given data set, we can find the mean, median, mode and range which is explained below.

Mean of data set:
         For finding the mean of all the values in a given data set, we have to find the sum of all numbers given in data set and we have to divide the total sum by the total number of elements.
Example:
Find the mean of data set: {8, 18, 23, 35, 51}
Solution:
         Mean      =    Total sum of data set  / Total number of elements in data set
                         =    (8 + 18 + 23 +  35 + 51) / 5
                         = 135 / 5   = 27
Median of a data set:
        In general, median is referred as the middle value of the given data set.
Example 1 :
Find the median for the given values: {5, 6, 9, 25, 34}
Solution:
      Here, the middle value is 9. Hence median = 9. 

How to translate word problems into algebraic expression

Introduction:

             Translating words into algebraic expression is the process of translating the word problems into an algebraic expression which can be used to solve the word problem and produce the solution for the given words problem. It is the simple way to solve words problem. Let us discuss about translating the words into algebraic expression.

Example problems to translating words into algebraic expression:

Problem 1:
Flowers shop has thirty Roses and forty Lilly. How many pieces of flowers does flowers shop have? 
Solution:
            Let a = Total number of Flowers in the flowers shop.
            The sum of thirty Roses and forty Lilly is equal to the total number of flowers in the flowers shop. It translates the words problem into an algebraic expression.
            a = 30 + 40
Solve this expression.
            Let a = Total number Pieces of Flowers in the flowers shop
                  a = 70.
There are 70 Pieces of Flowers in the flowers shop.

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Example on how to solve polynomial

Introduction:

          In math, variables and constants with the operations of addition, subtraction, multiplication, and non-negative, whole-number exponents of finite length is known as a polynomial or in-determinates. For example, x² − 4x + 7 is a polynomial, but x² − 4/x + 7x^3/2 is not, because its second term involves division by the variable x and because its third term contains an exponent that is not a whole number.

Example:

X²+4X-5=0,

                   X²-X+5X-5=0,

                   X(X-1) +5(X-1) =0,

                   (X-1)(X+5) =0.                         Therefore the roots are X =1, -5...

                       (OR)

USING THE FOMULAE:

                   X = ` (-B + sqrt (B^2 - 4AC))/(2A)`

                   X²+4X-5=0 -------------------> (2)

                    AX²+BX+C=0----------------> (3)

Comparing (2) and (3),

A = 1, B =4, C =-5 Substitute these values in one.

        x = `(-4 +- sqrt (4^2 - 4(1)(-5)))/(2(1))`

          x = `(-4 +- sqrt (16 +20))/(2)`

          x = `(-4 +- sqrt (36))/(2)`

         x = `(-4 +- 6)/(2)`

        x = `(-10)/(2)``(2)/(2)`

       x = -5, 1

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Numeric sequence and Types

Introduction:

The numeric sequence is called linear sequence. The numeric sequence are communicable to the constant rates of change and form the straight lines which is graphed.

        The numeric sequence are  15, 17, 19, 21, 23, 25… is a linear sequence that are represented in the table. The numeric sequence is used to draw graph, they provide straight line.

Example:

15, 17,19,21,23 this example numeric sequence the constant value  is 2.

Numeric Sequence Explanation and Types:

The numeric sequences contain any one number series. Some of then numeric sequences are given below.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ….etc the constant numeric sequences value is one

15, 17, 19, 21, 23, 25, 27, 29…..etc. the constant numeric sequences value is one

The above are simple explanation of the numeric sequence.

Different types of numeric sequences are :

• Arithmetic sequence
• Geometric sequence  
• Fibonacci Sequence

How to solve rational equation

Introduction:

Solving rational equations is one of the important topics that we need to know. What we do to one side of the equation, must do to the other. If we have fractions, we try to eliminate them by multiplying by the common denominator. Suppose, If there are quadratics involved, we must get all terms to one side with zero on the another.

Steps involved in solving a rational equation

Solving rational equation is an important to see at an equation that not having a variable in the denominator to make confident that we see the pattern for calculating rational equations. The following steps we will use in the solution process.

1. First we need to determine the least common denominator of the fractions in the given rational equation.

2. We need to take out the fractions by multiplying All terms by the least common denominator.

3. Then we have to simplify the terms in rational equation.

4. Solve the resulting equation.

5. Check the answers to make confident the solution does not make the fraction undefined.

Let’s look at first, how we would handle the equations like x/3 + 2x/2 = 4

Our first step in solving this equation is to multiply each term by its least common denominator of the fractions, which is 6

(6) x/3 + (6) 3x/2 = (6) 4

Simplifying the above equation results in the linear equation which we can solve to get our final answer.

2x + 6x = 24

8x = 24

x = 3

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Solve absolute value inequalities

Introduction:

        Other than algebra, there is one more important inequality. It is called as Absolute value inequality. The absolute value for any number is numerical value regardless of its sign.  For example, the absolute value of | -5 | is 5 and │+5 │ is 5. Here the vertical lines denotes absolute value.

It is a little more difficult when dealing with equations. There are three possible outcomes for absolute value inequalities. 

1. If x is any expression and a any positive number, and │x │= a, then it has either x = a or x = - a values.
2. If x is any expression and a any positive number, and │x │<>
3. If x is any expression and a any positive number, and │x │> a then it looks   x < -a and x > a.

Example

Solve: │4x – 4 │ = 8

Solution:

Use the result, if |x| = a, then x = a or x = -a

Here │4x – 4 │ = 8

So, it has either 4x – 4 = 8 or  4x – 4 = -8 values.

Solve each equation using the addition and multiplication principles.

4x = 12,    4x = -4

after solving the equation, we get

x = 3,     x = -1. Answer

Hope you like the above explanation to solve the absolute value inequalities, Please leave your comments, if you have any doubts.

Practice of Algebra Application

Introduction:

 Applications of algebra are found everywhere; the principles of algebra are applied in all branches of mathematics, for instance, calculus, geometry, and topology. They are applied every day by men and women working in all type of business. As a typical example of applying algebraic methods.

     Algebra is often referred to as a generalization of arithmetic, so it is collection of rules, then the rules for translating words into symbolic notation of mathematics, rules for formulating statements using symbolic notation, and rules for rewriting mathematical statements in  a manner that leaves their truth unchanged. 

Application of Algebra in Slopes of Straight Lines:

In general slope of equation is y=mx+b
Practice  Problem 1:
Y=x-3
Where m=1 (co efficient of x)
               B=-3
Equation of point-slope :
 Formula for point,slope
                (y-y1)=m(x-x1)

Practice  problem 1.  Find the equation of the line which passes through the point (2, 2) and has slope3.
Solution:
       Equation is    (y-y1)=m(x-x1)
Where, x1=2
                Y1=2
                M=3
So,
        y-2=3(x-2)
        y-2=3x-6
Answer is: 3x-y-4=0

Learn factor polynomials calculator

Introduction:

         Calculator is used to solve different types of problems. It is a web-based tool designed to solve different problems. Learning factor polynomials through calculators is simple. 

      Factor polynomials calculator are used to understand the polynomials factorization. Given expression can be factorized by using the greatest common factor. Let us discuss about the learn factor polynomials calculator.

Steps to learn factor polynomials:

The Steps to learn factor polynomials are as follows:

• Given expression can be arranged in the order of powers.
• Expression can be in the form of standard ax² + bx + c = 0.
• The expression should be factorized.
• Solve the given terms.

Example problems to learn factor polynomials calculator:

Factorize: 42ax + 36bx – 7ax²– 6by² using factor by group
Solution:
Step 1:
          Given expression 42ax + 36bx – 7ax²– 6by²
Step 2:
           Given expression in the standard form ax² + bx + c = 0
                     42ax + 36by – 7ax²– 6by² = 0
Step 3:
           Groups the terms
                      42ax – 7ax² +36by – 6by² = 0
                      7ax (6 – x) + 6by (6 – y) = 0
Step 4:
           The greatest common factor for the expression
                      7ax (6 – x) + 6by (6 – y) = 0
                     (7ax + 6by) (6 – x) (6 – y) = 0
Solution to the given equation is (5a – 6x) (3b – 4y) (b – 3y) = 0.

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About Polynomials

Introduction:

           An algebraic expression in which the variables have only non negative powers is called polynomial.  The monomial midterm study which contains finite number sum in x it said to be polynomial. The coefficients of the polynomial are said to be coefficients of the monomial in a polynomial. If coefficients of a polynomial are zero, then polynomial is said to be  zero polynomial. The highest coefficient power of x in a polynomial is called the leading coefficient for polynomial

Fundamental of Polynomial

Constants and variables are the fundamentals of polynomials.
Examples :In the formula for circumference of a circle, c=2π r ² and π are constants and c and r are variables

Degree and Types of Polynomials

Polynomials with more than one variable, the sum of powers of the variables taken up and the highest sum so obtained is called the degree of the polynomial.
Monomial, Binomial and Trinomials are three different types of Polynomials.

Properties of Polynomials

Addition and subtraction of two polynomials mean combining like terms.
We can perform multiplication and division also

Factorization of Polynomials

     You know that any polynomial of the form p(a) can also be written as
p(a) = g(a) x h(a) + R(a) it implies that Dividend = Quotient X Divisor + Remainder.
If the remainder is zero, then p(a) = g(a) x h(a). That is, the polynomial p(a) is a product of two other polynomials g(a) and h(a).

Order of operation using algebra

Introduction of order of operation using algebra:
 The Order of Operation using algebra is
P          - Parentheses,
E          - Exponents,
M         - Multiplication,
D          - Division,
A          - Addition and
S          - Subtraction

1. Calculations have to be finished from left to right in algebra.
2. First complete the operations inside a parenthesis.
3. Next Complete the operations of  exponents.
4. Then do multiplication and division, from left to right.
5. Then  do addition and subtraction, from left to right.

Step by step process for order of operation:

Parenthesis

                In algebra the parenthesis, combining symbols are always completed from the innermost set outward.

Exponents

                In algebra, an exponent is unlike than just multiplying. 

Example: 2^4 means 2 x 2 x 2 x 2 = 16

Evaluate 2 + 5 x (4 + 3) ÷ 2 - 6 using the order of operations.

Solution: 

Step 1:	2 + 5 x (4 + 3) ÷ 5 - 6	=	2 + 2 x 9 ÷ 3 – 6	Parentheses
Step 2:	2 + 5 x 7 ÷ 5 – 6	=	2 + 18 ÷ 3 – 6	Multiplication
Step 3:	2 + 35 ÷ 5 – 6	=	2 + 6 – 6	Division
Step 4:	2 + 7 - 7	=	2+6 – 6	Addition
Step 5:	2	=	2	Subtraction

Multiplication and Division in algebra

                These operations are completed in the order they show from left to right.  They are finished together because they have same importance.

Addition and Subtraction in algebra

         These two operation have two operations of the same importance. These are finished in the order that they show from left to right. 

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Learn Greatest Common Factor

Introduction:
        Greatest Common Factor (GCF) is the greatest number that is a factor of each of two or more numbers. To find either the Least Common Multiple (LCM) or Greatest Common Factor (GCF) of two numbers, you always start out the same way: you find the prime factorizations of the two numbers. Then you put the factors into a nice neat grid of rows and columns, and compare and contrast and take what you need.
Let us study the GCF of two numbers 24 and 15.
The factors of 24 are 1,2,3,4,6,8,12 and 24.
The factors of 15 are 1,3,5 and 15.
From this we observe that the greatest factor common to both these numbers are 3. In other words we can say that the greatest common factor of 24 and 15 is 3.

Greatest Common Factor Learning

Another method of finding out  the GCF is by using prime factorisation. The prime factors of 8 are 1,2,2,2 and that of 12 are 1,2,2,3. The prime factors that is  common for both numbers are 1,2,2. Multiply together all these numbers and we get 4. We can say that 4 is the greatest common factor of these numbers.

Least Common Multiple

Introduction:
   In arithmetic number of theory is the least common multiple or lowest common multiple (LCM) or smallest common multiple of two integers a and b is the smallest positive integer that is a multiple of both of a and of b. Since it was a multiple, it can be divided by a and b without a reminder. If either a or b is 0, so that number is no such positive integer, then LCM(a, b) is defined to be zero.

Example:

Multiples of 4 are
4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, ......................
Multiple of 7 are
7, 14, 21, ................
(add 4 to each get to next multiple)

Applications

When adding and subtracting, or comparing vulgar fractions, Useful to find the least common multiple of the denominators, is  called the lowest common denominator. For instance,
{3\over15}+{1\over4}={12\over60}+{15\over60}={9\over20},
where the denominator of 60 was used to least common multiple of 15 and 4.

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How to simplify Negative Exponents

Introduction:
      Exponents are also called Powers or Indices. Exponents containing negative sign are called negative exponents. The exponent of a number  says how many times to use the number in a multiplication. Dividing is the inverse (opposite) of Multiplying. A negative exponent number says how many times to divide by the number.
     Calculate the positive exponent (aⁿ)
       Then take the Reciprocal (i.e. 1/aⁿ)
      To change the sign ( minus to plus or plus to minus) of the exponent, use the Reciprocal (i.e. 1/aⁿ).

Simplifying Negative Exponents

Simplifying Negative Exponents:

            Exponent  is a small number written near the top of another number which show how many times a number or a variable is multiplied by itself. Negative exponents  indicate the inverse of the corresponding positive exponent.

Simplifying Variables with Negative Exponents:

Consider a problem like the one shown below.
x^-4  = 1/x⁴

Simplifying Fractions with Negative Exponents:

To increase a fraction to a power, raise the numerator and denominator to that power.
            a^-3b-⁴/b^-2=a-³b^-2


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