Polynomials Calculator

An Algebraic expression is of the form axn is called a monomial. The variable a is called the coefficient of xn and n, the degree of monomial. For example, 7x3 is monomial in x of degree 3 and 7 is the coefficient of x3. The combination of two monomials is called a binomial and the combination of three monomials is called a trinomial. For example, 2x3 + 3x is a binomial and 2x5 – 3x2 + 3 is trinomial. The sum of n number of monomials, where n is finite and x is called a polynomial in x.
Illustration to Polynomials:

Polynomial Calculator Example 1:

The polynomial calculator of the equation x2 + ax + b gives the remainder 18, when divided by x – 2 and leaves the polynomial calculator of remainder –2 when that is been divided by (x + 3).

Find the values of a and b.

Solution to the polynomial calculator:

P(x) = x2 + ax + b.

In tyh is polynomial calculator,

When x – 2 divides P(x) then the remainder is P (2).

∴P (2) = 4 + 2a + b.

But remainder = 18 ⇒ 4 + 2a + b = 18;

2a + b = 14 (1)

When (x + 3) divides P(x)

, the remainder is P (–3).     ∴ P (–3) = (–3)2 + a (–3) + b

= 9 – 3a + b.

But remainder = –2;      ∴ 9 – 3a + b = –2;

⇒ –3a + b = –11 (2)

(1) ⇒   2a + b = 14

(2) ⇒ –3a + b = –11 (subtracting)

5a        = 25

(Or) a     = 5

Substituting a = 5 in equation (1) we get

10 + b = 14; b = 4, ∴ a = 5, b = 4
Subtraction of Polynomials Calculator:

Example for Polynomials calculator:

Subtract        2x3 – 3x2 – 1 from x3 + 5x2 – 4x – 6.

Solution:

Using associative and distributive properties, we have

( x3 + 5x2 – 4x – 6) – (2x3 – 3x2 – 1) = x3 + 5x2 – 4x – 6 – 2x3 + 3x2 + 1

= x3 – 2x3 + 5x2 + 3x2 – 4x – 6 + 1

= (x3 – 2x3) + (5x2 + 3x2) + (–4x) + (–6+1)

= –x3 + 8x2 – 4x – 5.

The subtraction can also be performed in the following way:

Line (1): x3 + 5x2 – 4x – 6.

Line (2): 2x3 – 3x2 – 1.

Changing the signs of the polynomial in Line (2), we get

Line (3): –2x3 + 3x2 + 1.

Adding the polynomials in Line (1) and Line (3), we get

–x3 + 8x2 – 4x – 5.

Integral Part Definition

In geometrical and other applications of integral calculus it becomes necessary to find the difference in the valves of an integral of a function f(x) for two assigned values of the independent variable x, a and b. this difference is called the definite integral of f(x) over the range (a,b) and is denoted by $\int_{a}^{b}f(x)dx=F(b)-F(a)$, where F(x) is an integral of f(x), F(b) is the variable of F(x) at x=b, F(a) is the value of F(x) at x=a.

It is often written thus  : $\int_{a}^{b}f(x)dx=[F(x)]_{a}^{b}=F(b)-F(a)$.

Note 1. The integral $\int_{a}^{b}f(x)dx $ is read as the integral of f(x) from a to b.The number a is called the lower limit and the number b, the upper limit of integration.

Note 2. It should be seen that the value of a definite integral is perfectly unique and is independent of the particular form of integral which may employ to calculate it. Considering F(x)+c instead of F(x), we get

$\int_{a}^{b}f(x)dx=[F(x)+c]_{a}^{b}=[F(b)+c]-[F(a)+c]=F(b)-F(a)$

so that the arbitrary constant disappears in the process and we get the same values as on considering F(x). This is why the name is given as definite integral.

Note 3. It is assumed that a and b are finite.
Definition of Integration by Parts

When the given function cannot be integrated directly by using standard formulae, we try other methods. The process of integration is largely of tentative nature and no systematic procedure can be given as in differentiation. However, the following are two important methods opf integration.

1. Integration by Substitution           2. Integration by Parts

Here, we will discuss about Integration by parts

An Integration is the inverse process of differentiation.By differentiaon we find the derivative of the given function, whereas by integration we find the function whose derivative is known.

If the derivative of F(x) is f(x) then we say that the antiderivative or integral of f(x) is F(x),such that

int f( x ) dx= F( x )

Thusd/dx F(x) = f(x) => intf(x) dx = F(x)

Integration by Parts:

Theorem: if u and v are two differentiable function of x then

int ( u v ) dx = [ u * int v dx ] - int { du/dx * int v dx } dx .

We can express this result as given below:

Integral of product of two function

= ( 1st function ) * ( integral of 2nd function ) - int { ( derivative of 1st function ) * ( integral of 2nd function ) } dx .

We should choose u and v in such a way that the second function v on the right hand side is easy to integrate. Sometimes, this rule has to be used repeatedly. This rule is also useful in integrating logarithmic and inverse t - functions of the type log x, log(ax2 + bs + c), sin-1 x, tan-1 x etc.The following guidelines will help you to see which of the two functions in the product should be taken as the first function.

Notes:

If the integrand is of the form f(x) * xn, we consider xn as the first function and f(x)as the second function.
If the integrand contains a logarithmic or an inverse trigonometric function, we take it as the first function. If the second function is not given in any case we take that as one.

Example Problems on Integration by Parts:

Pro 1: Evaluate, int xsin 2x dx

Sol : Given,

int   xsin 2x dx

=     int sin 2x dx - int { d/dx ( x ) * int sin 2x dx } dx

=     x * ( -(cos 2x)/2 ) - int 1 * ( -(cos 2x)/2 ) dx

=  (-x cos 2x)/2 + 1/2 int cos 2x dx

"-(x cos  + 1/2 * "(sin

= -(x cos 2x)/2 + 1/4 sin 2x + C , which is the Answer.

Pro 2: int logx/x^2 dx

Sol :   Integrating by parts, taking logx as the fi rst function and 1/x^2  as the second function, we get

int "log   dx

=  int (log x) * 1/x^2 dx

=   ( log x ) * int 1/x^2 dx - int { d/dx ( log x ) * int 1/x^2 dx } dx

=    ( log x )(-1/x )   - int 1/x  * ( -1/x ) dx

= - log x/x + int 1/x^2 dx

= - log x/x - 1/x + C, which  is the required Answer.

Practice problems on Integration by Parts:

Pro 1: Evaluate, int (  x cos x) dx    ( Answer:   x sinx +cosx+c )

Pro 2: Evaluate, int e2x sin x dx            ( Answer: 1/5 e2x ( 2 sin x - cos x ) + C )

Natural Logarithmic Calculator

In mathematics, the natural logarithmic function is defined as the function contains three ports, namely the number, the base and logarithm itself,  now the natural logarithmic calculator is the function of  the logarithm function calculator which means we are using to the base of ‘e’ where e is constant, base of  ‘e’ value is given by

Understanding Definition of Logarithm is always challenging for me but thanks to all math help websites to help me out.

e = 2.718281828. and the numbers, and the logarithmic itself.

The logarithmic function calculator is represented as y = log_b^(x).

We take the value of base of ‘e’ is approximately e = 2.71.
Natural Logarithmic Calculator:

Here, we introduce the natural logarithmic calculator. The function of natural logarithmic calculator is represented as  y = ln(x)

Explanation about natural logarithmic calculator:

Input:

calcutor             given value

The given value is conversions by logarithmic calculator

Conversions:

 x =  conversion value.

Solution:

 y = calculated value will be displayed

Here we can the change the equation by x and y.

y = ln(x) for y i n natural logarithm

y = ln(x) for x  in natural logarithm

Note: we can calculate or perform the positive numbers only not negative numbers by natural logarithmic calculator.
Some Problems about Natural Logarithmic Calculator:

Problem1:

To solve the natural logarithmic function of ln(2)

Solution:

Input:

x = 2  given value

Conversion:

x = 2  the given value conversion by logarithmic calculator.

Solution:

y = 0.69 approximately

Problem 2:

To solve natural logarithmic function of  span style="font-size: small; " mce_style="font-size: small; ln(4)

Solution:

Input:

 x = 4 given value

Conversion:

 x = 4  the given value conversion by logarithmic calculator.

Solution:

 y = 1.38 approximately

Problem 3:

To solve natural logarithmic function of ln(6)

Solution

Input:

 x = 6 given value

Conversion:

 x = 6  the given value conversion by logarithmic calculator.

Solution:

 y = 1.79 approximately

Problem 4:

To solve the natural logarithmic function of in ln(10)

Solution:

Input:

x = 10 given value

Conversion:

 x = 10 the given value conversion by logarithmic calculator.

Solution:

 y = 2.30 approximately

Problem 5:

To solve the natural logarithmic function of ln(15)

Solution:

Input:

x = 15 given value

Conversion:

 x = 15 the given value conversion by logarithmic calculator.

Solution:

 y = 2.70  approximately


Practices problems for natural logarithmic function:

Problem1;

To solve the natural logarithmic of ln (20)

Answer is y = 2.99

Problem2

To solve natural logarithmic function of ln (50)

Answer is y = 3.91

Problem3

To solve natural logarithmic function of ln(100)

Answer is y = 4.60

Problem 4

To solve the natural logarithmic function of in ln(99)

Answer is y = 4.59

Hence, here we obtain the natural logarithmic calculator.

Scientific Notation Rules

 Often we come across abnormally large numbers or abnormally small numbers like 400,000,000,000 or very, very small number like0.00000000002. Especially in the field of science such abnormal size of numbers is quite common.

In calculations involving such numbers, the work is cumbersome and possible errors may occur.

A special type of notation is used to simplify the dealings with these type of numbers and it is named as scientific notation as such numbers are very predominant in the field of science.

However a scientific notation must be used following a set of rules, universally accepted.

Let us take a closer look.
Scientific Notation Rules – Description

Since very large or very small numbers can be rounded to the appropriate place value, the power of 10 is used as base in scientific notation. The established general form is,

                                       a x 10n, where,  1= I a I < 10

If  ‘n’ is positive, it represents the number of zeroes from ‘a’ to the right of unit place and hence  it denotes large numbers.

For very, very small decimal numbers ‘n’ is negative and represents the number of zeroes before ‘a’ up to the decimal point.
Scientific Notation Rules – Examples

The velocity of light is about 310,000,000 kilometers per second.

We find there are seven 0s after the digits 1. That is the number is same as 31 times 10 to the power 7. But as per scientific notation rules, 31x 107 is not allowed and hence the given is modified as 3.1 x 108

Thus in scientific notation, 310,000,000 = 3.1 x 108

Similarly it is equally incorrect to mention (as a scientific notation) 310,000,000 as 0..31 x 109

Let us consider another example with a very  small number.

The wave lengths of a certain color is experimentally found to be 0.0000015 millimeter.

There are 6 decimal places to cross the digit 1from right. That the given number is (1/1000000)th of 1.5.

Hence in scientific notation it is written as 1.5 x 10-6

Importance of Derivatives of Trigonometric Functions

Trigonometric Functions 
In Mathematics, trigonometric functions or circular functions are ratios of the sides of a right angled triangle, having the given angle.  Trigonometry deals with functions of an angle present in a triangle. It relates the angles of the given triangle with the length of its three sides. It also helps to find the length and angle of a triangle. Trigonometric functions are widely used in real life in various fields such as carpentry, surveys, engineering, physics, navigation, astronomy etc.

Trigonometric functions are sin p, cos p, tan p, cot p, sec p, csc p.  Among them sine, cosine and tangent are the most familiar and basic trigonometric functions. Before going into derivatives of trigonometric functions, let us first see what these trigonometric functions refer to.

Let us consider a circle of radius 1. If a ray originates at the origin which makes an angle with x axis, what will the sine, cosine and tangent of this angle provide?
• The sine of this angle gives the length to which the triangle has risen on the other side i.e. the y component

• The cosine of this angle gives the x component length

• The tangent of this angle gives the slope of the triangle, which is obtained by dividing Y component by the X component.

Thus, trigonometric functions can also be defined as the lengths of all line segments from a given unit circle.

Derivative of Trigonometric Function
With this basic understanding on trigonometric functions, let us explore on the derivatives of trigonometric functions namely sin p, cos p, tan p, cot p, sec p, csc p.

The derivatives for each of these trigonometric functions can be represented as follows:

Derivative of Sinx
The symbolic expression of Sinx derivative is given by

d/dx (sin x) = cos x   or sin’ x = cos x


It is easy to find the derivatives of sin x. This is because, the derivative of sin x can be obtained by just calculating the cos x.

Derivative of Cosx
The Cosx derivative is given by

d/dx (cos x) = - sin x or cos’ x = - sin x


Derivative of Tan p
The derivative of tan p is given by “tan-1 p = sec^2   p”

Derivative of Cot p
The derivative of cot p is given by “cot-1 p = - coesc^2 p”

Derivative of Sec p
The derivative of sec p is given by “sec’ p = sec p tan p”

Derivative of Csc p
The derivative of csc p is given by “csc’ p = - csc p cot p”

Different types of Matrices

Matrix is a rectangular array of numbers or functions which are called the elements or the entries of the matrix. The elements or entries are arranged in rows and columns in either () or [] brackets. The elements of a matrix are denoted by aij where ‘i’ is the row number and ‘j’ is the column number. We can only add or subtract two matrices if they both have same number of rows and columns.

Order of matrices: An order of the matrices is written as m X n where m is the number of rows and n is the number of columns in it. That means a matrix of order 2 X 2 can only be added to a matrix of order 2 X 2 and a matrix of order 3 X 3 can only be subtracted from a matrix of order 3 x 3.

Types of matrices

1. Column matrix: A matrix is said to be a column matrix if has only one column. For example: - a matrix of order 4 X 1 is said to be a column matrix as it has four rows but only one column. In general, a matrix of order m X 1 is considered as column matrix where m can be any real number.
2. Row matrix: A matrix is said to be row matrix if it has only one row. For example: - a matrix of order 1 X 4 is said to be a row matrix as it has four columns but only one row. In general, a matrix of order 1 X n is considered as row matrix where n can be any real number. A example of row matrix is [2 , 6, 11, 7].
3. Zero matrix: A matrix is said to be zero matrix if all its elements or entries are zero. It is also called a null matrix. For example: - [0], [0 0] are zero matrices. We denote zero matrix by O.
4. Square matrix: A matrix is said to be square matrix if its number of rows are equal to number of columns. Thus m X n is a square matrix if m = n and is known as a square matrix of order ‘n’.
5. Unit matrix: A matrix is said to be unit matrix if all its elements or entries are one. For example: -
[1], [1 1], [1 1 1] are all unit matrices.

Polynomial Factoring

Factoring polynomial is similar to factoring numbers, but you will be dealing with expressions instead of numbers. In order to factor a polynomial, you have to determine the polynomial that divides the original polynomial evenly.

Polynomial Factoring

How to factor a polynomial?
Different approaches are used to factor the polynomial; most common approaches are simple factoring, and factoring in pairs. In this article, you will find examples for these two approaches of polynomial factoring.

Simple Factoring
In simple factoring, you identify a common factor for the expression and place it before the parenthesis. What is a common factor? To know that, look at the example below:
5x + 60 = 5 (x) + 5(6) = 5 (x+12)

Now the common factor of the above expression is 5. To cross verify, multiply the factor with each term in the expression and ensure if you are getting the original polynomial back. For example,

5(x+6) = 5 multiplied with x + 5 multiplied with 12 = 5x + 60 (the original polynomial)

Factoring in Pairs
“Factoring in Pairs” is yet another factoring polynomials solver. This approach is used when you cannot find any common factor for the terms in the expression. In this approach, the expression is split into pairs of terms and then each pair will be factored separately. Here is an example:

Find the factors of ab-5b-2a+10.

This polynomial algebraic expression includes 2 variables namely a and b. There is no common factor for these 4 terms. In this case, take the first two terms and find a common factor. Similarly, take the last 2 terms and find a common factor, as shown below:

ab-5b-2a+10 = b (a-5) -2 (a-5) = (b-2) (a-5)

In the above step, the first two terms ab-5b are taken into consideration. Take the common factor out, it is b. Now it becomes b (a-5). Similarly take the next two terms -2a+10. The common factor is 2 here. When you put it as -2, the expression becomes -2 (a-5). Now the expression can be further simplified as (b-2) (a-5).

If the consecutive terms do not have any common factors, then you can rearrange the terms based on the commonality and then do the factoring.