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Remainder Theorem Calculator + Online Solver With Free Steps

The Remainder Theorem Calculator is an online tool that is used to calculate the reminder for polynomials P(x). The Remainder Theorem Calculator works on the remainder theorem formula which divides a polynomial P(x) with a linear polynomial in order to obtain the desired remainder. 

The Remainder Theorem Calculator is a very effective online calculator that solves the issue of long division by providing the solution to the user in a matter of a few seconds. The results obtained by this calculator are quick and always accurate. 

The Remainder Theorem Calculator is very easy to use as it simply takes the input from the user and presents the solution in a detailed manner.

What Is the Remainder Theorem Calculator?

The Remainder Theorem Calculator is an online calculator that is used to obtain the remainder for any polynomial P(x) when that polynomial is divided by a linear polynomial. 

In simple words, the Remain Theorem Calculator performs the division of two polynomials and presents a remainder.

The Remainder Theorem Calculator is a free calculator available online used to perform the long division of polynomials. The procedure of division of polynomials to obtain the desired remainder is quite lengthy and tedious but the Remainder Theorem Calculator takes care of this problem.

The Remainder Theorem Calculator provides speedy and accurate results by dividing the two polynomials and presenting the remainder. 

This calculator makes use of the concept that if there exists a polynomial P(x) divided by a linear polynomial x-a then the remainder that is obtained is P(a), which is the value of the polynomial P(x) at x=a. 

The formula that is used by the Remainder Theorem Calculator to obtain the remainder for a polynomial P(x) divided by a linear polynomial x-a is given as:

$\frac{P(x)}{x-a}$ = Q(x) + R(x) 

In this formula, P(x) is the polynomial and x-a is the divisor. The polynomial Q(x) obtained is the quotient polynomial, whereas, R(x) is the remainder.

How To Use the Remainder Theorem Calculator?

You can use this calculator by simply entering the numerator and denominator in the specified fields.

The Remainder Theorem Calculator is fairly easy to use due to its simple and direct interface. The interface for the Remainder Theorem Calculator is very user-friendly as the user can easily navigate through it to obtain the designated results.

The interface of the Remainder Theorem Calculator consists of two input boxes. The first input box is labeled with “Enter the Numerator Polynomial” and it prompts the user to insert the polynomial whose division needs to be conducted.

The second input box has the title “Enter the Denominator Polynomial” which prompts the user to enter the linear polynomial which acts as the divisor. 

Once these two input values have been inserted, all there is left to do for the user is to simply click on the button that says “Divide” and the calculator will begin processing the solution.

The best feature of the Remainder Theorem Calculator is its interface because it is very simple and the user can conveniently insert the input values without much hassle. 

For an enhanced understanding of using this calculator, given below is a step-by-step guide. 

Step 1

The first step for using the Remainder Theorem Calculator is to analyze your polynomials. You can choose polynomials of any degree as the input. Make sure that the denominator polynomial is a linear polynomial.

Step 2

The next step is to insert the first input value. The first input value is the polynomial P(x) whose division is required. Enter this polynomial into the input box with the title “Enter the Numerator Polynomial.”

Step 3

Next up, move on to the second input box. The second input box prompts the user to enter the linear polynomial which will acts as the divisor for P(x). This polynomial is in the form x-a. Insert this polynomial in the input box with the title “Enter the Denominator Polynomial.”

Step 4

Now that you have your polynomials in their fixated input boxes, the final step is to click on the button that says “Divide” to trigger the Remainder Theorem Calculator to begin the solution. 

Output of the Remainder Theorem Calculator

Once the Remainder Theorem Calculator has been triggered to obtain the solution, the output will be presented after a few seconds. The calculator makes use of the following formula for obtained the remainder:

$\frac{P(x)}{x-a}$ = Q(x) + R(x) 

Thus, the Remainder Theorem Calculator presents the output of the division of the polynomial P(x) in the form of its quotient Q(x) and its remainder R(x). 

How Does the Remainder Theorem Calculator Work?

The Remainder Theorem Calculator works on the principle of the division of polynomials. It is one of the most fundamental algebraic concepts because it deals with the long division of two polynomials with each other.

To understand the working of the Remainder Theorem Calculator, let’s revise the concept of the Remainder Theorem.

Remainder Theorem

The Remainder Theorem is one of the most crucial algebraic concepts as it deals with the division of two polynomials. It states that if a polynomial P(x) is divided by a liner polynomial x-a then the remainder is obtained by calculating P(a).

The remainder P(a) is calculated by substituting the value x=a into the polynomial P(x). It can also be determined with the help of the following formula:

$\frac{P(x)}{x-a}$ = Q(x) + R(x)

Where R(x) is the remainder and Q(x) is the quotient.

Factor Theorem

The factor theorem is an extension of the remainder theorem. The factor theorem states that if the remainder obtained after the division of two polynomials is zero, the then linear polynomial is said to be a factor of P(x).

In other words, we can say that if P(x) is divided by x-a and the remainder P(a) = 0 then x-a is a factor of the polynomial P(x).

The factor theorem is a special case of the remainder theorem where the end product or the remainder is always zero.

Solved Examples

To develop a much better understanding of the working of the Remainder Theorem Calculator, a few examples are given below to help you strengthen your concepts on the remainder theorem. 

Example 1

Determine the remainder when the following polynomial is divided by x-3. The polynomial P(x) is given below:

\[ P(x) = 2x^{2} – 5x -1 \]

Solution

The first step for using the Remainder Theorem Calculator is to analyze our polynomials. The polynomial P(x) is given below:

\[ P(x) = 2x^{2} -5x-1\]

The linear polynomial or the divisor is given below:

x-3 

Enter the polynomial P(x) into the first input box. Similarly, enter the linear polynomial x-3 in the second input box of the Remainder Theorem Calculator.

Once these input values have been entered, click on “Divide.”

The Remainder Theorem Calculator will take a few moments to load the solution. The calculator will present the solution in the following manner:

$\frac{P(x)}{x-a}$ = Q(x) + R(x)

The solution presented by the Remainder Theorem Calculator for the polynomial P(x) is shown below:

Input

\[ \frac{2x^{2} – 5x-1}{x-3} \]

Output

\[ 2x^{2} -5x – 1 = (2x+1)(x-3) + 2\]

According to this output presented by the Remainder Theorem Calculator, the quotient Q(x) is (2x+1) and the remainder R(x) is 2. 

Example 2

A polynomial P(x) is given as:

\[ P(x) = x^{3} -4x^{2} -7x+10 \]

Determine the remainder for this polynomial when P(x) is divided by x-2. 

Solution

To begin the solution of this polynomial P(x) with the aid of the Reminder Theorem Calculator, firstly, analyze the two polynomials. The polynomial which needs to undergo division is given below:

\[ P(x) = x^{3} -4x^{2} -7x+10 \]

Similarly, the linear polynomial which acts as the divisor is given below:

 x-2 

Now, let’s take a look at the inputs that we have for the Remainder Calculator Theorem. The polynomial P(x) acts as our first input. Insert this polynomial into the input box with the label “Enter the Numerator Polynomial”.

Next up, move on to the second input box with the label “Enter the Denominator Polynomial.” This input box is for the divisor so enter the linear polynomial into the second input box.

Now that both the input boxes have been filled, the next step is to simply click on the button that says “Divide”. Upon doing so, the calculator begins the solution. The Remainder Theorem Calculator takes a few seconds before displaying the solution.

The solution is displayed in two tabs which are given below:

Input

\[ \frac{x^{3} -4x^{2} -7x+10}{x-2} \]

Output

\[ x^{3} -4x^{2} -7x+10 = (x^{2} – 2x -11)(x-2) + (-12) \]

Where in this solution, $(x^{2} -2x -11)$ acts as the quotient Q(x) and (-12) acts as the remainder R(x).

Hence, the division of the two polynomials is successfully conducted.

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