Find Remainder Of Polynomial Division Calculator

Easily calculate the remainder and quotient when dividing two polynomials using our find remainder of polynomial division calculator. Input coefficients to get instant results.

Polynomial Division Calculator

Results Overview

Summary of Polynomials and their Degrees

Polynomial	Coefficients	String Representation	Degree
Dividend P(x)	–	–	–
Divisor D(x)	–	–	–
Quotient Q(x)	–	–	–
Remainder R(x)	–	–	–

What is the Find Remainder of Polynomial Division Calculator?

The find remainder of polynomial division calculator is a tool designed to perform polynomial long division, specifically to find the quotient and, most importantly, the remainder when one polynomial (the dividend P(x)) is divided by another (the divisor D(x)). This process is analogous to long division with integers.

Anyone studying algebra, calculus, or fields like engineering and computer science that use polynomial manipulations should use this calculator. It's particularly helpful for understanding the Remainder Theorem and Factor Theorem, or when simplifying rational expressions. A common misconception is that the remainder is always a constant; however, the remainder is itself a polynomial whose degree is less than the degree of the divisor.

Our find remainder of polynomial division calculator automates the division steps, providing the remainder and quotient polynomials instantly.

Find Remainder of Polynomial Division Formula and Mathematical Explanation

When a polynomial P(x) (dividend) is divided by a non-zero polynomial D(x) (divisor), we get a unique quotient polynomial Q(x) and a unique remainder polynomial R(x) such that:

P(x) = D(x) * Q(x) + R(x)

where the degree of R(x) is less than the degree of D(x), or R(x) is the zero polynomial.

The process used by the find remainder of polynomial division calculator mimics polynomial long division:

1. Arrange both P(x) and D(x) in descending powers of x. If any power is missing, insert it with a coefficient of 0.
2. Divide the first term of P(x) by the first term of D(x) to get the first term of Q(x).
3. Multiply D(x) by this first term of Q(x) and subtract the result from P(x) to get a new polynomial (the first intermediate remainder).
4. Repeat steps 2 and 3 with the new polynomial as the dividend until the degree of the resulting polynomial is less than the degree of D(x). This final polynomial is the remainder R(x).

Variables Table:

Variable	Meaning	Unit/Type	Typical Range
P(x)	Dividend Polynomial	Coefficients (Array)	Real numbers
D(x)	Divisor Polynomial	Coefficients (Array)	Real numbers (non-zero polynomial)
Q(x)	Quotient Polynomial	Coefficients (Array)	Real numbers
R(x)	Remainder Polynomial	Coefficients (Array)	Real numbers
deg(P)	Degree of P(x)	Non-negative integer	0, 1, 2, …
deg(D)	Degree of D(x)	Non-negative integer	0, 1, 2, …

For a more detailed walkthrough, see our guide on polynomial long division steps.

Practical Examples (Real-World Use Cases)

The find remainder of polynomial division calculator is useful in various mathematical contexts.

Example 1: Dividing x² + 3x + 5 by x + 1

• P(x) coefficients: 1, 3, 5
• D(x) coefficients: 1, 1

Using the calculator, we input these coefficients. The division P(x) = (x+1)(x+2) + 3 gives:

• Quotient Q(x): x + 2 (Coefficients: 1, 2)
• Remainder R(x): 3 (Coefficients: 3)

Here, the remainder is a constant because the divisor was degree 1.

Example 2: Dividing 2x³ – x² + x – 1 by x² – 1

• P(x) coefficients: 2, -1, 1, -1
• D(x) coefficients: 1, 0, -1 (Note the 0 for the x term)

Inputting into the find remainder of polynomial division calculator:

• Quotient Q(x): 2x – 1 (Coefficients: 2, -1)
• Remainder R(x): 3x – 2 (Coefficients: 3, -2)

The remainder has a degree of 1, which is less than the divisor's degree of 2.

How to Use This Find Remainder of Polynomial Division Calculator

1. Enter Dividend Coefficients: In the "Dividend P(x) Coefficients" field, enter the coefficients of your dividend polynomial, starting from the term with the highest power down to the constant term, separated by commas. For example, for 2x³ + x – 5, enter "2,0,1,-5" (including 0 for the missing x² term).
2. Enter Divisor Coefficients: In the "Divisor D(x) Coefficients" field, enter the coefficients of your divisor polynomial similarly. For x – 2, enter "1,-2".
3. Calculate: Click the "Calculate" button.
4. Read Results: The calculator will display:
   • The remainder polynomial R(x) (both coefficients and string form).
   • The quotient polynomial Q(x).
   • The degrees of P(x), D(x), Q(x), and R(x).
   • A table and chart summarizing the results.
5. Reset: Click "Reset" to clear the fields to their default values.
6. Copy: Click "Copy Results" to copy the main results and inputs to your clipboard.

The find remainder of polynomial division calculator updates automatically if you change the input values after the first calculation.

Key Factors That Affect Polynomial Division Results

Several factors influence the outcome of polynomial division and the remainder you find:

• Degree of the Dividend P(x): A higher degree dividend generally leads to a longer division process and potentially a higher degree quotient.
• Degree of the Divisor D(x): This dictates the maximum possible degree of the remainder (it must be less than deg(D)). If deg(D) > deg(P), the quotient is 0 and the remainder is P(x).
• Leading Coefficients: The leading coefficients of P(x) and D(x) are crucial in determining each term of the quotient during the division steps.
• Zero Coefficients: Missing terms (with zero coefficients) in either polynomial must be accounted for to maintain proper alignment during long division. Our find remainder of polynomial division calculator handles these.
• The Remainder Theorem: If the divisor is of the form (x – c), the remainder is P(c). Our calculator works even when the divisor is not linear, but for linear divisors, you can cross-check with the Remainder Theorem.
• Factor Theorem: A special case where the remainder is 0, meaning the divisor is a factor of the dividend. You might be interested in factoring polynomials if you get a zero remainder.

Frequently Asked Questions (FAQ)

What is the remainder when a polynomial P(x) is divided by (x-c)?

According to the Remainder Theorem, the remainder is P(c). You can use the find remainder of polynomial division calculator with divisor coefficients "1, -c" or directly evaluate P(c).

What if the degree of the divisor is greater than the degree of the dividend?

If deg(D) > deg(P), the quotient Q(x) will be 0, and the remainder R(x) will be the dividend P(x) itself. The calculator handles this.

What if the remainder is zero?

If the remainder R(x) is 0, it means the divisor D(x) is a factor of the dividend P(x). P(x) = D(x)Q(x).

Can I use this calculator for polynomials with complex coefficients?

This specific calculator is designed for polynomials with real number coefficients. The logic for complex numbers is similar but requires handling complex arithmetic.

How do I enter a polynomial like x³ – 1?

You would enter the coefficients as "1,0,0,-1" (for 1x³ + 0x² + 0x – 1).

Is this calculator the same as a synthetic division calculator?

Synthetic division is a shortcut for polynomial division when the divisor is linear (x-c). This calculator uses a method equivalent to long division, which works for any divisor degree. It will give the same remainder as synthetic division when the divisor is linear.

Why is the degree of the remainder always less than the degree of the divisor?

This is a fundamental property of the division algorithm for polynomials, analogous to how the remainder in integer division is always less than the divisor.

Where is polynomial division used?

It's used in finding roots of polynomials (if the remainder is 0), simplifying rational expressions, partial fraction decomposition in calculus, and in coding theory (e.g., CRC). Our polynomial roots finder can be useful after division.