108 LearnersLast updated on June 25th, 2025
Remainder Theorem Calculator
Calculators are reliable tools for solving simple mathematical problems and advanced calculations like trigonometry. Whether you’re cooking, tracking BMI, or planning a construction project, calculators will make your life easy. In this topic, we are going to talk about the remainder theorem calculator.
What is Remainder Theorem Calculator?
A remainder theorem calculator is a tool used to find the remainder when a polynomial is divided by a linear divisor. This calculator simplifies the process of finding the remainder, making it quicker and more efficient, saving time and effort.
How to Use the Remainder Theorem Calculator?
Given below is a step-by-step process on how to use the calculator:
Step 1: Enter the polynomial: Input the polynomial expression into the given field.
Step 2: Enter the divisor: Input the linear divisor by which you want to divide the polynomial.
Step 3: Click on calculate: Click on the calculate button to find the remainder.
Step 4: View the result: The calculator will display the remainder instantly.
How to Apply the Remainder Theorem?
To apply the remainder theorem, substitute the root of the divisor into the polynomial. For a divisor of the form (x - a), substitute x = a into the polynomial. The result of this substitution is the remainder. For example, if P(x) is the polynomial and (x - a) is the divisor, then: Remainder = P(a)
Tips and Tricks for Using the Remainder Theorem Calculator
When using a remainder theorem calculator, there are a few tips and tricks that can make the process easier and help avoid mistakes:
Familiarize yourself with polynomial expressions and their components.
Understand the divisor format (x - a) to correctly find the root.
Ensure that you input the polynomial and divisor correctly, checking for signs and coefficients.
Common Mistakes and How to Avoid Them When Using the Remainder Theorem Calculator
Even when using a calculator, mistakes can happen. Here are some common errors users make when using a remainder theorem calculator.
Mistake 1
Incorrectly identifying the divisor root
Ensure you correctly identify the root of the divisor. For example, for (x - 3), the root is 3. Incorrect roots lead to incorrect remainders.
Mistake 2
Misplacing signs in the polynomial or divisor
Pay attention to the signs in both the polynomial and the divisor. A misplaced sign can alter the remainder significantly.
Mistake 3
Inputting the polynomial coefficients incorrectly
Double-check the coefficients of your polynomial to ensure they are entered correctly. An incorrect coefficient can lead to a wrong remainder.
Mistake 4
Forgetting to adjust for complex numbers
If your divisor’s root is a complex number, ensure your calculations account for this. This involves using complex arithmetic where necessary.
Mistake 5
Assuming all calculators accommodate all polynomial types
Not all calculators can handle every polynomial form or degree. Double-check your calculator’s capabilities and, if needed, verify manually or use a more advanced tool.
Remainder Theorem Calculator Examples
Problem 1
What is the remainder when dividing P(x) = 3x^3 + 5x^2 - 6x + 4 by (x - 2)?
Use the remainder theorem:
Remainder = P(2) = 3 × 2³ + 5 × 2² − 6 × 2 + 4
Remainder = 3 × 8 + 5 × 4 − 12 + 4
Remainder = 24 + 20 − 12 + 4
Remainder = 36
Therefore, the remainder is 36.
Explanation
By substituting x = 2 into the polynomial, we calculate the remainder as 36.
Problem 2
Find the remainder when P(x) = x^4 - 4x^3 + 6x - 5 is divided by (x + 1).
Use the remainder theorem:
Remainder = P(−1) = (−1)⁴ − 4 × (−1)³ + 6 × (−1) − 5
Remainder = 1 + 4 − 6 − 5
Remainder = −6
Therefore, the remainder is −6.
Explanation
Substituting x = -1 into the polynomial, the remainder is calculated as -6.
Problem 3
Calculate the remainder of P(x) = 2x^3 - 7x^2 + x + 8 when divided by (x - 3).
Use the remainder theorem:
Remainder = P(3) = 2 × 3³ − 7 × 3² + 3 + 8
Remainder = 2 × 27 − 7 × 9 + 3 + 8
Remainder = 54 − 63 + 3 + 8
Remainder = 2
Therefore, the remainder is 2.
Explanation
By substituting x = 3 into the polynomial, the remainder is determined to be 2.
Problem 4
What is the remainder when P(x) = 5x^4 + 2x^3 - x + 6 is divided by (x - 5)?
Use the remainder theorem:
Remainder = P(5) = 5 × 5⁴ + 2 × 5³ − 5 + 6
Remainder = 5 × 625 + 2 × 125 − 5 + 6
Remainder = 3125 + 250 − 5 + 6
Remainder = 3376
Therefore, the remainder is 3376.
Explanation
Substituting x = 5 into the polynomial allows us to calculate the remainder as 3376.
Problem 5
Determine the remainder of P(x) = 4x^2 - 9x + 7 when divided by (x + 2).
Use the remainder theorem:
Remainder = P(−2) = 4 × (−2)² − 9 × (−2) + 7
Remainder = 4 × 4 + 18 + 7
Remainder = 16 + 18 + 7
Remainder = 41
Therefore, the remainder is 41.
Explanation
Substituting x = -2 into the polynomial gives a remainder of 41.
FAQs on Using the Remainder Theorem Calculator
1.How do you calculate the remainder using the remainder theorem?
2.Can the remainder theorem be used for non-linear divisors?
3.Why is it important to check the polynomial’s coefficients?
4.How do I use a remainder theorem calculator?
5.Is the remainder theorem calculator accurate?
Glossary of Terms for the Remainder Theorem Calculator
- Remainder Theorem: A mathematical principle used to find the remainder of a polynomial division by a linear divisor.
- Polynomial: An algebraic expression consisting of variables and coefficients.
- Divisor: A linear expression of the form (x - a) used in polynomial division.
- Root: The value of x that makes the divisor equal to zero.
- Coefficient: Numerical or constant factors in the terms of a polynomial.
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Seyed Ali Fathima S
About the Author
Seyed Ali Fathima S a math expert with nearly 5 years of experience as a math teacher. From an engineer to a math teacher, shows her passion for math and teaching. She is a calculator queen, who loves tables and she turns tables to puzzles and songs.
Fun Fact
: She has songs for each table which helps her to remember the tables
