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Remainder Theorem Calculator: Find the Remainder of Any Polynomial

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How to Use the Remainder Theorem Calculator for Polynomials

Remainder Theorem Calculator

What is Remainder Theorem Calculator?

The Remainder Theorem Calculator is an online algebraic tool that helps you quickly find the remainder when a polynomial is divided by a linear divisor of the form (x – a). Thanks to the Remainder Theorem, you don’t need to perform lengthy polynomial division or synthetic division—just enter your polynomial and the value of ‘a’ to get the remainder instantly, along with step-by-step working. This calculator is ideal for maths students studying polynomials, algebra, or preparing for competitive exams.

Formula or Logic Behind Remainder Theorem Calculator

The Remainder Theorem states: The remainder when a polynomial \( f(x) \) is divided by \( x - a \) is simply \( f(a) \). That means you just substitute \( x = a \) into the original polynomial and calculate the value. If the result is zero, then (x – a) is a factor of the polynomial. This approach replaces the need for time-consuming polynomial division and makes checking factors or verifying remainders very easy.

Remainder Theorem: Example Calculations Table

Polynomial f(x)	Divisor (x–a)	a	Calculation (f(a))	Remainder
2x³ – 5x² + x – 7	x – 2	2	2×(2)³ – 5×(2)² + 2 – 7 = 3	3
x² + x + 1	x – 1	1	1² + 1 + 1 = 3	3
x³ – 3x + 2	x + 1	-1	(–1)³ – 3×(–1) + 2 = 4	4
3x⁴ – 2x³ + x – 9	x – 0	0	3×0 – 2×0 + 0 – 9 = –9	-9
4x² – 7x + 5	x – 2	2	4×(2)² – 7×2 + 5 = 7	7

Steps to Use the Remainder Theorem Calculator

Enter the polynomial expression in standard algebraic form (e.g., 2x^3-5x^2+x-7).
Input the value of x (a) for which you want the remainder.
Click on the 'Calculate' button.
Get instant results with detailed step-by-step calculation below the button.

Why Use Vedantu’s Remainder Theorem Calculator?

Vedantu’s Remainder Theorem Calculator is designed for ease-of-use, accuracy, and maximum convenience. Its interactive, mobile-optimized design works smoothly across all devices, giving instant and reliable results for school, college, or competitive exams. Students love its stepwise explanations and the time it saves when checking answers or revising concepts. It’s trusted by lakhs of learners across India as a revision and learning companion.

Real-life Applications of Remainder Theorem Calculator

The Remainder Theorem has practical uses in:

Quickly checking if (x – a) is a factor of any polynomial (remainder is zero)
Solving high school and board exam problems, especially in CBSE/ICSE/NCERT syllabi
Math Olympiads and scholarship/entrance exams where time matters
Learning the connection between roots and factors for higher maths
Coding and computer science, e.g., when working with polynomials in error-checking or cryptography

You can also use this calculator for fast homework help and seamless self-study corrections.

Read more about related concepts here: Factor Theorem, Polynomial, and Algebra Topics. For prime numbers and factors, try our HCF Calculator.

This tool and the content above are created and reviewed by Vedantu’s certified maths teachers, fully updated as per latest CBSE/ICSE/NCERT standards and examination trends.

FAQs on Remainder Theorem Calculator: Find the Remainder of Any Polynomial

1. What is the Remainder Theorem and how does it work?

The Remainder Theorem provides a quick way to find the remainder when a polynomial is divided by a linear expression (x - a). It states that the remainder is simply the value of the polynomial when x = a; that is, f(a). This avoids the lengthy process of polynomial long division. To use it, substitute 'a' into the polynomial and evaluate the expression. The result is the remainder.

2. What is the formula for the Remainder Theorem?

The Remainder Theorem doesn't have a formula in the traditional sense. Instead, it's a statement: When a polynomial f(x) is divided by (x - a), the remainder is f(a). This means you directly substitute a for x in the polynomial to find the remainder.

3. How do I use the Remainder Theorem to find the remainder when dividing a polynomial by (x-2)?

To find the remainder when dividing a polynomial, say f(x), by (x - 2), simply substitute x = 2 into the polynomial f(x) and evaluate the expression. The result is the remainder.

4. What is the remainder when 3x² - 5x + 2 is divided by (x - 1)?

Substitute x = 1 into the polynomial: 3(1)² - 5(1) + 2 = 3 - 5 + 2 = 0. Therefore, the remainder is 0. This also means that (x - 1) is a factor of 3x² - 5x + 2.

5. Can the Remainder Theorem be used with any divisor?

No, the Remainder Theorem is specifically designed for divisors of the form (x - a), where 'a' is a constant. It doesn't directly apply to divisors that are not linear factors.

6. What is the difference between the Remainder Theorem and the Factor Theorem?

Both theorems deal with polynomials. The Remainder Theorem states that when f(x) is divided by (x-a), the remainder is f(a). The Factor Theorem is a specific case of the Remainder Theorem; it states that (x - a) is a factor of f(x) if and only if f(a) = 0 (i.e., the remainder is 0).

7. How can I use the Remainder Theorem to check if (x+3) is a factor of x³ + 2x² - 5x - 6?

If (x + 3) is a factor, then the remainder when dividing by (x + 3) should be 0. Substitute x = -3 into the polynomial: (-3)³ + 2(-3)² - 5(-3) - 6 = -27 + 18 + 15 - 6 = 0. Since the remainder is 0, (x + 3) is indeed a factor.

8. What are some real-world applications of the Remainder Theorem?

While not directly used in everyday scenarios, the Remainder Theorem finds application in various areas of mathematics and computer science. It's a fundamental concept in polynomial algebra, serving as a building block for more advanced topics such as solving polynomial equations and finding roots.

9. Find the remainder when x⁴ + 3x³ - 2x + 5 is divided by (x + 1).

Substitute x = -1 into the polynomial: (-1)⁴ + 3(-1)³ - 2(-1) + 5 = 1 - 3 + 2 + 5 = 5. The remainder is 5.

10. Explain the Remainder Theorem in simple terms.

Imagine dividing a number. The remainder is what's left over. The Remainder Theorem does the same for polynomials. To find the remainder when dividing a polynomial by (x-a), just plug in 'a' for x in the polynomial. The result is your remainder.

11. Why is the Remainder Theorem useful?

The Remainder Theorem simplifies the process of finding remainders of polynomial division, especially when the divisor is of the form (x - a). It avoids the longer and more complex method of polynomial long division. It is also crucial for understanding concepts like the Factor Theorem and polynomial factorization.