Remainder Theorem formula proof solved examples and step by step method
The concept of Remainder Theorem plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Whether you're learning polynomial division in school or preparing for competitive exams, understanding the remainder theorem can help you quickly solve problems without lengthy calculations. Let's dive into the meaning, formula, examples, and application tricks for this essential Maths concept.
What Is Remainder Theorem?
The Remainder Theorem is a shortcut in algebra. It states that when any polynomial f(x) is divided by a linear term of the form (x – a), the remainder is simply the value f(a). You'll find this concept applied in areas such as polynomial division, synthetic division, and quick factor checks in higher classes. This theorem is part of the polynomial chapter in NCERT Class 9-11, IIT JEE, CBSE, ICSE, and other exam boards.
Key Formula for Remainder Theorem
Here’s the standard formula:
\( \text{If } f(x) \text{ is divided by } (x-a), \text{ then remainder } = f(a) \)
Cross-Disciplinary Usage
Remainder Theorem is not only useful in Maths but also plays an important role in Physics (for polynomial motion equations), Computer Science (for algorithmic checks and cryptography), and logical reasoning. Students preparing for JEE, NEET, NTSE, and Olympiads will see its relevance in various questions around roots, factors, and quick evaluative calculations.
Step-by-Step Illustration
Example: Find the remainder when \( f(x) = x^3 - 4x^2 + 6x - 8 \) is divided by \( x - 2 \).
1. Identify the value of 'a' in \( x-a \): Here, \( a = 2 \ )2. Substitute \( x = 2 \) into the polynomial:
\( f(2) = (2)^3 - 4(2)^2 + 6(2) - 8 \)
3. Calculate step by step:
\( 8 - 16 + 12 - 8 = -4 \)
4. Final answer: **Remainder = -4**
Speed Trick or Vedic Shortcut
Here’s a quick way to use the Remainder Theorem, especially during exams:
- Always identify the zero of the divisor (set \( x-a = 0 \), so \( x = a \)).
- Plug this value directly into every term of your polynomial (no need to expand or write long division steps).
- Sum up the results — the answer is your remainder!
Example Trick: What is the remainder when \( f(x) = 3x^2 + x + 7 \) is divided by \( x+2 \)?
Set \( x+2=0 \Rightarrow x=-2 \). Put into \( f(x) \):
\( f(-2) = 3\times(-2)^2 + (-2) + 7 = 12 - 2 + 7 = 17 \)
Remainder = 17
Such tricks are taught in Vedantu’s live and recorded classes to build exam speed and accuracy.
Try These Yourself
- Find the remainder when \( x^2 + 5x + 6 \) is divided by \( x - 1 \).
- If \( f(x) = x^4 - 3x^3 + x - 1 \), what is the remainder when divided by \( x + 2 \)?
- Which divisors will always give the remainder zero for \( x^2 - 4 \)?
- Find the remainder when \( 2x^3 + 3x - 5 \) is divided by \( x-0 \).
Frequent Errors and Misunderstandings
- Forgetting to change the sign in \( x-a \) (e.g. using \( a = -2 \) instead of \( a = 2 \)).
- Plugging the wrong value of \( x \) into the formula.
- Confusing remainder theorem with factor theorem (remainder zero does NOT always mean divisor is a factor unless remainder = 0).
Relation to Other Concepts
The idea of Remainder Theorem connects closely with Factor Theorem and Polynomial Division. Mastering this helps with understanding complex algebra, divisibility of polynomials, and fast factorization in senior classes.
| Remainder Theorem | Factor Theorem |
|---|---|
| Finds the remainder when dividing by (x–a) | Tells if (x–a) is a factor if the remainder is zero |
| Remainder = f(a) | If f(a)=0, then (x–a) is a factor |
| All divisors (x–a), for any 'a' | Focus on zero-remainder only (factor check) |
Classroom Tip
A quick way to remember the Remainder Theorem is: "Substitute the root of the divisor into the polynomial!" Many Vedantu teachers use visual tables and mobile-friendly formula boxes to help students save time and avoid errors during calculation.
Wrapping It All Up
We explored Remainder Theorem—covering its definition, formula, stepwise illustrations, common mistakes, and links to concepts like factor theorem and polynomial division. Practice these methods or use Vedantu’s Remainder Theorem Calculator for fast checks. Mastering it now makes all advanced polynomial topics easier later!
Explore related topics:
Factor Theorem |
Polynomial Division |
Remainder Theorem Calculator |
NCERT Class 10 Maths Important Topics
FAQs on Remainder Theorem Explained with Proof and Applications
1. What is the Remainder Theorem?
The Remainder Theorem states that when a polynomial f(x) is divided by (x − a), the remainder is equal to f(a).
- If f(x) is a polynomial and you substitute x = a, the value obtained is the remainder.
- It only applies when dividing by a linear divisor of the form (x − a).
- This theorem simplifies polynomial division without long division.
2. What is the formula for the Remainder Theorem?
The formula for the Remainder Theorem is: if f(x) is divided by (x − a), then the remainder is f(a).
- Divisor: (x − a)
- Remainder: f(a)
- Example: If f(x) = x² + 3x + 2 and divisor is (x − 1), remainder = f(1).
3. How do you use the Remainder Theorem to find the remainder?
To use the Remainder Theorem, substitute the value of a into the polynomial f(x).
- Step 1: Identify the divisor (x − a).
- Step 2: Set x = a.
- Step 3: Compute f(a).
- The result is the remainder.
4. Can you give an example of the Remainder Theorem?
Yes, for example, if f(x) = x² + 5x + 6 is divided by (x − 2), the remainder is 20.
- Substitute x = 2.
- f(2) = (2)² + 5(2) + 6
- = 4 + 10 + 6 = 20
- Therefore, the remainder is 20.
5. What is the difference between the Remainder Theorem and the Factor Theorem?
The Factor Theorem is a special case of the Remainder Theorem where the remainder equals zero.
- Remainder Theorem: Remainder = f(a).
- Factor Theorem: If f(a) = 0, then (x − a) is a factor.
- Factor Theorem helps find roots of polynomials.
6. When is the remainder zero in the Remainder Theorem?
The remainder is zero when f(a) = 0, meaning (x − a) is a factor of the polynomial.
- Substitute x = a into f(x).
- If the result equals 0, division leaves no remainder.
- This indicates a root or solution of the polynomial equation.
7. Can the Remainder Theorem be used for divisors other than (x − a)?
The Remainder Theorem applies only when the divisor is a linear expression of the form (x − a).
- It does not directly apply to quadratic or higher-degree divisors.
- For higher-degree divisors, polynomial long division or synthetic division is required.
8. How is the Remainder Theorem related to synthetic division?
The Remainder Theorem is the theoretical basis of synthetic division when dividing by (x − a).
- Synthetic division is a shortcut method.
- The final number obtained in synthetic division equals f(a).
- This number is the remainder.
9. How do you find the value of a constant using the Remainder Theorem?
You find the unknown constant by setting f(a) equal to the given remainder and solving the equation.
- Step 1: Substitute x = a into the polynomial.
- Step 2: Set the result equal to the given remainder.
- Step 3: Solve for the unknown constant.
10. Why is the Remainder Theorem important in algebra?
The Remainder Theorem is important because it simplifies polynomial division and helps identify factors and roots quickly.
- It avoids lengthy polynomial long division.
- It connects directly to the Factor Theorem.
- It is widely used in solving polynomial equations and algebraic proofs.
