

How to Use the Remainder Theorem to Solve Polynomial Problems?
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 Formula, Steps & Examples
1. What is the Remainder Theorem in Maths?
The Remainder Theorem states that when a polynomial p(x) is divided by a linear polynomial (x - a), the remainder is equal to p(a). This means you can find the remainder without performing long division by simply substituting a into the polynomial.
2. How do you use the Remainder Theorem to solve problems?
To use the Remainder Theorem:
- Identify the linear divisor (x - a).
- Find the value of a (by setting the divisor to zero).
- Substitute this value of a into the polynomial p(x).
- The result of this substitution is the remainder.
3. What is the formula for the Remainder Theorem?
The Remainder Theorem doesn't have a single formula, but the core concept can be expressed as: If p(x) is divided by (x - a), then the remainder is p(a). More generally, if p(x) is divided by (ax + b), the remainder is p(-b/a).
4. How is the Remainder Theorem different from the Factor Theorem?
The Remainder Theorem finds the remainder when a polynomial is divided by a linear expression. The Factor Theorem states that if the remainder is 0, then the linear expression is a factor of the polynomial. In essence, the Factor Theorem is a specific case of the Remainder Theorem where the remainder is zero.
5. Can the Remainder Theorem be used for non-integer values of a?
Yes, the Remainder Theorem works for all real values of a, not just integers. You can substitute any real number into the polynomial to find the remainder when divided by the corresponding linear factor (x - a).
6. How does synthetic division relate to the Remainder Theorem?
Synthetic division provides a quicker method for polynomial division, particularly when the divisor is linear. The remainder obtained from synthetic division is the same remainder given by the Remainder Theorem.
7. What if the divisor is not in the form (x – a)?
If the divisor is of the form (ax + b), then find the value of x that makes the divisor zero (x = -b/a) and substitute this value into the polynomial p(x) to obtain the remainder.
8. Why is the Remainder Theorem important for competitive exams?
The Remainder Theorem provides a fast way to check for factors and find remainders, saving valuable time during competitive exams. It's frequently used in questions involving polynomial manipulation and equation solving.
9. Can the Remainder Theorem be applied to polynomials with rational or complex coefficients?
Yes, the Remainder Theorem applies to polynomials with rational or complex coefficients. The process of substituting the value of a remains the same regardless of the type of coefficients.
10. What are some common mistakes students make when applying the Remainder Theorem?
Common mistakes include incorrect substitution of a, misinterpreting the divisor, and confusing the Remainder Theorem with the Factor Theorem. Careful attention to detail and practice are essential to avoid these errors.
11. How can I use the Remainder Theorem to quickly check my polynomial division?
After performing polynomial long division, use the Remainder Theorem to calculate the expected remainder. If your calculated remainder matches the remainder from your long division, it acts as a quick check for accuracy.











