Video Explanation for Remainder Theorem:

Introduction:
The remainder theorem is a fundamental concept in polynomial algebra. It helps us find the remainder when a polynomial is divided by a linear factor. Understanding this theorem is crucial for solving various problems related to polynomials. In this video, we will explore the remainder theorem in detail and discuss its applications.

Explanation:
1. Definition of Remainder Theorem:
- The remainder theorem states that if a polynomial f(x) is divided by the linear factor (x - a), the remainder will be equal to f(a).
- Mathematically, if f(x) = (x - a) * q(x) + r, where q(x) is the quotient and r is the remainder, then f(a) = r.

2. Example:
- Let's consider an example to understand the concept better. Suppose we have a polynomial f(x) = 2x^3 - 5x^2 + 4x + 3, and we want to find the remainder when f(x) is divided by (x - 2).
- Applying the remainder theorem, we substitute x = 2 in the polynomial: f(2) = 2(2)^3 - 5(2)^2 + 4(2) + 3 = 16 - 20 + 8 + 3 = 7.
- Therefore, the remainder when f(x) is divided by (x - 2) is 7.

3. Applications:
- The remainder theorem has various applications in mathematics, particularly in solving polynomial equations and finding factors of polynomials.
- It helps us determine whether a given value is a root of a polynomial or not.
- By finding the remainder, we can verify if a given polynomial is a factor of another polynomial or not.

4. Conclusion:
- The remainder theorem is a powerful tool in polynomial algebra that allows us to find the remainder when a polynomial is divided by a linear factor.
- Understanding this theorem is essential for solving polynomial-related problems.
- By applying the remainder theorem, we can simplify complex polynomial expressions and solve equations efficiently.

References:
- EduRev website: [Link to EduRev Polynomials](

https://www.edurev.in/subjects/mathematics/7f00b1d1) https://www.edurev.in/subjects/mathematics/7f00b1d1)