Remainder Theorem and NCERT Solutions for Polynomials

The Remainder Theorem is an important concept in algebra that helps to solve polynomial equations. It states that if a polynomial p(x) is divided by a linear factor (x-a), the remainder is equal to p(a).

Example: If p(x) = x^2 + 3x - 4 is divided by x-1, then the remainder is p(1) = 1^2 + 3(1) - 4 = 0.

The NCERT Solutions for Polynomials contains exercises related to the Remainder Theorem. Let's take a look at an example from Ex 2.1:

Question: Find the remainder when x^3 + x^2 - 2x - 1 is divided by x-2.

Solution:
1. First, we need to apply the Remainder Theorem. Let a=2, since we are dividing by x-2.
2. Substitute a into the polynomial: p(2) = 2^3 + 2^2 - 2(2) - 1 = 8 + 4 - 4 - 1 = 7.
3. Therefore, the remainder when x^3 + x^2 - 2x - 1 is divided by x-2 is 7.

Summary:
- The Remainder Theorem states that the remainder when a polynomial is divided by a linear factor (x-a) is equal to p(a).
- The NCERT Solutions for Polynomials includes exercises related to the Remainder Theorem.
- To use the Remainder Theorem, substitute the value of a into the polynomial and solve for the remainder.