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Remainder Theorem of Polynomials - Polynomials, Class 9, Mathematics

Introduction:
The remainder theorem is a fundamental concept in polynomial algebra. It establishes a relationship between the polynomial division process and the evaluation of polynomials. Understanding the remainder theorem is crucial in solving polynomial equations and finding factors of polynomials.

Statement of the Remainder Theorem:
The remainder theorem states that if a polynomial f(x) is divided by a linear polynomial (x - a), then the remainder obtained is equal to f(a).

Explanation:
To understand the remainder theorem, let's consider a polynomial f(x) and a linear polynomial (x - a) as the divisor. When we divide f(x) by (x - a), we obtain a quotient q(x) and a remainder r(x). The remainder theorem states that the remainder r(x) is equal to f(a).

Example:
Let's illustrate the remainder theorem with an example. Consider the polynomial f(x) = 2x^3 - 5x^2 + 3x - 1 and the divisor (x - 2). If we divide f(x) by (x - 2), we obtain a quotient q(x) and a remainder r(x).

By polynomial division, we find that the quotient q(x) = 2x^2 - x - 1 and the remainder r(x) = 3.

Now, let's evaluate f(2) and compare it with the remainder r(x) obtained. When we substitute x = 2 in f(x), we get f(2) = 2(2)^3 - 5(2)^2 + 3(2) - 1 = 8 - 20 + 6 - 1 = -7.

As per the remainder theorem, the remainder r(x) is equal to f(a), where a = 2. So, r(x) = f(2), which in our case is 3. Hence, the remainder theorem is validated.

Applications of the Remainder Theorem:
1. Polynomial Factorization: The remainder theorem helps in finding factors of polynomials. If the remainder of dividing a polynomial f(x) by (x - a) is zero, then (x - a) is a factor of f(x).

2. Solving Polynomial Equations: The remainder theorem allows us to solve polynomial equations. By equating the remainder to zero, we can find the values of x that make the polynomial equal to zero.

3. Evaluating Polynomials: The remainder theorem provides a shortcut to evaluate polynomials at specific values. Instead of performing polynomial division, we can directly substitute the value of x in the polynomial to find the result.

Conclusion:
The remainder theorem is a powerful tool in polynomial algebra. It establishes a connection between polynomial division and the evaluation of polynomials. Understanding and applying the remainder theorem can assist in factorizing polynomials, solving polynomial equations, and evaluating polynomials at specific values.