RESIDUE THEOREM

The remainder theorem in algebra is a fundamental theorem that states that if a polynomial f(x) is divided by a linear polynomial (x - a), then the remainder is equal to the value of f(a).

Statement of the theorem

If f(x) is a polynomial and a is a number, then the remainder of f(x) divided by (x - a) is equal to f(a).

Proof of the theorem

The proof of the remainder theorem is based on polynomial division. If f(x) is divided by (x - a), then we can write:

f(x) = q(x)(x - a) + r

where q(x) is the quotient and r is the remainder.

Substituting x = a in the previous equation, we obtain:

f(a) = q(a)(a - a) + r
f(a) = r

Therefore, the remainder is equal to the value of f(a).

Applications of the remainder theorem

The remainder theorem has several important applications in algebra and other areas of mathematics, such as:

1. _Calculation of residues_: The residue theorem is used to calculate residues in polynomial division.
2. _Factoring polynomials_: The remainder theorem is used to factorize polynomials.
3. _Solving equations_: The remainder theorem is used to solve polynomial equations.
4. _Analysis of functions_: The remainder theorem is used to analyze the properties of polynomial functions.

Example

Suppose we want to calculate the remainder of f(x) = x^2 + 3x + 2 divided by (x - 1).

According to the remainder theorem, the remainder is equal to f(1).

f(1) = 1^2 + 3(1) + 2
f(1) = 6

Therefore, the remainder is 6.