Remainder Theorem

Remainder Theorem

Definition

The Remainder Theorem is a powerful tool used to find the remainder when a polynomial is divided by a linear polynomial.

Statement

If a polynomial f(x) is divided by (x - a), then the remainder is f(a).

Explanation

• The theorem states that when a polynomial f(x) is divided by (x - a), the remainder is the value of the polynomial at x = a.
• In other words, the remainder is f(a), which is the value of the polynomial when x is equal to a.
• This theorem can be used to evaluate polynomials at specific points without having to perform long division.

Example

• Find the remainder when f(x) = x^3 - 2x^2 - 5x + 6 is divided by (x - 2).
• Using the Remainder Theorem, the remainder is f(2) = 2^3 - 2(2)^2 - 5(2) + 6 = 4.
• Therefore, the remainder is 4.

Applications

• The Remainder Theorem is used to:
  • Evaluate polynomials at specific points.
  • Find the remainder when a polynomial is divided by a linear polynomial.
  • Factor polynomials.
  • Solve polynomial equations.

Remainder Theorem

• The Remainder Theorem is a powerful tool used to find the remainder when a polynomial is divided by a linear polynomial.

Statement of the Theorem

• If a polynomial f(x) is divided by (x - a), then the remainder is f(a).

Explanation and Key Points

• The theorem states that the remainder is the value of the polynomial at x = a.
• The remainder is f(a), which is the value of the polynomial when x is equal to a.
• The theorem can be used to evaluate polynomials at specific points without performing long division.

Example Application

• To find the remainder when f(x) = x^3 - 2x^2 - 5x + 6 is divided by (x - 2), use the Remainder Theorem.
• The remainder is f(2) = 2^3 - 2(2)^2 - 5(2) + 6 = 4.

Applications of the Remainder Theorem

• Evaluate polynomials at specific points.
• Find the remainder when a polynomial is divided by a linear polynomial.
• Factor polynomials.
• Solve polynomial equations.