1. What is the Remainder and Factor Theorem?

The Remainder Theorem states that when a polynomial f(x) is divided by (x - a), the remainder is equal to f(a). The Factor Theorem is a special case where if f(a) = 0, then (x - a) is a factor of f(x).

2. How Does the Calculator Work?

The calculator evaluates the polynomial at the given value a:

Where:

• \( f(x) \) — Polynomial function
• \( a \) — Value to evaluate the polynomial at
• \( f(a) \) — Result of substituting x = a into the polynomial

3. Importance of Remainder and Factor Theorems

Details: These theorems are fundamental in polynomial algebra, helping to find roots of polynomials, factor polynomials, and solve polynomial equations.

4. Using the Calculator

Tips: Enter the polynomial in standard form (e.g., x^3 - 2x^2 + 3x - 4) and the value a. The calculator will compute f(a) and determine if (x - a) is a factor.

5. Frequently Asked Questions (FAQ)

Q1: What is the difference between the Remainder and Factor Theorems?
A: The Remainder Theorem gives the remainder when dividing by (x - a), while the Factor Theorem specifically addresses when (x - a) is a factor (remainder is 0).

Q2: Can these theorems be used for polynomials of any degree?
A: Yes, both theorems apply to polynomials of any degree.

Q3: What if the polynomial has multiple variables?
A: These theorems specifically apply to single-variable polynomials.

Q4: How are these theorems related to polynomial division?
A: Both theorems are consequences of polynomial division, providing shortcuts to find remainders and factors without performing full division.

Q5: Can these theorems help find all factors of a polynomial?
A: Yes, by testing possible roots (values of a that make f(a) = 0), you can find all linear factors of a polynomial.