1. What Is The Factor Theorem?

The Factor Theorem states that for a polynomial f(x), if f(a) = 0, then (x - a) is a factor of the polynomial. Conversely, if (x - a) is a factor, then f(a) = 0. This calculator computes f(a) to determine the remainder when dividing f(x) by (x - a).

2. How Does The Calculator Work?

The calculator uses the Factor Theorem formula:

Where:

• \( f(x) \) — The polynomial function
• \( a \) — The value at which to evaluate the polynomial

Explanation: The calculator evaluates the polynomial at x = a to find the remainder when f(x) is divided by (x - a). If the remainder is zero, (x - a) is a factor of the polynomial.

3. Importance Of Remainder Calculation

Details: Calculating remainders using the Factor Theorem is essential for polynomial factorization, finding roots of polynomials, and solving polynomial equations in algebra and calculus.

4. Using The Calculator

Tips: Enter the polynomial in standard form (e.g., "2x^3 + 3x^2 - 5x + 7") and the value a. The calculator will compute f(a) and display the remainder.

5. Frequently Asked Questions (FAQ)

Q1: What does a remainder of zero mean?
A: If f(a) = 0, then (x - a) is a factor of the polynomial, meaning a is a root of the equation f(x) = 0.

Q2: How should I format the polynomial?
A: Use standard algebraic notation with terms like "3x^2", "-5x", "+7". Spaces are optional but recommended for clarity.

Q3: Can I use fractions or decimals for the value a?
A: Yes, the calculator accepts any real number for a, including fractions and decimals.

Q4: What if my polynomial has multiple variables?
A: This calculator is designed for single-variable polynomials in x. Multi-variable polynomials are not supported.

Q5: How accurate are the calculations?
A: The calculator provides results with 4 decimal places precision, suitable for most algebraic applications.