1. What is the Polynomial Factor Theorem?

The Polynomial Factor Theorem states that for a polynomial f(x), (x - a) is a factor if and only if f(a) = 0. This theorem provides a method to determine factors of polynomials and find their roots.

2. How Does the Calculator Work?

The calculator uses the Factor Theorem:

Where:

• \( f(x) \) — Polynomial function
• \( a \) — Constant value
• \( f(a) \) — Value of polynomial when x = a

Explanation: If substituting x = a into the polynomial gives zero, then (x - a) is a factor of the polynomial.

3. Importance of Factor Theorem

Details: The Factor Theorem is fundamental in algebra for factoring polynomials, finding roots, and solving polynomial equations. It's essential for polynomial division and simplification.

4. Using the Calculator

Tips: Enter the polynomial in standard form (e.g., x^3 - 2x^2 - 5x + 6) and the value a to test. The calculator will determine if (x - a) is a factor by evaluating f(a).

5. Frequently Asked Questions (FAQ)

Q1: What is the difference between Factor Theorem and Remainder Theorem?
A: The Factor Theorem is a special case of the Remainder Theorem. If f(a) = 0, then (x - a) is a factor (Factor Theorem). The Remainder Theorem states that f(a) equals the remainder when f(x) is divided by (x - a).

Q2: Can the Factor Theorem be used for polynomials of any degree?
A: Yes, the Factor Theorem applies to polynomials of any degree, from linear equations to higher-degree polynomials.

Q3: What if f(a) is not exactly zero but very close?
A: For exact factorization, f(a) must be exactly zero. Very small values may indicate approximate roots but not exact factors.

Q4: How do I find possible values of a to test?
A: Use the Rational Root Theorem to find possible rational roots to test as values of a.

Q5: Can this theorem be used with complex numbers?
A: Yes, the Factor Theorem works with complex numbers. If f(a) = 0 for a complex number a, then (x - a) is a factor.