Factor Theorem

Algebra-2

1. Fundamental Concepts

Definition: The Factor Theorem states that a polynomial \(f(x)\) has a factor \((x - c)\) if and only if \(f(c) = 0\).
Zero of a Polynomial: A number \(c\) is a zero of the polynomial \(f(x)\) if \(f(c) = 0\).
Linear Factors: Linear factors are factors of the form \((x - c)\) where \(c\) is a constant.

2. Key Concepts

Basic Rule: $$\text{If } f(c) = 0, \text{ then } (x - c) \text{ is a factor of } f(x).$$

Degree Preservation: The degree of the polynomial remains unchanged when factored into linear factors.

Application: Used to find roots and factor polynomials in algebraic equations.

3. Examples

Example 1 (Basic)

Problem: Determine if \(x - 2\) is a factor of \(f(x) = x^3 - 4x^2 + 5x - 2\).

Step-by-Step Solution:

Evaluate \(f(2)\): \(f(2) = 2^3 - 4 \cdot 2^2 + 5 \cdot 2 - 2\)
Simplify: \(8 - 16 + 10 - 2 = 0\)

Validation: Since \(f(2) = 0\), \(x - 2\) is a factor of \(f(x)\).

Example 2 (Intermediate)

Problem: Find all zeros of \(g(x) = x^3 - 7x + 6\).

Step-by-Step Solution:

Test possible rational zeros using the Rational Root Theorem: ±1, ±2, ±3, ±6
Test \(g(1)\): \(g(1) = 1^3 - 7 \cdot 1 + 6 = 0\). So, \(x - 1\) is a factor.
Divide \(g(x)\) by \(x - 1\) to get \(x^2 + x - 6\).
Factor \(x^2 + x - 6 = (x + 3)(x - 2)\).
The zeros are \(x = 1\), \(x = -3\), and \(x = 2\).

Validation: Substitute each zero back into \(g(x)\) to confirm they satisfy the equation.

4. Problem-Solving Techniques

Rational Root Theorem: Use to list possible rational zeros.
Synthetic Division: Efficient for dividing polynomials by binomials of the form \(x - c\).
Graphical Method: Use graphs to approximate zeros and verify with algebraic methods.