The Fundamental Theorem of Algebra - Algebra-2

1. Fundamental Concepts

Definition: The Fundamental Theorem of Algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root.
Complex Numbers: Complex numbers are of the form \[a + bi\] where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit such that \(i^2 = -1\).
Polynomial Degree: The degree of a polynomial is the highest power of the variable in the polynomial.

2. Key Concepts

Roots and Factors: If \(r\) is a root of the polynomial \(P(x)\), then \((x - r)\) is a factor of \(P(x)\).

Number of Roots: A polynomial of degree \(n\) has exactly \(n\) roots, counting multiplicities.

Application: Used to find all roots of a polynomial, which is essential in solving polynomial equations and understanding their behavior.

3. Examples

Example 1 (Basic)

Problem: Find all the roots of the polynomial \[P(x) = x^2 - 4.\]

Step-by-Step Solution:

Set the polynomial equal to zero: \[x^2 - 4 = 0.\]
Factor the polynomial: \[(x - 2)(x + 2) = 0.\]
Solve for \(x\): \[x - 2 = 0 \quad \text{or} \quad x + 2 = 0.\]
The roots are: \[x = 2 \quad \text{and} \quad x = -2.\]

Validation: Substitute \(x = 2\) and \(x = -2\) into \(P(x)\): \[P(2) = 2^2 - 4 = 0,\] \[P(-2) = (-2)^2 - 4 = 0.\]

Example 2 (Intermediate)

Problem: Find all the roots of the polynomial \[Q(x) = x^3 - 6x^2 + 11x - 6.\]

Step-by-Step Solution:

Use the Rational Root Theorem to test possible rational roots: \(\pm 1, \pm 2, \pm 3, \pm 6\).
Test \(x = 1\): \[Q(1) = 1^3 - 6 \cdot 1^2 + 11 \cdot 1 - 6 = 0.\]
Since \(x = 1\) is a root, factor \(Q(x)\) as \((x - 1)(x^2 - 5x + 6)\).
Factor the quadratic: \[x^2 - 5x + 6 = (x - 2)(x - 3).\]
The roots are: \[x = 1, \quad x = 2, \quad x = 3.\]

Validation: Substitute \(x = 1\), \(x = 2\), and \(x = 3\) into \(Q(x)\): \[Q(1) = 1^3 - 6 \cdot 1^2 + 11 \cdot 1 - 6 = 0,\] \[Q(2) = 2^3 - 6 \cdot 2^2 + 11 \cdot 2 - 6 = 0,\] \[Q(3) = 3^3 - 6 \cdot 3^2 + 11 \cdot 3 - 6 = 0.\]

4. Problem-Solving Techniques

Rational Root Theorem: Use the Rational Root Theorem to list possible rational roots and test them.
Factoring: Factor the polynomial using known roots or by grouping terms.
Synthetic Division: Use synthetic division to divide the polynomial by a linear factor and reduce the degree of the polynomial.
Graphical Analysis: Use graphing tools to visualize the polynomial and estimate the roots.