Conjugate Root Theorem

Algebra-2

1. Fundamental Concepts

Definition: The Conjugate Root Theorem states that if a polynomial has real coefficients and a complex number is a root, then its conjugate is also a root.
Polynomial with Real Coefficients: A polynomial where all the coefficients are real numbers.
Complex Conjugates: Pairs of complex numbers of the form and .

2. Key Concepts

Conjugate Root Theorem: If is a polynomial with real coefficients and is a root, then is also a root.

Implication for Roots: Non-real roots of polynomials with real coefficients always occur in conjugate pairs.

Application: Used to find all roots of a polynomial when some roots are known to be complex.

3. Examples

Example 1 (Basic)

Problem: Given the polynomial and one of its roots is , find the other roots.

Step-by-Step Solution:

Since is a root, by the Conjugate Root Theorem, is also a root.
The polynomial can be factored as , where is the third root.
Multiply the factors :
So, .
By comparing coefficients, we find .

Validation: Substitute into : . ✓

Example 2 (Intermediate)

Problem: Given the polynomial and one of its roots is , find the other roots.

Step-by-Step Solution:

Since is a root, by the Conjugate Root Theorem, is also a root.
The polynomial can be factored as .
Multiply the factors :
So, .
By comparing coefficients, we find and .
Thus, the quadratic factor is .
Solve using the quadratic formula: .

Validation: Substitute into : . ✓

4. Problem-Solving Techniques

Identify Known Roots: Use the given roots to apply the Conjugate Root Theorem.
Factorization: Factor the polynomial using the known roots and their conjugates.
Compare Coefficients: Compare the coefficients of the factored form with the original polynomial to find the remaining roots.
Quadratic Formula: Use the quadratic formula to solve for the remaining roots if necessary.