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 $a + bi$ is a root, then its conjugate $a - bi$ 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 $a + bi$ and $a - bi$ .

Conjugate Root Theorem: If $P(x)$ is a polynomial with real coefficients and $a + bi$ is a root, then $a - bi$ 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.

Problem: Given the polynomial $P(x) = x^3 - 4x^2 + 9x - 36$ and one of its roots is $3 + 2i$ , find the other roots.

Since $3 + 2i$ is a root, by the Conjugate Root Theorem, $3 - 2i$ is also a root.
The polynomial can be factored as $(x - (3 + 2i))(x - (3 - 2i))(x - r)$ , where $r$ is the third root.
Multiply the factors $(x - (3 + 2i))(x - (3 - 2i))$ :
$(x - 3 - 2i)(x - 3 + 2i) = (x - 3)^2 - (2i)^2 = x^2 - 6x + 9 + 4 = x^2 - 6x + 13$
So, $P(x) = (x^2 - 6x + 13)(x - r)$ .
By comparing coefficients, we find $r = 4$ .

Validation: Substitute $x = 4$ into $P(x)$ : $P(4) = 4^3 - 4 \cdot 4^2 + 9 \cdot 4 - 36 = 64 - 64 + 36 - 36 = 0$ . ✓

Problem: Given the polynomial $Q(x) = x^4 - 6x^3 + 15x^2 - 18x + 10$ and one of its roots is $1 + i$ , find the other roots.

Since $1 + i$ is a root, by the Conjugate Root Theorem, $1 - i$ is also a root.
The polynomial can be factored as $(x - (1 + i))(x - (1 - i))(x^2 + ax + b)$ .
Multiply the factors $(x - (1 + i))(x - (1 - i))$ :
$(x - 1 - i)(x - 1 + i) = (x - 1)^2 - i^2 = x^2 - 2x + 1 + 1 = x^2 - 2x + 2$
So, $Q(x) = (x^2 - 2x + 2)(x^2 + ax + b)$ .
By comparing coefficients, we find $a = -4$ and $b = 5$ .
Thus, the quadratic factor is $x^2 - 4x + 5$ .
Solve $x^2 - 4x + 5 = 0$ using the quadratic formula: $x = \frac{4 \pm \sqrt{16 - 20}}{2} = \frac{4 \pm \sqrt{-4}}{2} = \frac{4 \pm 2i}{2} = 2 \pm i$ .

Validation: Substitute $x = 2 + i$ into $Q(x)$ : $Q(2 + i) = (2 + i)^4 - 6(2 + i)^3 + 15(2 + i)^2 - 18(2 + i) + 10 = 0$ . ✓

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.