Understanding Algebra's Theorems

Polynomial Theorems and Solutions

The Conjugate Root Theorem tells us that roots always come in matching pairs. If you find an imaginary or irrational solution, its conjugate must also be a solution. For example, if $1+i\sqrt{3}$ is a root, then $1-i\sqrt{3}$ must also be a root. Similarly, if $2-3i$ is a solution, then $2+3i$ is also a solution.

The Fundamental Theorem of Algebra states that a polynomial of degree n has exactly n roots. This means a quartic (degree 4) equation has 4 solutions, and a quintic (degree 5) equation has 5 solutions. When solving polynomials, we can use synthetic division to find roots one by one.

Let's see this in action with the equation $x^4+x^3-7x^2-9x-18=0$. First, we try factors of the constant term (±1, ±2, ±3, ±6, etc.) using synthetic division. Once we find that 3 is a root, we divide the polynomial by $x-3$ and continue factoring. The final solution shows roots at x = 3, x = -3, and $x = \frac{-1 \pm i\sqrt{7}}{2}$.

Quick Tip: When solving polynomials, first identify the degree to know exactly how many solutions to expect, then look for the simplest rational roots using factors of the constant term.

For another example, the equation $x^5-x^4-3x^3+3x^2-4x+4=0$ has degree 5, so it has 5 solutions. Through systematic factoring, we find the solutions are x = 1, x = -2, x = 2, and x = ±i.