Lesson 6: The Fundamental Theorem of Algebra

Property

The Fundamental Theorem of Algebra states that every polynomial equation of degree $n \geq 1$ with complex coefficients has at least one complex root. The corollary states that a polynomial of degree $n$ has exactly $n$ roots (counting multiplicities) in the complex number system.

Examples

Property

A complex conjugate pair is of the form $a + bi, a - bi$.

Product of Complex Conjugates
If $a$ and $b$ are real numbers, then

Examples

• Multiply $(4 - 5i)(4 + 5i)$ using the pattern: $a=4$ and $b=5$. The result is $a^2 + b^2 = 4^2 + 5^2 = 16 + 25 = 41$.

Property

The discriminant of a quadratic equation is

1. If $D > 0$, there are two unequal real solutions.
2. If $D = 0$, there is one solution of multiplicity two.
3. If $D < 0$, there are two complex conjugate solutions.

Examples

• For $y = x^2 - x - 3$, the discriminant is $D = (-1)^2 - 4(1)(-3) = 13$. Since $D > 0$, the equation has two distinct real solutions and the graph has two x-intercepts.

• For $y = 2x^2 + x + 1$, the discriminant is $D = 1^2 - 4(2)(1) = -7$. Since $D < 0$, the equation has two complex solutions and the graph has no x-intercepts.