2.5 The Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra:

If has a polynomial degree of , the number of zeros is also

Conjugate Pairs:

and

Finding a Polynomial with Given Zeros:

Example:

The Imaginary Unit, i

The equation to the left has no real solution because there is no real number x that be squared to equal -3. For that reason, there is an imaginary unit, i.

You can create a set of complex numbers by adding real numbers to real multiples of i, which
can be written in standard form.

Standard form: a+bi

i equals:

There is a pattern to this, i to the fifth degree equals i, i to the sixth degree equals -1, and so on.

This pattern will help you solve equations like i^170. You can divide 170 by four (because i to the fourth equals one), and the remainder is two. i to the second power equals -1, which is your answer.

Examples Problems

Put i in standard form:

1.

** the -2i^2 turns into a +2 because you times it by (-1)

2.