Complex Numbers

The imaginary number i is defined as

1i so that 12 i

Complex numbers are in the form a + bi

where a is called the real part and bi is the imaginary part.

Complex Numbers

If a + bi is a complex number, its complex conjugate is a – bi.

To add or subtract complex numbers, add or subtract the real parts and add or subtract the imaginary parts.

To multiply two complex numbers, use FOIL, taking advantage of the fact that to simplify.12 i

To divide two complex numbers, multiply top and bottom by the complex conjugate of the bottom.

Complex Numbers

Complex solutions to the Quadratic Formula

When using the Quadratic Formula to solve a quadratic equation, you may obtain a result like , which you should rewrite as

4.i2144

In general if a is positive.iaa

Polynomial Roots (zeros)

If f(x) is a polynomial of degree n, then f has precisely n linear factors:

nn cx...cxcxcxaxf 321

where c1, c2, c3,… cn are complex numbers.

This means that c1, c2, c3,… cn are all roots of f(x), so

that f(c1) = f(c2) = f(c3) = … =f(cn) = 0

Note: some of these roots may be repeated.

Polynomial Roots (zeros)

For polynomial equations with real coefficients, any complex roots will occur in conjugate pairs.

(If a + bi is a root, then a - bi is also a root)