1. SECTION 3.4 Complex Zeros and the Fundamental Theorem of Algebra

2. I NTRODUCTION Consider the polynomial p(x) = x 2 + 1 The zeros of p are the solutions to x 2 + 1 = 0, or x 2 = -1 This equation has no real solutions, but you may recall from Intermediate Algebra that we can formally extract the square roots of both sides to get i is the so-called imaginary unit

3. I MAGINARY U NIT The imaginary unit i satisfies the two following properties 1. 2. If c is a real number with c 0 then

4. C OMPLEX NUMBERS A complex number is a number of the form a + bi, where a and b are real numbers and i is the imaginary unit

5. E XAMPLE Perform the indicated operations and simplify. Write your final answer in the form a + bi. 1. (1 - 2 i ) - (3 + 4 i ) 2. (1 - 2 i )(3 + 4 i ) 3. (1 - 2 i )/(3 - 4 i ) 4. 5.

6. C ONJUGATE The conjugate of a complex number a + bi is the number a - bi The notation commonly used for conjugation is abar: For example,

7. P ROPERTIES OF CONJUGATE Suppose z and w are complex numbers. for n = 1, 2, 3, …. z is a real number if and only if

8. E XAMPLE Suppose we wish to find the zeros of f(x) = x 2 - 2x + 5 Two things are important to note 1. The zeros 1+2 i and 1 - 2 i are complex conjugates 2. (x - [1+2i]) (x - [1 - 2i]) = x 2 - 2x + 5