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SECTION 3.4 Complex Zeros and the Fundamental Theorem of Algebra.

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Presentation on theme: "SECTION 3.4 Complex Zeros and the Fundamental Theorem of Algebra."— Presentation transcript:

1 SECTION 3.4 Complex Zeros and the Fundamental Theorem of Algebra

2 I NTRODUCTION Consider the polynomial p(x) = x The zeros of p are the solutions to x = 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 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 i are complex conjugates 2. (x - [1+2i]) (x - [1 - 2i]) = x 2 - 2x + 5


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