4 4 The Fundamental Theorem of Algebra Perform

4. 4 The Fundamental Theorem of Algebra ♦ Perform arithmetic operations on complex numbers ♦ Solve quadratic equations having complex solutions ♦ Apply the fundamental theorem of algebra ♦ Factor polynomials having complex zeros ♦ Solve polynomial equations having complex solutions Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Complex Numbers A complex number can be written in standard form as a + bi, where a and b are real numbers. The real part is a and the imaginary part is b. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4 - 2

Example Write each expression in standard form. Support your results using a calculator. a) ( 4 + 2 i) + (6 3 i) b) ( 9 i) (4 7 i) c) ( 2 + 5 i)2 d) Solution Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4 - 3

Solution continued Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4 - 4

Quadratic Equations with Complex Solutions We can use the quadratic formula to solve quadratic equations if the discriminant is negative. There are no real solutions, and the graph does not intersect the x-axis. The solutions can be expressed as imaginary numbers. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4 - 5

Example Solve the quadratic equation 4 x 2 – 12 x = – 11. Solution Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4 - 6

Fundamental Theorem of Algebra The polynomial f(x) of degree n 1 has at least one complex zero. Number of Zeros Theorem A polynomial of degree n has at most n distinct zeros. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4 - 7

Example Represent a polynomial of degree 4 with leading coefficient 3 and zeros of 2, 4, i and i in complete factored form and expanded form. Solution Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4 - 8

Conjugate Zeros Theorem If a polynomial f(x) has only real coefficients and if a + bi is a zero of f(x), then the conjugate a bi is also a zero of f(x). Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4 - 9

Example Find the zeros of f(x) = x 4 + 5 x 2 + 4 given one zero is i. Solution Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4 - 10

Solution continued Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4 - 11

Example Solve x 3 = 2 x 2 5 x + 10. Solution Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4 - 12

Solution continued Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4 - 13