2. Warm Up  Divide the complex number 3 – 2i 1 + i  Multiply the complex number (3 -2i)(1+i)

9. Math IV Lesson11 Complex numbers 2.5 Essential Question: How do you find roots of polynomials? Standard: MM4A4. Students will investigate functions. a. Compare and contrast properties of functions within and across the following types: linear, quadratic, polynomial, power, rational, exponential, logarithmic, trigonometric, and piecewise.

10.  The degree of a polynomial with one variable is the largest exponent of that variable.  Root: where the polynomial is equal to zero.  A quadratic factor with no real zeros is said to be prime.

11. The degree of a polynomial with one variable is the largest exponent of that variable.

12. Roots A root is where the polynomial is equal to zero So, a polynomial of degree 3 will have 3 complex roots (places where the polynomial is equal to zero). A polynomial of degree 4 will have 4 complex roots. And so on.

13. A polynomial can be rewritten like this: The factors like (x-r 1 ) are called Linear Factors, because they make a line when you plot them.

14. Example: what are the roots of x 2 - 9? x 2 - 9 has a degree of 2, so there will be 2 roots. Let us solve it. We want it to be equal to zero: x 2 - 9 = 0 First move the -9 to the other side: x 2 = +9 Then take the square root of both sides: x = ±3 So the roots are -3 and +3

15. Graph of a Polynomial Function Here is the graph of our polynomial function: The Zeros of the Polynomial are the values of x when the polynomial equals zero. In other words, the Zeros are the x-values where y equals zero.

16. Factors, Roots, Zeros For our Polynomial Function: The Factors are:(x + 5) & (x - 3) The Roots/Solutions are:x = -5 and 3 The Zeros are at:(-5, 0) and (3, 0)

17. Just because an equation has a degree of n doesn’t mean there are n real roots. They can be imaginary roots.

18. Roots & Zeros of Polynomials II Finding the Roots/Zeros of Polynomials: The Fundamental Theorem of Algebra Descartes’ Rule of Signs The Complex Conjugate Theorem

19. Fundamental Theorem Of Algebra Every Polynomial Equation with a degree higher than zero has at least one root in the set of Complex Numbers. A Polynomial Equation of the form P(x) = 0 of degree ‘n’ with complex coefficients has exactly ‘n’ Roots in the set of Complex Numbers. COROLLARY:

20. Polynomials can have imaginary roots

21. Real/Imaginary Roots Just because a polynomial has ‘n’ complex roots doesn’t mean that they are all Real! In this example, however, the degree is still n = 3, but there is only one Real x-intercept or root at x = -1, the other 2 roots must have imaginary components.

22. Descartes’ Rule of Signs Arrange the terms of the polynomial P(x) in descending degree: The number of times the coefficients of the terms of P(x) change sign = the number of Positive Real Roots (or less by any even number) The number of times the coefficients of the terms of P(-x) change sign = the number of Negative Real Roots (or less by any even number) In the examples that follow, use Descartes’ Rule of Signs to predict the number of + and - Real Roots!

23. Find Roots/Zeros of a Polynomial We can find the Roots or Zeros of a polynomial by setting the polynomial equal to 0 and factoring. Some are easier to factor than others! The roots are: 0, -2, 2

24. Find Roots/Zeros of a Polynomial If we cannot factor the polynomial, but know one of the roots, we can divide that factor into the polynomial. The resulting polynomial has a lower degree and might be easier to factor or solve with the quadratic formula. We can solve the resulting polynomial to get the other 2 roots: (x - 5) is a factor

25. Complex Conjugates Theorem Roots/Zeros that are not Real are Complex with an Imaginary component. Complex roots with Imaginary components always exist in Conjugate Pairs. If a + bi (b ≠ 0) is a zero of a polynomial function, then its Conjugate, a - bi, is also a zero of the function.

26. Complex roots always come in pairs Example: x 2 -x+1 Had these roots: 0.5 - 0.866iand0.5 + 0.866i You can either have: No complex roots 2complex roots 4 complex roots 6 complex roots …

27. Find Roots/Zeros of a Polynomial If the known root is imaginary, we can use the Complex Conjugates Theorem. Ex: Find all the roots of If one root is 4 - i. Because of the Complex Conjugate Theorem, we know that another root must be 4 + i. Can the third root also be imaginary? Consider…Descartes: # of Pos. Real Roots = 2 or 0 Descartes: # of Neg. Real Roots = 1

28. Example (con’t) Ex: Find all the roots of If one root is 4 - i. If one root is 4 - i, then one factor is [x - (4 - i)], and Another root is 4 + i, & another factor is [x - (4 + i)]. Multiply these factors:

29. Example (con’t) Ex: Find all the roots of If one root is 4 - i. If the product of the two non-real factors is then the third factor (that gives us the neg. real root) is the quotient of P(x) divided by : The third root is x = -3

30. Finding Roots/Zeros of Polynomials We use the Fundamental Thm. Of Algebra, Descartes’ Rule of Signs and the Complex Conjugate Thm. to predict the nature of the roots of a polynomial. We use skills such as factoring, polynomial division and the quadratic formula to find the zeros/roots of polynomials.

31.  # of complex roots= the highest degree of the polynomial  Possible number of positive real roots = number of times the sign changes in P(x)  Possible number of negative real roots= the number of times the sign changes in P(-x).  Possibly number of imaginary roots= all of the even numbers at and below the highest degree of the polynomial. DETERMINING POSSIBLE ROOTS

32. Factoring a polynomial

33. Use the quadratic formula to solve find the zeros of F(x) = x 2 -12x + 26 QUADRATIC FORMULA

34. P144 # 1-4, 11-19 odd HOMEWORK