Fundamental Theorem of Algebra (NEW): A polynomial function of degree n > 0 has n complex zeros. Some of these zeros may be repeated.

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1 .5 and.6 Comple Numbers, Comple Zeros and the Fundamental Theorem of Algebra Pre Calculus.5 COMPLEX NUMBERS 1. Understand that - 1 is an imaginary number denoted by the letter i.. Evaluate the square root of negative numbers.. Understand that comple numbers look like a + bi, where a is real and bi is imaginary. 4. Know how to Add and Subtract Comple Numbers. 5. Know how to Multiply Comple Numbers and simplify powers of i. 6. Know how to find a comple conjugate. 7. Know how to use the comple conjugate to Divide comple numbers. 8. Know how to solve a quadratic equation using a domain of comple numbers. See the Worksheet for eamples and practice with these targets..6 COMPLEX ZEROS AND THE FUNDAMENTAL THEOREM OF ALGEBRA 1. Know how many and what type (real or non-real) of zeros a polynomial can have.. Understand that comple zeros come in pairs.. Given the zeros (real and comple) of a polynomial, find the standard form of the polynomial. 4. Given a polynomial, find all the real and comple zeros. 5. Use the zeros of a polynomial to write a polynomial as a product of linear and irreducible quadratic factors. First you should become familiar with the following theorems and concepts Fundamental Theorem of Algebra (NEW): A polynomial function of degree n > 0 has n comple zeros. Some of these zeros may be repeated. Multiplicity (Ch A): The number of times a zero or a factor is repeated. Comple Conjugate Zeros (NEW): If = a + bi is a zero of a polynomial, then = a bi (the conjugate) is also a zero. Factor Theorem (Ch A): If c is a zero, then ( c) is a factor and vice-versa. Factors of a Polynomial with Real Coefficients: Every polynomial function with real coefficients can be written as a product of linear factors and irreducible quadratic factors*, each with real coefficients. *An irreducible quadratic factor is a quadratic with NO real solutions (i.e. comple solutions). Eample 1: Suppose you have a polynomial function with zeros = and = + i. a) What is the minimum degree of this polynomial? b) Write a polynomial function in factored form with these given zeros. Use linear factors. c) Write your polynomial function as the product of linear and irreducible quadratic factors with real coefficients. - 9

2 .5 and.6 Comple Numbers, Comple Zeros and the Fundamental Theorem of Algebra Pre Calculus d) Write your polynomial function in standard form (with with zeros = and = + i.) Now that we have gone from the zeros to the equation of the polynomial in standard form, let s revisit using the standard form of a polynomial to find the zeros as they relate to multiplicity and comple zeros. Eample : Given the function 4 f( ) = a) List the possible rational zeros (like lesson -4). b) Use your graphing calculator to determine which of these possible zeros could be zeros. Verify them. Then, find ALL zeros of the function. c) Write the function from part (b) as a product of linear and irreducible quadratic factors. - 10

3 .7 Graphs of Rational Functions PreCalculus.7 Graphs of Rational Functions Day 1 1. Find the following attributes of rational functions: i) End Behavior (including Horizontal Asymptotes or Slant Asymptotes) DAY 1 ii) Vertical Asymptotes (distinguish between holes and vertical asymptotes) DAY 1 iii) -intercept(s) DAY iv) y-intercept DAY. Graph a rational function by hand including all of the above attributes (if they eist). DAY All of the functions we will deal with for the remainder of this chapter will be Rational Functions. Let s begin with a definition and a review of end behavior f ( ) Rational Function: y g( ) Eample 1: Find the end behavior of the function using limit notation. NOTE: For ALL rational functions, IF the end behavior is a numeric value, that value will be the SAME as. In addition, the end behavior for all rational functions can fit into three categories. Eample are provided below. A) f 54 B) g 4 7 C) h 4 5 Eample : Find the end behavior for each rational function above. a) b) c) Eample : Since end behavior only uses the 1 st term of a polynomial and a rational function is simply the ratio of two polynomials, we can use the ratio of the end behaviors to find the end behavior of the rational function. How does the ratio of the end behaviors compare to the end behavior of the graphs? - 11

4 .7 Graphs of Rational Functions PreCalculus Here s why this works Step 1: Consider the parent function 1 h. As, where does h () approach? Step : Let s rewrite 4 g 7 We need to put all this information together with the what we already know about Domain, Vertical Asymptotes and Holes from previous chapters yep, you were supposed to remember that!! -18 Eample 4: Use the equation below to answer the following questions: h( ) = a) Find the end behavior. Include HA or slant asymptotes. b) Find any vertical asymptotes. Distinguish between holes and vertical asymptotes. Eample 5: Use the equation below to answer the following questions: g( ) a) Find the end behavior. Include HA or slant asymptotes = b) Find any vertical asymptotes. Distinguish between holes and vertical asymptotes. - 1

5 .7 Graphs of Rational Functions PreCalculus.7 Graphs of Rational Functions Day 1. Find the following attributes of rational functions: iii) -intercept(s) DAY iv) y-intercept DAY. Graph a rational function by hand including all of the above attributes (if they eist). DAY An -intercept is any -value whose y-value =. We will write our answer as. A y-intercept is any y-value whose -value =. We will write our answer as. Eample 1: Find the -intercept and the y-intercept of f ( ) 4 7. Let s see how these definitions apply to Rational Functions When does a fraction = 0? How do we find the y-value of a fraction when =0? Eample : Find the -intercept and the y-intercept of f ( ) Now to put this information with what we learned about asymptotes in -7 day 1. We will revisit the same eamples ( + )( -) Eample : Given h( ) = = (-5)( -) a) Find the end behavior. End Behavior: lim h ( ) 1 HA: y = 1 b) Find any vertical asymptotes. Hole at = ; VA: = 5 c) Find the -intercept(s). y d) Find the y-intercept. e) Graph the function using the information above and any additional points as needed. - 1

6 .7 Graphs of Rational Functions PreCalculus For eamples 4-5, for the given function find the following (if they eist). a. End Behavior including the equations of horizontal or slant asymptotes b. Vertical Asymptote(s). Distinguish between holes and vertical asymptotes. c. - intercept(s) d. y-intercept and additional points in each region to determine the shape. e. Graph without a calculator. Eample 4: g( ) ( + 4)( + ) = = + 1 a) Find the end behavior. End Behavior: lim g ( ) SA: y 5 b) Find any vertical asymptotes. VA: = 1 lim g ( ) y 1 k = Eample 5: ( ) y - 14

7 .9 Solving Inequalities in One Variable PreCalculus.9 Solving Inequalities in One Variable 1. Make a sign chart of a function using the values of where the function is zero or undefined. Use a sign chart (or a graph) to determine the intervals where a function is positive or negative.. Understand how ( and [ relate to < and < when solving inequalities. In section., we graphed polynomials like ( ) ( ) ( ) Polynomial functions can be positive, negative or zero. To solve f( ) 0 To solve f ( ) 0 f = - + ( - 5) using the zeros, multiplicity, and end behavior. Eample 1: Use the graph above to answer the following inequalities. Write your answer in interval notation. a) Find the values of so that ( ) ( ) - + ( - 5) > 0. b) Find the values of so that ( ) ( ) - + ( - 5) ³ 0. c) Find the values of so that ( ) ( ) - + ( - 5) < 0. d) Find the values of so that ( ) ( ) - + ( - 5) 0. Sign charts are simply number lines with (pos) or (neg) signs on them to represent whether or not the function s y-values are positive or negative. For polynomial functions, the critical numbers for the sign chart are the zeros of the function. Eample : Make a sign chart for the function ( ) ( ) ( ) f = -1-7 (+ 5) Eample : Solve the following inequality: ( ) ( ) (Use your sign chart from eample ) -1-7 (+ 5) > 0-15

8 .9 Solving Inequalities in One Variable PreCalculus For the remainder of this lesson, we will deal with rational functions (a.k.a. fraction of polynomials). Rational functions can be positive, negative, zero or undefined. : A fraction equals 0 when the numerator equals 0. A fraction is undefined when the denominator equals 0. **Always use ( ) where the fraction is undefined. For this reason, we need to include BOTH of these values in our sign charts. Eample 4: Make a sign chart for g( ) =. Indicate which values are zeros and which values are undefined. - + ( 1)( ) Eample 5: Answer the following inequalities using the sign chart from eample 5: a) Find the values of so that ( - 1 )( + ) < 0. b) Find the values of so that ( - 1 )( + ) 0. c) Find the values of so that ( - 1 )( + ) > 0. d) Find the values of so that ( - 1 )( + ) ³ 0 In order to make the sign chart, one side of the inequality MUST be written as a single fraction. Eample 6: Solve the following inequality: