Unit 6: Polynomials. 1 Polynomial Functions and End Behavior. 2 Polynomials and Linear Factors. 3 Dividing Polynomials

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1 Date Period Unit 6: Polynomials DAY TOPIC 1 Polynomial Functions and End Behavior Polynomials and Linear Factors 3 Dividing Polynomials 4 Synthetic Division and the Remainder Theorem 5 Solving Polynomial Equations 6 7 Solving Polynomial Inequalities 8 Roots of Polynomial Equations 9 The Fundamental Theorem of Algebra 10 The Binomial Theorem 11 Review

2 U6 D1: Polynomial Functions & End Behavior 1. An epression that is a real number, a variable, or a product of a real number and a variable with wholenumber eponents is known as. a. A is a monomial or the sum of monomials. Standard form is written in descending order of eponents. b. The eponent of the variable in a term is the of that term. P ( ) 5 5 constant Leading coefficient cubic term quadratic term linear term Facts about polynomials: 1. classify by the number of terms it contains. A polynomial of more than three terms does not usually have a special name 3. Polynomials can also be classified by degree. 4. the degree of a polynomial is: Degree Name using degree Polynomial eample Number of terms 0 constant 1 linear quadratic 3 cubic 4 quartic Name using number of terms 5 quintic Page 1 of 3

3 Eamples: 1. Write each polynomial in standard form. Then classify it by degree and by the number of terms. a b c d Use a graphing calculator to determine whether the data best fits a linear model, a quadratic model, or a cubic model. Predict y when 100. y The table shows world gold production for several years. Find a quartic function that models the data. Use it to estimate production since Year Production (millions of troy ounces) Page of 3

4 End Behavior: describes the far left and far right portions of a graph Behavior Up and Up, Down and Down, Down and Up, Up and Down, Graph As y As y Equation Eamples: Determine the end behavior of the graphs of each function below 1) y 3 ) 3 y 3) gt () t t 6 4) h Closure: Simplify the epression below (in standard form), and then provide all of the information a) Classify by # of terms b) Find the degree c) Find the end behavior Page 3 of 3

5 U6 D: Polynomials and Linear Factors Warm up: Write (7 8 5) (9 9) in standard form, and then determine the following: Name of polynomial (degree), Name (# of terms) End behavior: Standard Form: Factored form: 1 3 Standard Form: A polynomial can be written as a product of linear factors (degree ). A linear factor is like a number, meaning it cannot be factored anymore. Factored form is etremely important because it helps us to find the of the polynomial functions. Remember, these are the - intercepts! Eample #: 3 Work backwards: Factor! Remember, think GCF first. a) 10 1 b) Standard form: Standard form is helpful because it tells us the easily. Factored form is helpful because it tells us the, and therefore helps us to graph. Eample: Find the zeros and graph! 3 1 y * This technique is known as the property. Page 4 of 3

6 Eample #: Find the zeros and graph. Label zeros on the graph! f When a zero is repeated, it is said to have a. A multiple zero has a equal to the number of times the zero occurs. So for this eample, has a multiplicity of. Directions: Find the zeros of each polynomial and state the multiplicity of any multiple zeros. a) y 3 3 b) f 4 ( ) 6 8 c) f ( ) ( )( 1)( 1) d) 4 4 Sometimes you might know the zeros of a function, but need to find the equation in standard form. (Review: why do we need an equation in standard form sometimes?) a) zeros are -, 3, and 3 b) zeros are -4,, 1 Closure: What are standard form and factored form of a polynomial and why are they useful? Page 5 of 3

7 U6 D3: Dividing Polynomials (Long Division) Question #1: Is 8 a factor of 76? Question #: Is 4 a factor of 3 1 Important fact: To be a factor, division must yield a remainder of! Directions: Determine whether each divisor is a factor of each dividend. (Do this by division!) a) b) Write your final answers in quotient times divisor plus remainder form: a) b) HW Start! After the class activity, you can start on your homework. The first problems are listed below Divide using long division: Check your answers 1) ) ) ) Page 6 of 3

8 U6 D4: Synthetic Division & the Remainder Theorem Today we will learn an alternative to long division: division. In this process, you omit all variables and eponents. You must remember to the sign of the divisor, so you can add throughout the process Eample #1: Eample #: Divide 4 6 by 1 The Remainder Theorem: constant, then the remainder is If a polynomial Pa. P of degree n 1 is divided by a, where a is a What the heck does that mean? 4 Use synthetic division to find P 1 for P Page 7 of 3

9 Practice: 1) Divide using synthetic division: ) Use synthetic division and the remainder theorem to find Pa ( ). a) P 7 159; a 3 b) P 4 109; a 3 3) Etension: Divide using synthetic division, then check by using long division. Write your final answer in quotient times divisor plus remainder form Page 8 of 3

10 U6 D5: Solving Polynomial Equations Warm up questions: a) Name 4 ways we have learned to solve quadratic equations. b) How does the solution to a quadratic equation relate to the graph of a quadratic function? c) What does it mean for 5 to be a solution to a higher degree polynomial (cubic, quartic, etc.)? Today we are going to practice factoring & solving (or finding zeros) of higher degree polynomials. From Day : 7 18 Today: 3 8 Some of our solving will require that we can factor the sum and difference of two cubes. a b 3 3 a b 3 3 Perfect Cubes Practice: Factor only (for now ) Page 9 of 3

11 3. Solve Solve Now we need to be able factor quartic (degree 4) trinomials Let a, substitute, factor, then back substitute. Make sure that you find factors (or roots)! Solve Solve Closure: factor y and 3 3 y 3 3 Etension: How can we solve 3 on a graph? Page 10 of 3

12 U6 D7: Solving Polynomial Inequalities Page 11 of 3

13 U6 D7: Worksheet Directions: Solve each polynomial inequality algebraically on the number line. Check your solution graphically. Remember you do not need to find imaginary roots because they are not on the number line. Real answers only!! Page 1 of 3

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15 U6 D8: Roots of Polynomial Equations So far we have looked at several ways to find the roots of an equation. Today, we ll learn another one The Rational Root Theorem p If is in simplest form and is a rational root of the polynomial equation n n1 an an 1... a1 a0 0 q with integer coefficients, then p must be a factor of a 0 and q must be a factor of a n. Eample (this will be the easiest way to eplain). Find the rational roots of Possible rational roots are constant leading coef Find the leading coefficient: Factors: Find the constant term: Factors: Now, test all the potential points: Plug them back into the original equation and see if it s true! Eample #: Page 14 of 3

16 You can use the Rational Root Theorem to find all the roots of a polynomial equation (Hooray!) a) 1 0 b) Step 1: Find all possible rational roots Step : Test each of the possible roots Step 3: Use synthetic division to find the quotient Step 4: Find the roots of the quotient Irrational Root Theorem: If so is its conjugate a b is a root, then its conjugate is also a root. If a b is a root, then 6. A polynomial equation with integer coefficients has the roots 1 3 and 11. Find two additional roots. 7. A polynomial equation with rational coefficients has the roots 7 and 5. Find two additional roots. Page 15 of 3

17 Imaginary Root Theorem: If the imaginary number a bi is a root of a polynomial with real coefficients, then the conjugate a bi is also a root. 8. If a polynomial equation with real coefficients has 3i and i among its roots, then what two other roots must it have? 9. A polynomial equation with integer coefficients has the roots 3 i and i. Find two additional roots. 10. Find a third degree polynomial equation with rational coefficients that has roots 3 and 1 i. 11. Find a cubic polynomial equation with rational coefficients that has roots 1 and i. Summing it all up: When/how do we use the following? a. Remainder Theorem: d. Irrational Root Theorem: b. Long Division/Synthetic Division: e. Imaginary Root Theorem: c. Rational Root Theorem: Page 16 of 3

18 U6 D9: The Fundamental Theorem of Algebra You have solved polynomial equations and found that their roots are included in the set of comple numbers. In other words, the roots have been integers, rational numbers, irrational numbers, and imaginary numbers. But, the question remains, can all polynomial equations be solved using comple numbers? Carl Friedrich Gauss ( ) proved that the answer to this question is yes! The roots of every polynomial equation, even those with imaginary coefficients, are comple numbers. The answer to this question was so important that it is now known as thee Fundamental Theorem of Algebra. More info about gradeamathhelp.com. Fundamental Theorem of Algebra: If Pis ( ) a polynomial of degree n 1 with comple coefficients, then P ( ) 0 has at least one comple root. Corollary to the Fundamental Theorem of Algebra: Including imaginary roots and multiple roots, an nth degree polynomial equation has eactly n roots; the related polynomial function has eactly n zeros. In other words, if the leading coefficient of a polynomial is 8, then there are comple roots! Eample #1: Find the number of comple roots of the equation below. Then, break up those roots into the number of possible real roots and the number of possible imaginary roots. 3 1 Eample #: Review of our number system: COMPLEX NUMBERS REAL NUMBERS RATIONAL NUMBERS INTEGERS WHOLE NUMBERS NATURAL NUMBERS I R R A T I O N A L IMAGINARY NUMBERS Page 17 of 3

19 Let s put some of our theorems together: Directions: For each equation, state the number of comple roots, the possible number of real roots, and the possible rational roots. 1) 4 ) ) Find the number of comple zeros of f ( ) 3 3. Find all the zeros. Step #1: use synthetic division to find a rational zero Step #: Solve the remaining quadratic using one of our methods 4) Find the number of comple zeros of y Find all the zeros. Closure: How can you determine the number of comple roots of a polynomial? Number or real roots? Page 18 of 3

20 U6 D10: The Binomial Theorem (Combinations Review) Warm up: Simplify the following. You may use your calculator where possible. 1) 6 ) 5 C 3 3) 8! 3!5! 4) n n! 4! 5) We will need to use combinations for the binomial theorem. If you did not take finite math and need etra help with this, please seek it out. n C r or n n! r r! n r! and n! The binomial theorem is used to epand (reverse or factor) binomials Long way 3 3 Short cut 3 3 Eample #1: Use Pascal s triangle to epand 5 a b Eample #: Use Pascal s triangle to epand 4 3y The eponents of each term add up to. Do not forget parentheses when you have y Page 19 of 3

21 Sometimes you might not want to write out all of the rows of Pascal s triangle, but you still need the coefficients. We can use combinations to skip to any row of Pascal s triangle. Find Look row This is particular useful if we only need to find one term: For eample, find the 5 th term of 6 3 y. Let s find the pattern (it s easier to look at eamples than to give a formula with all variables) a) Third term of 3 1 b) Second term of 3 9 c) Eighth term of 15 y d) Seventh term of 11 y Wrap up: One term of a binomial epression is 7 5 y. What is the term just before that term? Etension: Eample 7 5y Page 0 of 3

22 U6 D11: Review 1. Directions: Give the end behavior of each of the functions below. Write your answer as one of the following:,,,,,,, a) y b) y 4 c) y 14 3 ( 8). Directions: Simplify each of the following a) 7 b) 9 C 5 c) C y 3) Find the zeros of ) Classify with degree and # of terms: a) b) y 4 y 10 5) Which of the following is a rational zero of y = ? a. = 35 b. = -35 c. = -7 d. =7 3 6) State the zeros and the multiplicity for each for the function f 5 3 7) Give the number of possible imaginary roots for each polynomial below a) y b) 5 4 y 7 8) If 5 4i is a root of a polynomial, name another root of that same polynomial Page 1 of 3

23 9) The roots of a cubic polynomial are 1 and 5i. Write the polynomial in standard form. Show all work. 10) Use cubic regression to find the cubic model for the following data y a. Write the equation of the cubic model. Round each number to three decimal places. y = b. Use your model to predict the value of y when 1. y = 11) Epand 6 3y 1) Find the 3 rd term of ) Find allllll zeros algebraicially Page of 3

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Anchorage School District/Alaska Sr. High Math Performance Standards Algebra Anchorage School District/Alaska Sr. High Math Performance Standards Algebra Algebra 1 2008 STANDARDS PERFORMANCE STANDARDS A1:1 Number Sense.1 Classify numbers as Real, Irrational, Rational, Integer,

Algebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 2012-13 school year. This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra 2.5 ZEROS OF POLYNOMIAL FUNCTIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions.

Algebra 1 If you are okay with that placement then you have no further action to take Algebra 1 Portion of the Math Placement Test Dear Parents, Based on the results of the High School Placement Test (HSPT), your child should forecast to take Algebra 1 this fall. If you are okay with that placement then you have no further action

Section 3-3 Approximating Real Zeros of Polynomials - Approimating Real Zeros of Polynomials 9 Section - Approimating Real Zeros of Polynomials Locating Real Zeros The Bisection Method Approimating Multiple Zeros Application The methods for finding zeros

LINEAR INEQUALITIES. less than, < 2x + 5 x 3 less than or equal to, greater than, > 3x 2 x 6 greater than or equal to, LINEAR INEQUALITIES When we use the equal sign in an equation we are stating that both sides of the equation are equal to each other. In an inequality, we are stating that both sides of the equation are PRE-CALCULUS GRADE 12 [C] Communication Trigonometry General Outcome: Develop trigonometric reasoning. A1. Demonstrate an understanding of angles in standard position, expressed in degrees and radians.

3-17 15-25 5 15-10 25 3-2 5 0. 1b) since the remainder is 0 I need to factor the numerator. Synthetic division tells me this is true Section 5.2 solutions #1-10: a) Perform the division using synthetic division. b) if the remainder is 0 use the result to completely factor the dividend (this is the numerator or the polynomial to the

BookTOC.txt. 1. Functions, Graphs, and Models. Algebra Toolbox. Sets. The Real Numbers. Inequalities and Intervals on the Real Number Line College Algebra in Context with Applications for the Managerial, Life, and Social Sciences, 3rd Edition Ronald J. Harshbarger, University of South Carolina - Beaufort Lisa S. Yocco, Georgia Southern University

Polynomial. Functions. 6A Operations with Polynomials. 6B Applying Polynomial. Functions. You can use polynomials to predict the shape of containers. Polynomial Functions 6A Operations with Polynomials 6-1 Polynomials 6- Multiplying Polynomials 6-3 Dividing Polynomials Lab Explore the Sum and Difference of Two Cubes 6-4 Factoring Polynomials 6B Applying

Factors and Products CHAPTER 3 Factors and Products What You ll Learn use different strategies to find factors and multiples of whole numbers identify prime factors and write the prime factorization of a number find square

MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education) MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers,

The degree of a polynomial function is equal to the highest exponent found on the independent variables. DETAILED SOLUTIONS AND CONCEPTS - POLYNOMIAL FUNCTIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE NOTE

PERT Computerized Placement Test PERT Computerized Placement Test REVIEW BOOKLET FOR MATHEMATICS Valencia College Orlando, Florida Prepared by Valencia College Math Department Revised April 0 of 0 // : AM Contents of this PERT Review

Chapter 4 -- Decimals Chapter 4 -- Decimals \$34.99 decimal notation ex. The cost of an object. ex. The balance of your bank account ex The amount owed ex. The tax on a purchase. Just like Whole Numbers Place Value - 1.23456789

2.5 Zeros of a Polynomial Functions .5 Zeros of a Polynomial Functions Section.5 Notes Page 1 The first rule we will talk about is Descartes Rule of Signs, which can be used to determine the possible times a graph crosses the x-axis and

Algebra 2 Year-at-a-Glance Leander ISD 2007-08. 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks Algebra 2 Year-at-a-Glance Leander ISD 2007-08 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks Essential Unit of Study 6 weeks 3 weeks 3 weeks 6 weeks 3 weeks 3 weeks

Algebra I Vocabulary Cards Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression

Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important

Mathematics Placement Mathematics Placement The ACT COMPASS math test is a self-adaptive test, which potentially tests students within four different levels of math including pre-algebra, algebra, college algebra, and trigonometry. hsn.uk.net Higher Mathematics UNIT OUTCOME 1 Polnomials and Quadratics Contents Polnomials and Quadratics 64 1 Quadratics 64 The Discriminant 66 3 Completing the Square 67 4 Sketching Parabolas 70 5 Determining 1.3 Polynomials and Factoring Polynomials Constant: a number, such as 5 or 27 Variable: a letter or symbol that represents a value. Term: a constant, variable, or the product or a constant and variable. Polynomial Functions Lessons 7-1 and 7-3 Evaluate polynomial functions and solve polynomial equations. Lessons 7- and 7-9 Graph polynomial and square root functions. Lessons 7-4, 7-5, and 7-6 Find factors