Day 6: 6.4 Solving Polynomial Equations Warm Up: Factor. 1. x 2-2x x 2-9x x 2 + 6x + 5
|
|
- Malcolm Stone
- 5 years ago
- Views:
Transcription
1 Day 6: 6.4 Solving Polynomial Equations Warm Up: Factor. 1. x 2-2x x 2-9x x 2 + 6x + 5 Solving Equations by Factoring Recall the factoring pattern: Difference of Squares: Note: There is no such thing sum of squares! Sum of Cubes: Difference of Cubes: Verify these 2 patterns by multiplying: 3 Example - Factor. a. x 3-8 b. 8x 3-1 c s 3 d. 64x e. 216x 3-1 f. 8x
2 4 Example - Solve. Find all complex roots. a. 27x = 0 b. 8x Example - Factor. (Hint: a quadratic in disguise...) a. x 4-2x 2-8 b. x 4 + 7x Example - Solve the higher degree polynomial equation. a. x 4 - x 2 = 12 b. x x = 0
3 Quick Check Factor. 1. x 4-5x x 4-3x 2-10 Solve the equation. (See example 4) 3. 27x 3-1 = 0 4. x = 0 5. x 4 + 3x 2-28 = = -x 4-12x 2
4 Day 7: 6.5 Theorems about Roots of Polynomial Equations Warm Up List all the integer factors of each number Multiply. 4. (x - 5)(x 2 + 7) 5. (x + 2)(x + 3)(x - 3) Recall rational numbers can be written as a Their decimal representations or 8. irrational numbers cannot be written as a quotient of integers. Their decimal representations do not repeat and do not terminate. 9. imaginary numbers are non-real numbers and written in the form of 10. root is also known as an or a (graphically) (algebraically) I. Rational Root Theorem If is in simplest form and is a rational root of the polynomial equation... with integer coefficients, then p must be a factor of a 0, and q must be a factor of a n. What does this mean in English? 1 Example - Find the rational roots. x 3 + x 2-3x - 3 = 0 Step 1: List the possible rational roots. Step 2: Test each of them.
5 2 Example - Use the rational root theorem to find all roots of the equation. 2x 3 - x 2 + 2x - 1 = 0 Step 1: List the possible rational roots. Step 2: Test each of them. Step 3: Use synthetic division with the root you found in step 2 to find the quotient. Step 4: Find the roots of the quotient. Quick Check. For 1-2, find the rational roots. 1. x 3-4x 2-2x + 8 = 0 2. x 3-2x 2-5x + 10 = x 3 + x 2 - x + 1 = 0 (Find all roots)
6 Recall: Solve using the quadratic formula. x 2-4x - 1 = 0 II. Irrational Roots In problem above, the 2 solutions are and 2-5. This pair is called... Conjugates - are solutions to polynomial equations. They ALWAYS come in pairs of the form a + b and a - b. Therefore, if you find one conjugate, you automatically know the other conjugate. Irrational Root Theorem Let a and b be rational numbers and b be an irrational number. If a + b is a root (solution) or a polynomial equation with rational coefficients, then the conjugate a - b is also a root. Notice that you conjugate the b. 3 Example (Irrational Roots) Given are roots of a polynomial equation with integer coefficients. Find two additional roots. a and - 11 b. 2-7 and 5 III. Imaginary Roots Complex Conjugates - number pairs of the form a + bi and a - bi. You can use complex conjugates to find an equation s imaginary roots. Imaginary Root Theorem - If the imaginary number a + bi is a root of a polynomial equation with real coefficients, then the conjugate a - bi also is a root. Notice that you conjugate bi. 4 Example (Imaginary Roots) Given are roots of a polynomial equation with integer coefficients. Find two additional roots. a. 3 - i and 2i b. 3i and -2 + i
7 5 Example Find a third degree polynomial equation with rational coefficients that has roots 3 and 1 + i. Step 1: Find the other root using Imaginary Root Theorem. Step 2: Write the polynomial in factored form using Factor Theorem. Step 3: Multiply the factors to write the polynomial in standard form. Quick Check Given are roots of a polynomial equation with integer coefficients. Find two additional roots. a. 2-5 and 7 b. 3i and -2 + i c. Find a fourth degree polynomial equation with rational coefficients that has roots i and 2i. Day Fundamental Theorem of Algebra Warm Up State the degree of each polynomial. 1. 3x 2 - x x x x Solve each equation using the quadratic formula. 4. x = 0 5. x 2-2x + 3 = x 2 + 5x + 4 = 0
8 I. The Fundamental Theorem of Algebra Activity: Zeros a. Find the solutions. b. Are the solutions real or imaginary? c. How many solutions are there? 1. x 4-5x = 0 2. x 4 + 7x = 0 3. Make a conjecture about the number of zeros of a fourth degree polynomial, regardless of the types of zeros. A fourth degree polynomial function has 4 zeros regardless of type. All along we have been solving polynomial equations; we have been finding their roots. The roots have turned out to be integers, rational, irrational, and imaginary. In other words, roots can be complex numbers. Diagram of numbers: Fundamental Theorem of Algebra (FTA) Corollary Including complex roots and multiple roots, an nth degree polynomial equation has exactly n roots; the related polynomial function has exactly n zeros. In other words, you can factor a polynomial of degree n into n linear factors. The number n includes multiple roots. For example, x 3 = 0 has roots and can be written as or (linear factor form) The equation has 3 same linear factors. The 3 roots are the same, - they are all.
9 1 Example Find the number of complex roots, the possible number of real roots, and the possible rational roots. a. x 3 + 2x 2-4x - 6 = 0 By Fundamental Theorem of Algebra, there are complex roots. By Imaginary Root Theorem, there are either imaginary roots or imaginary roots ( conjugate pair). So there are either real roots or real root. By the Rational Root Theorem, the possible rational roots are,,, and. b. x 4-3x 3 + x 2 - x + 3 = 0 2 Example (Must know!) State the number of complex zeros. Then find all the zeros. (Note: You often have to use a combination of graphing, the Factor Theorem, polynomial division, the Remainder Theorem, and the Quadratic Formula!) a. f(x) = x 3 + x 2 - x + 2
10 b. f(x) = x 3-2x 2 + 4x - 8 Day Fundamental Theorem of Algebra (continued) Warm Up 1. Find the number of complex roots, the possible number of real roots, and the possible rational roots. x 4-3x 3 + 4x + 1 = 0 Complex roots: Possible number of real roots: Possible rational roots: 2. State the number of complex roots. Then find all the zeros. f(x) = x 5 + 3x 4 - x - 3 Day 10 Review Chapter 6 Day 11 Test Chapter 6
Dividing Polynomials: Remainder and Factor Theorems
Dividing Polynomials: Remainder and Factor Theorems When we divide one polynomial by another, we obtain a quotient and a remainder. If the remainder is zero, then the divisor is a factor of the dividend.
More informationMore Polynomial Equations Section 6.4
MATH 11009: More Polynomial Equations Section 6.4 Dividend: The number or expression you are dividing into. Divisor: The number or expression you are dividing by. Synthetic division: Synthetic division
More information3.4. ZEROS OF POLYNOMIAL FUNCTIONS
3.4. ZEROS OF POLYNOMIAL FUNCTIONS What You Should Learn Use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions. Find rational zeros of polynomial functions. Find
More informationZEROS OF POLYNOMIAL FUNCTIONS ALL I HAVE TO KNOW ABOUT POLYNOMIAL FUNCTIONS
ZEROS OF POLYNOMIAL FUNCTIONS ALL I HAVE TO KNOW ABOUT POLYNOMIAL FUNCTIONS TOOLS IN FINDING ZEROS OF POLYNOMIAL FUNCTIONS Synthetic Division and Remainder Theorem (Compressed Synthetic Division) Fundamental
More informationSection 4.1: Polynomial Functions and Models
Section 4.1: Polynomial Functions and Models Learning Objectives: 1. Identify Polynomial Functions and Their Degree 2. Graph Polynomial Functions Using Transformations 3. Identify the Real Zeros of a Polynomial
More informationLesson 7.1 Polynomial Degree and Finite Differences
Lesson 7.1 Polynomial Degree and Finite Differences 1. Identify the degree of each polynomial. a. 1 b. 0.2 1. 2 3.2 3 c. 20 16 2 20 2. Determine which of the epressions are polynomials. For each polynomial,
More informationSection 3.6 Complex Zeros
04 Chapter Section 6 Complex Zeros When finding the zeros of polynomials, at some point you're faced with the problem x = While there are clearly no real numbers that are solutions to this equation, leaving
More informationLesson 7.1 Polynomial Degree and Finite Differences
Lesson 7.1 Polynomial Degree and Finite Differences 1. Identify the degree of each polynomial. a. 3x 4 2x 3 3x 2 x 7 b. x 1 c. 0.2x 1.x 2 3.2x 3 d. 20 16x 2 20x e. x x 2 x 3 x 4 x f. x 2 6x 2x 6 3x 4 8
More informationx 2 + 6x 18 x + 2 Name: Class: Date: 1. Find the coordinates of the local extreme of the function y = x 2 4 x.
1. Find the coordinates of the local extreme of the function y = x 2 4 x. 2. How many local maxima and minima does the polynomial y = 8 x 2 + 7 x + 7 have? 3. How many local maxima and minima does the
More informationSkills Practice Skills Practice for Lesson 10.1
Skills Practice Skills Practice for Lesson.1 Name Date Higher Order Polynomials and Factoring Roots of Polynomial Equations Problem Set Solve each polynomial equation using factoring. Then check your solution(s).
More information5.1 Monomials. Algebra 2
. Monomials Algebra Goal : A..: Add, subtract, multiply, and simplify polynomials and rational expressions (e.g., multiply (x ) ( x + ); simplify 9x x. x Goal : Write numbers in scientific notation. Scientific
More informationSolving Quadratic Equations Review
Math III Unit 2: Polynomials Notes 2-1 Quadratic Equations Solving Quadratic Equations Review Name: Date: Period: Some quadratic equations can be solved by. Others can be solved just by using. ANY quadratic
More informationRoots & Zeros of Polynomials. How the roots, solutions, zeros, x-intercepts and factors of a polynomial function are related.
Roots & Zeros of Polynomials How the roots, solutions, zeros, x-intercepts and factors of a polynomial function are related. A number a is a zero or root of a function y = f (x) if and only if f (a) =
More informationSkills Practice Skills Practice for Lesson 4.1
Skills Practice Skills Practice for Lesson.1 Name Date Thinking About Numbers Counting Numbers, Whole Numbers, Integers, Rational and Irrational Numbers Vocabulary Define each term in your own words. 1.
More informationWarm-Up. Simplify the following terms:
Warm-Up Simplify the following terms: 81 40 20 i 3 i 16 i 82 TEST Our Ch. 9 Test will be on 5/29/14 Complex Number Operations Learning Targets Adding Complex Numbers Multiplying Complex Numbers Rules for
More informationComplex Numbers: Definition: A complex number is a number of the form: z = a + bi where a, b are real numbers and i is a symbol with the property: i
Complex Numbers: Definition: A complex number is a number of the form: z = a + bi where a, b are real numbers and i is a symbol with the property: i 2 = 1 Sometimes we like to think of i = 1 We can treat
More informationSection 6.6 Evaluating Polynomial Functions
Name: Period: Section 6.6 Evaluating Polynomial Functions Objective(s): Use synthetic substitution to evaluate polynomials. Essential Question: Homework: Assignment 6.6. #1 5 in the homework packet. Notes:
More information6x 3 12x 2 7x 2 +16x 7x 2 +14x 2x 4
2.3 Real Zeros of Polynomial Functions Name: Pre-calculus. Date: Block: 1. Long Division of Polynomials. We have factored polynomials of degree 2 and some specific types of polynomials of degree 3 using
More informationCP Algebra 2. Unit 3B: Polynomials. Name: Period:
CP Algebra 2 Unit 3B: Polynomials Name: Period: Learning Targets 10. I can use the fundamental theorem of algebra to find the expected number of roots. Solving Polynomials 11. I can solve polynomials by
More informationChapter 2 Polynomial and Rational Functions
Chapter 2 Polynomial and Rational Functions Section 1 Section 2 Section 3 Section 4 Section 5 Section 6 Section 7 Quadratic Functions Polynomial Functions of Higher Degree Real Zeros of Polynomial Functions
More informationA repeated root is a root that occurs more than once in a polynomial function.
Unit 2A, Lesson 3.3 Finding Zeros Synthetic division, along with your knowledge of end behavior and turning points, can be used to identify the x-intercepts of a polynomial function. This information allows
More informationChapter 2 Polynomial and Rational Functions
Chapter 2 Polynomial and Rational Functions Overview: 2.2 Polynomial Functions of Higher Degree 2.3 Real Zeros of Polynomial Functions 2.4 Complex Numbers 2.5 The Fundamental Theorem of Algebra 2.6 Rational
More information!!! 1.! 4x 5 8x 4 32x 3 = 0. Algebra II 3-6. Fundamental Theorem of Algebra Attendance Problems. Identify all the real roots of each equation.
Page 1 of 15 Fundamental Theorem of Algebra Attendance Problems. Identify all the real roots of each equation. 1. 4x 5 8x 4 32x 3 = 0 2. x 3 x 2 + 9 = 9x 3. x 4 +16 = 17x 2 Page 2 of 15 4. 3x 3 + 75x =
More informationTheorems About Roots of Polynomial Equations. Rational Root Theorem
8-6 Theorems About Roots of Polynomial Equations TEKS FOCUS TEKS (7)(E) Determine linear and quadratic factors of a polynomial expression of degree three and of degree four, including factoring the sum
More informationSolving Quadratic Equations by Formula
Algebra Unit: 05 Lesson: 0 Complex Numbers All the quadratic equations solved to this point have had two real solutions or roots. In some cases, solutions involved a double root, but there were always
More information3.4 The Fundamental Theorem of Algebra
333371_0304.qxp 12/27/06 1:28 PM Page 291 3.4 The Fundamental Theorem of Algebra Section 3.4 The Fundamental Theorem of Algebra 291 The Fundamental Theorem of Algebra You know that an nth-degree polynomial
More informationTheorems About Roots of Polynomial Equations. Theorem Rational Root Theorem
- Theorems About Roots of Polynomial Equations Content Standards N.CN.7 Solve quadratic equations with real coefficients that have complex solutions. Also N.CN.8 Objectives To solve equations using the
More informationNAME DATE PERIOD. Operations with Polynomials. Review Vocabulary Evaluate each expression. (Lesson 1-1) 3a 2 b 4, given a = 3, b = 2
5-1 Operations with Polynomials What You ll Learn Skim the lesson. Predict two things that you expect to learn based on the headings and the Key Concept box. 1. Active Vocabulary 2. Review Vocabulary Evaluate
More informationUnit 4 Polynomial/Rational Functions Zeros of Polynomial Functions (Unit 4.3)
Unit 4 Polynomial/Rational Functions Zeros of Polynomial Functions (Unit 4.3) William (Bill) Finch Mathematics Department Denton High School Lesson Goals When you have completed this lesson you will: Find
More informationNatural Numbers Positive Integers. Rational Numbers
Chapter A - - Real Numbers Types of Real Numbers, 2,, 4, Name(s) for the set Natural Numbers Positive Integers Symbol(s) for the set, -, - 2, - Negative integers 0,, 2,, 4, Non- negative integers, -, -
More informationA. Incorrect! Apply the rational root test to determine if any rational roots exist.
College Algebra - Problem Drill 13: Zeros of Polynomial Functions No. 1 of 10 1. Determine which statement is true given f() = 3 + 4. A. f() is irreducible. B. f() has no real roots. C. There is a root
More informationSummary for a n = b b number of real roots when n is even number of real roots when n is odd
Day 15 7.1 Roots and Radical Expressions Warm Up Write each number as a square of a number. For example: 25 = 5 2. 1. 64 2. 0.09 3. Write each expression as a square of an expression. For example: 4. x
More informationPower and Polynomial Functions. College Algebra
Power and Polynomial Functions College Algebra Power Function A power function is a function that can be represented in the form f x = kx % where k and p are real numbers, and k is known as the coefficient.
More informationChapter 2. Polynomial and Rational Functions. 2.5 Zeros of Polynomial Functions
Chapter 2 Polynomial and Rational Functions 2.5 Zeros of Polynomial Functions 1 / 33 23 Chapter 2 Homework 2.5 p335 6, 8, 10, 12, 16, 20, 24, 28, 32, 34, 38, 42, 46, 50, 52 2 / 33 23 3 / 33 23 Objectives:
More informationH-Pre-Calculus Targets Chapter I can write quadratic functions in standard form and use the results to sketch graphs of the function.
H-Pre-Calculus Targets Chapter Section. Sketch and analyze graphs of quadratic functions.. I can write quadratic functions in standard form and use the results to sketch graphs of the function. Identify
More informationSection 1.3 Review of Complex Numbers
1 Section 1. Review of Complex Numbers Objective 1: Imaginary and Complex Numbers In Science and Engineering, such quantities like the 5 occur all the time. So, we need to develop a number system that
More informationThe standard form for a general polynomial of degree n is written. Examples of a polynomial in standard form
Section 4 1A: The Rational Zeros (Roots) of a Polynomial The standard form for a general polynomial of degree n is written f (x) = a n x n + a n 1 x n 1 +... + a 1 x + a 0 where the highest degree term
More informationMath 1302 Notes 2. How many solutions? What type of solution in the real number system? What kind of equation is it?
Math 1302 Notes 2 We know that x 2 + 4 = 0 has How many solutions? What type of solution in the real number system? What kind of equation is it? What happens if we enlarge our current system? Remember
More information6.1 Using Properties of Exponents 1. Use properties of exponents to evaluate and simplify expressions involving powers. Product of Powers Property
6.1 Using Properties of Exponents Objectives 1. Use properties of exponents to evaluate and simplify expressions involving powers. 2. Use exponents and scientific notation to solve real life problems.
More informationOhio s Learning Standards-Extended. Mathematics. The Real Number System Complexity a Complexity b Complexity c
Ohio s Learning Standards-Extended Mathematics The Real Number System Complexity a Complexity b Complexity c Extend the properties of exponents to rational exponents N.RN.1 Explain how the definition of
More informationMIDTERM REVIEW. Write an algebraic expression to represent the following verbal expressions. 1) Double the difference of a number and 7.
NAME MIDTERM REVIEW DATE Write an algebraic epression to represent the following verbal epressions. 1) Double the difference of a number and 7. ) Find the value of the epression 0. Solve each equation.
More informationA2 HW Imaginary Numbers
Name: A2 HW Imaginary Numbers Rewrite the following in terms of i and in simplest form: 1) 100 2) 289 3) 15 4) 4 81 5) 5 12 6) -8 72 Rewrite the following as a radical: 7) 12i 8) 20i Solve for x in simplest
More informationPolynomials and Polynomial Functions
Unit 5: Polynomials and Polynomial Functions Evaluating Polynomial Functions Objectives: SWBAT identify polynomial functions SWBAT evaluate polynomial functions. SWBAT find the end behaviors of polynomial
More informationSect Complex Numbers
161 Sect 10.8 - Complex Numbers Concept #1 Imaginary Numbers In the beginning of this chapter, we saw that the was undefined in the real numbers since there is no real number whose square is equal to a
More informationBeginning Algebra. 1. Review of Pre-Algebra 1.1 Review of Integers 1.2 Review of Fractions
1. Review of Pre-Algebra 1.1 Review of Integers 1.2 Review of Fractions Beginning Algebra 1.3 Review of Decimal Numbers and Square Roots 1.4 Review of Percents 1.5 Real Number System 1.6 Translations:
More informationMaintaining Mathematical Proficiency
Chapter Maintaining Mathematical Proficiency Simplify the expression. 1. 8x 9x 2. 25r 5 7r r + 3. 3 ( 3x 5) + + x. 3y ( 2y 5) + 11 5. 3( h 7) 7( 10 h) 2 2 +. 5 8x + 5x + 8x Find the volume or surface area
More informationCommon Core Algebra 2 Review Session 1
Common Core Algebra 2 Review Session 1 NAME Date 1. Which of the following is algebraically equivalent to the sum of 4x 2 8x + 7 and 3x 2 2x 5? (1) 7x 2 10x + 2 (2) 7x 2 6x 12 (3) 7x 4 10x 2 + 2 (4) 12x
More informationAlgebra II Chapter 5: Polynomials and Polynomial Functions Part 1
Algebra II Chapter 5: Polynomials and Polynomial Functions Part 1 Chapter 5 Lesson 1 Use Properties of Exponents Vocabulary Learn these! Love these! Know these! 1 Example 1: Evaluate Numerical Expressions
More informationRadical Expressions and Graphs 8.1 Find roots of numbers. squaring square Objectives root cube roots fourth roots
8. Radical Expressions and Graphs Objectives Find roots of numbers. Find roots of numbers. The opposite (or inverse) of squaring a number is taking its square root. Find principal roots. Graph functions
More informationUsing Properties of Exponents
6.1 Using Properties of Exponents Goals p Use properties of exponents to evaluate and simplify expressions involving powers. p Use exponents and scientific notation to solve real-life problems. VOCABULARY
More informationLearning Objectives. Zeroes. The Real Zeros of a Polynomial Function
The Real Zeros of a Polynomial Function 1 Learning Objectives 1. Use the Remainder and Factor Theorems 2. Use the Rational Zeros Theorem to list the potential rational zeros of a polynomial function 3.
More informationP.6 Complex Numbers. -6, 5i, 25, -7i, 5 2 i + 2 3, i, 5-3i, i. DEFINITION Complex Number. Operations with Complex Numbers
SECTION P.6 Complex Numbers 49 P.6 Complex Numbers What you ll learn about Complex Numbers Operations with Complex Numbers Complex Conjugates and Division Complex Solutions of Quadratic Equations... and
More informationAlgebra III Chapter 2 Note Packet. Section 2.1: Polynomial Functions
Algebra III Chapter 2 Note Packet Name Essential Question: Section 2.1: Polynomial Functions Polynomials -Have nonnegative exponents -Variables ONLY in -General Form n ax + a x +... + ax + ax+ a n n 1
More informationOctober 28, S4.4 Theorems about Zeros of Polynomial Functions
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial
More informationMAT116 Final Review Session Chapter 3: Polynomial and Rational Functions
MAT116 Final Review Session Chapter 3: Polynomial and Rational Functions Quadratic Function A quadratic function is defined by a quadratic or second-degree polynomial. Standard Form f x = ax 2 + bx + c,
More informationSCORE BOOSTER JAMB PREPARATION SERIES II
BOOST YOUR JAMB SCORE WITH PAST Polynomials QUESTIONS Part II ALGEBRA by H. O. Aliu J. K. Adewole, PhD (Editor) 1) If 9x 2 + 6xy + 4y 2 is a factor of 27x 3 8y 3, find the other factor. (UTME 2014) 3x
More informationThe final is cumulative, but with more emphasis on chapters 3 and 4. There will be two parts.
Math 141 Review for Final The final is cumulative, but with more emphasis on chapters 3 and 4. There will be two parts. Part 1 (no calculator) graphing (polynomial, rational, linear, exponential, and logarithmic
More informationHomework. Basic properties of real numbers. Adding, subtracting, multiplying and dividing real numbers. Solve one step inequalities with integers.
Morgan County School District Re-3 A.P. Calculus August What is the language of algebra? Graphing real numbers. Comparing and ordering real numbers. Finding absolute value. September How do you solve one
More informationSecondary Math 3 Honors - Polynomial and Polynomial Functions Test Review
Name: Class: Date: Secondary Math 3 Honors - Polynomial and Polynomial Functions Test Review 1 Write 3x 2 ( 2x 2 5x 3 ) in standard form State whether the function is even, odd, or neither Show your work
More informationPowers, Roots and Radicals. (11) Page #23 47 Column, #51, 54, #57 73 Column, #77, 80
Algebra 2/Trig Unit Notes Packet Name: Period: # Powers, Roots and Radicals () Homework Packet (2) Homework Packet () Homework Packet () Page 277 # 0 () Page 277 278 #7 6 Odd (6) Page 277 278 #8 60 Even
More informationb) 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
More informationCHAPTER EIGHT: SOLVING QUADRATIC EQUATIONS Review April 9 Test April 17 The most important equations at this level of mathematics are quadratic
CHAPTER EIGHT: SOLVING QUADRATIC EQUATIONS Review April 9 Test April 17 The most important equations at this level of mathematics are quadratic equations. They can be solved using a graph, a perfect square,
More informationPolynomial Functions
Polynomial Functions Polynomials A Polynomial in one variable, x, is an expression of the form a n x 0 a 1 x n 1... a n 2 x 2 a n 1 x a n The coefficients represent complex numbers (real or imaginary),
More informationJust DOS Difference of Perfect Squares. Now the directions say solve or find the real number solutions :
5.4 FACTORING AND SOLVING POLYNOMIAL EQUATIONS To help you with #1-1 THESE BINOMIALS ARE EITHER GCF, DOS, OR BOTH!!!! Just GCF Just DOS Difference of Perfect Squares Both 1. Break each piece down.. Pull
More informationChapter 7 Polynomial Functions. Factoring Review. We will talk about 3 Types: ALWAYS FACTOR OUT FIRST! Ex 2: Factor x x + 64
Chapter 7 Polynomial Functions Factoring Review We will talk about 3 Types: 1. 2. 3. ALWAYS FACTOR OUT FIRST! Ex 1: Factor x 2 + 5x + 6 Ex 2: Factor x 2 + 16x + 64 Ex 3: Factor 4x 2 + 6x 18 Ex 4: Factor
More informationAlgebra 2 Notes AII.7 Polynomials Part 2
Algebra 2 Notes AII.7 Polynomials Part 2 Mrs. Grieser Name: Date: Block: Zeros of a Polynomial Function So far: o If we are given a zero (or factor or solution) of a polynomial function, we can use division
More informationChapter 2 notes from powerpoints
Chapter 2 notes from powerpoints Synthetic division and basic definitions Sections 1 and 2 Definition of a Polynomial Function: Let n be a nonnegative integer and let a n, a n-1,, a 2, a 1, a 0 be real
More informationWarm Up Lesson Presentation Lesson Quiz. Holt Algebra 2 2
6-5 Warm Up Lesson Presentation Lesson Quiz 2 Warm Up Factor completely. 1. 2y 3 + 4y 2 30y 2y(y 3)(y + 5) 2. 3x 4 6x 2 24 Solve each equation. 3(x 2)(x + 2)(x 2 + 2) 3. x 2 9 = 0 x = 3, 3 4. x 3 + 3x
More informationChapter Five Notes N P U2C5
Chapter Five Notes N P UC5 Name Period Section 5.: Linear and Quadratic Functions with Modeling In every math class you have had since algebra you have worked with equations. Most of those equations have
More informationRational Numbers. a) 5 is a rational number TRUE FALSE. is a rational number TRUE FALSE
Fry Texas A&M University!! Math 150!! Chapter 1!! Fall 2014! 1 Chapter 1A - - Real Numbers Types of Real Numbers Name(s) for the set 1, 2,, 4, Natural Numbers Positive Integers Symbol(s) for the set, -,
More informationChapter P. Prerequisites. Slide P- 1. Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide P- 1 Chapter P Prerequisites 1 P.1 Real Numbers Quick Review 1. List the positive integers between -4 and 4.. List all negative integers greater than -4. 3. Use a calculator to evaluate the expression
More information3.3 Dividing Polynomials. Copyright Cengage Learning. All rights reserved.
3.3 Dividing Polynomials Copyright Cengage Learning. All rights reserved. Objectives Long Division of Polynomials Synthetic Division The Remainder and Factor Theorems 2 Dividing Polynomials In this section
More informationNotice that we are switching from the subtraction to adding the negative of the following term
MTH95 Day 6 Sections 5.3 & 7.1 Section 5.3 Polynomials and Polynomial Functions Definitions: Term Constant Factor Coefficient Polynomial Monomial Binomial Trinomial Degree of a term Degree of a Polynomial
More informationChapter 2 Formulas and Definitions:
Chapter 2 Formulas and Definitions: (from 2.1) Definition of Polynomial Function: Let n be a nonnegative integer and let a n,a n 1,...,a 2,a 1,a 0 be real numbers with a n 0. The function given by f (x)
More informationZeros and Roots of a Polynomial Function. Return to Table of Contents
Zeros and Roots of a Polynomial Function Return to Table of Contents 182 Real Zeros of Polynomial Functions For a function f(x) and a real number a, if f (a) = 0, the following statements are equivalent:
More informationUnit 5 Evaluation. Multiple-Choice. Evaluation 05 Second Year Algebra 1 (MTHH ) Name I.D. Number
Name I.D. Number Unit Evaluation Evaluation 0 Second Year Algebra (MTHH 039 09) This evaluation will cover the lessons in this unit. It is open book, meaning you can use your textbook, syllabus, and other
More informationChapter 3-1 Polynomials
Chapter 3 notes: Chapter 3-1 Polynomials Obj: SWBAT identify, evaluate, add, and subtract polynomials A monomial is a number, a variable, or a product of numbers and variables with whole number exponents
More informationU7 - Polynomials. Polynomial Functions - In this first section, we will focus on polynomial functions that have a higher degree than 2.
U7 - Polynomials Name 1 Polynomial Functions - In this first section, we will focus on polynomial functions that have a higher degree than 2. - A one-variable is an expression that involves, at most, the
More informationNAME DATE PERIOD. A negative exponent is the result of repeated division. Extending the pattern below shows that 4 1 = 1 4 or 1. Example: 6 4 = 1 6 4
Lesson 4.1 Reteach Powers and Exponents A number that is expressed using an exponent is called a power. The base is the number that is multiplied. The exponent tells how many times the base is used as
More informationUnit 4, Ongoing Activity, Little Black Book of Algebra II Properties
Unit 4, Ongoing Activity, Little Black Book of Algebra II Properties Little Black Book of Algebra II Properties Unit 4 - Radicals & the Complex Number System 4.1 Radical Terminology define radical sign,
More informationWarm Up Lesson Presentation Lesson Quiz. Holt McDougal Algebra 2
Warm Up Lesson Presentation Lesson Quiz Algebra 2 Warm Up Factor each expression. 1. 3x 6y 2. a 2 b 2 3(x 2y) (a + b)(a b) Find each product. 3. (x 1)(x + 3) 4. (a + 1)(a 2 + 1) x 2 + 2x 3 a 3 + a 2 +
More informationChapter 3: Polynomial and Rational Functions
Chapter 3: Polynomial and Rational Functions 3.1 Polynomial Functions and Their Graphs A polynomial function of degree n is a function of the form P (x) = a n x n + a n 1 x n 1 + + a 1 x + a 0 The numbers
More informationCh 7 Summary - POLYNOMIAL FUNCTIONS
Ch 7 Summary - POLYNOMIAL FUNCTIONS 1. An open-top box is to be made by cutting congruent squares of side length x from the corners of a 8.5- by 11-inch sheet of cardboard and bending up the sides. a)
More informationPre-Calculus Assignment Sheet Unit 8-3rd term January 20 th to February 6 th 2015 Polynomials
Pre-Calculus Assignment Sheet Unit 8- rd term January 0 th to February 6 th 01 Polynomials Date Topic Assignment Calculator Did it Tuesday Multiplicity of zeroes of 1/0/1 a function TI-nspire activity
More informationcorrelated to the Utah 2007 Secondary Math Core Curriculum Algebra 1
correlated to the Utah 2007 Secondary Math Core Curriculum Algebra 1 McDougal Littell Algebra 1 2007 correlated to the Utah 2007 Secondary Math Core Curriculum Algebra 1 The main goal of Algebra is to
More informationPreCalculus Notes. MAT 129 Chapter 5: Polynomial and Rational Functions. David J. Gisch. Department of Mathematics Des Moines Area Community College
PreCalculus Notes MAT 129 Chapter 5: Polynomial and Rational Functions David J. Gisch Department of Mathematics Des Moines Area Community College September 2, 2011 1 Chapter 5 Section 5.1: Polynomial Functions
More informationHow many solutions are real? How many solutions are imaginary? What are the solutions? (List below):
1 Algebra II Chapter 5 Test Review Standards/Goals: F.IF.7.c: I can identify the degree of a polynomial function. F.1.a./A.APR.1.: I can evaluate and simplify polynomial expressions and equations. F.1.b./
More informationHonors Advanced Mathematics November 4, /2.6 summary and extra problems page 1 Recap: complex numbers
November 4, 013.5/.6 summary and extra problems page 1 Recap: complex numbers Number system The complex number system consists of a + bi where a and b are real numbers, with various arithmetic operations.
More informationChapter 1A -- Real Numbers. iff. Math Symbols: Sets of Numbers
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1A! Page 1 Chapter 1A -- Real Numbers Math Symbols: iff or Example: Let A = {2, 4, 6, 8, 10, 12, 14, 16,...} and let B = {3, 6, 9, 12, 15, 18,
More informationChapter 3: Polynomial and Rational Functions
Chapter 3: Polynomial and Rational Functions 3.1 Polynomial Functions A polynomial on degree n is a function of the form P(x) = a n x n + a n 1 x n 1 + + a 1 x 1 + a 0, where n is a nonnegative integer
More informationElementary Algebra
Elementary Algebra 978-1-63545-008-8 To learn more about all our offerings Visit Knewton.com/highered Source Author(s) (Text or Video) Title(s) Link (where applicable) Flatworld Text John Redden Elementary
More informationChapter 8. Exploring Polynomial Functions. Jennifer Huss
Chapter 8 Exploring Polynomial Functions Jennifer Huss 8-1 Polynomial Functions The degree of a polynomial is determined by the greatest exponent when there is only one variable (x) in the polynomial Polynomial
More information171S4.4 Theorems about Zeros of Polynomial Functions. March 27, 2012
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial
More informationInstructional Units Plan Algebra II
Instructional Units Plan Algebra II This set of plans presents the topics and selected for ACT s rigorous Algebra II course. The topics and standards are arranged in ten units by suggested instructional
More informationPre-Calculus: Functions and Their Properties (Solving equations algebraically and graphically, matching graphs, tables, and equations, and
Pre-Calculus: 1.1 1.2 Functions and Their Properties (Solving equations algebraically and graphically, matching graphs, tables, and equations, and finding the domain, range, VA, HA, etc.). Name: Date:
More informationZeros of Polynomial Functions
OpenStax-CNX module: m49349 1 Zeros of Polynomial Functions OpenStax College This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 4.0 In this section, you will:
More informationChapter 4: Radicals and Complex Numbers
Section 4.1: A Review of the Properties of Exponents #1-42: Simplify the expression. 1) x 2 x 3 2) z 4 z 2 3) a 3 a 4) b 2 b 5) 2 3 2 2 6) 3 2 3 7) x 2 x 3 x 8) y 4 y 2 y 9) 10) 11) 12) 13) 14) 15) 16)
More informationMath 175 MT#1 Additional Material Study Sheet
Math 175 MT#1 Additional Material Study Sheet Use the following functions for this worksheet : 1 2 3 2 w( x) = ; f ( x) = 3x 11x 4 ; p( x) = 2x x 17x + 12 ; 2 + x 4 3 2 ( ) 3 ; ( ) 6 22 48 40 ; ( ) 2 k
More informationReference Material /Formulas for Pre-Calculus CP/ H Summer Packet
Reference Material /Formulas for Pre-Calculus CP/ H Summer Packet Week # 1 Order of Operations Step 1 Evaluate expressions inside grouping symbols. Order of Step 2 Evaluate all powers. Operations Step
More information5.6 Logarithmic and Exponential Equations
SECTION 5.6 Logarithmic and Exponential Equations 305 5.6 Logarithmic and Exponential Equations PREPARING FOR THIS SECTION Before getting started, review the following: Solving Equations Using a Graphing
More information