200 300 400 500 100 200 300 400 500 100 200 300 400 500 100 200 300 400 500 100 200 300 400 500 100 Remainder/ Factor Theorem End Behavior Zeros / Graphs.

Slides:

Advertisements

Similar presentations

Notes 6.6 Fundamental Theorem of Algebra

Advertisements

A POLYNOMIAL is a monomial or a sum of monomials.

Remainder/ Factor Theorem End Behavior Zeros PolynomialsGrab.

5.5: Polynomial Long Division and Synthetic Division

Remainder/ Factor Theorem End Behavior Zeros PolynomialsAsymptotes.

Creating Polynomials Given the Zeros.. What do we already know about polynomial functions? They are either ODD functions They are either EVEN functions.

Section 5.5 – The Real Zeros of a Rational Function

EXAMPLE 2 Find all zeros of f (x) = x 5 – 4x 4 + 4x x 2 – 13x – 14. SOLUTION STEP 1 Find the rational zeros of f. Because f is a polynomial function.

Chapter 4 – Polynomials and Rational Functions

C. Higher Functions Pre-Calculus 30.

Remainder and Factor Theorem Unit 11. Definitions Roots and Zeros: The real number, r, is a zero of f(x) iff: 1.) r is a solution, or root of f(x)=0 2.)

EXAMPLE 2 Find all real zeros of f (x) = x 3 – 8x 2 +11x SOLUTION List the possible rational zeros. The leading coefficient is 1 and the constant.

Zeros of Polynomials PolynomialType of Coefficient 5x 3 + 3x 2 + (2 + 4i) + icomplex 5x 3 + 3x 2 + √2x – πreal 5x 3 + 3x 2 + ½ x – ⅜rational 5x 3 + 3x.

Warm Up Solve using synthetic OR long division Polynomial Functions A polynomial is written in standard form when the values of the exponents are.

The Rational Zero Theorem The Rational Zero Theorem gives a list of possible rational zeros of a polynomial function. Equivalently, the theorem gives all.

Bell Ringer 1. What is the Rational Root Theorem (search your notebook…Unit 2). 2. What is the Fundamental Theorem of Algebra (search your notebook…Unit.

2 pt 3 pt 4 pt 5pt 1 pt 2 pt 3 pt 4 pt 5 pt 1 pt 2pt 3 pt 4pt 5 pt 1pt 2pt 3 pt 4 pt 5 pt 1 pt 2 pt 3 pt 4pt 5 pt 1pt Roly Poly Divide and Conquer! Get.

Fundamental Theorem of Algebra If f(x) is a polynomial of degree n, where n≥1, then the equation f(x) = 0 has at least one complex root. Date: 2.6 Topic:

4-5, 4-6 Factor and Remainder Theorems r is an x intercept of the graph of the function If r is a real number that is a zero of a function then x = r.

GUIDED PRACTICE for Example How many solutions does the equation

7.5.1 Zeros of Polynomial Functions

6.6 The Fundamental Theorem of Algebra

Quick Crisp Review Zeros of a polynomial function are where the x-intercepts or solutions when you set the equation equal to zero. Synthetic and long division.

6.9 Rational Zero Theorem Parts of a polynomial function f(x) oFactors of the leading coefficient = q oFactors of the constant = p oPossible rational roots.

Ch 2.5: The Fundamental Theorem of Algebra

Polynomials Integrated Math 4 Mrs. Tyrpak. Definition.

Lesson 2.5, page 312 Zeros of Polynomial Functions Objective: To find a polynomial with specified zeros, rational zeros, and other zeros, and to use Descartes’

1. Describe the end behavior of the graph y = 2x 5 – 3x Sketch a graph of 3 rd degree with a zero at -5 (multiplicity 2) and a zero at 0 (multiplicity.

Section 3.3 Theorems about Zeros of Polynomial Functions.

Warm Up. Find all zeros. Graph.. TouchesThrough More on Rational Root Theorem.

Introduction Synthetic division, along with your knowledge of end behavior and turning points, can be used to identify the x-intercepts of a polynomial.

4.5 Quadratic Equations Zero of the Function- a value where f(x) = 0 and the graph of the function intersects the x-axis Zero Product Property- for all.

Factor Theorem Using Long Division, Synthetic Division, & Factoring to Solve Polynomials.

Characteristics of Polynomials: Domain, Range, & Intercepts

1. Describe the end behavior of the graph y = 2x 5 – 3x Sketch a graph of 3 rd degree with a zero at -5 (multiplicity 2) and a zero at 0 (multiplicity.

7.1 Polynomial Functions Evaluate Polynomials

UNIT 2, LESSON 1 POLYNOMIAL FUNCTIONS. WHAT IS A POLYNOMIAL FUNCTION? Coefficients must be real numbers. Exponents must be whole numbers.

Polynomial Functions Characteristics The Remainder Theorem The Factor Theorem Equations and Graphs Math.

LESSON 5.6 Rational Zeros of Polynomial Functions.

Real Zeros of Polynomial Functions. Solve x 3 – 2x + 1 = 0. How? Can you factor this? Can you use the quadratic formula? Now what if I tell you that one.

Slide Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc.

Remainder and Factor Theorems Unit 11. Definitions The real number, r, is a zero of f(x) iff:  r is a solution, or root, of f(x)=0  x-r is a factor.

1 Algebra 2: Section 6.2 Evaluating and Graphing Polynomial Functions (Day 1)

Advanced Algebra Notes Section 5.2: Evaluate and Graph Polynomial Functions A __________________ is a number, a variable, or the product of numbers and.

I am able to solve a quadratic equation using complex numbers. I am able to add, subtract, multiply, and divide complex numbers. Solve the equation.

Polynomials. DegreeNameExample 0Constant 1Linear 2Quadratic 3Cubic 4Quartic 5Quintic Some of the Special Names of the Polynomials of the first few degrees:

Chapter 2 – Polynomial and Rational Functions 2.5 – The Fundamental Theorem of Algebra.

Zeros of Polynomial Functions A-APR.2 Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is.

Zeros (Solutions) Real Zeros Rational or Irrational Zeros Complex Zeros Complex Number and its Conjugate.

Warm-ups Week 8 10/8/12 Find the zeros of f(x) = x3 + 2x2 – 13x + 10 algebraically (without a graphing calculator). What if I told.

Warm Up Compute the following by using long division.

Roly Poly Divide and Conquer! Get to the root of the Problem!

Polynomial Function Review

Where am I now? Review for quiz 4.1.

Pre-Calculus Section 2.3 Synthetic Division

Review Chapter 2 Sections

Real Zeros Intro - Chapter 4.2.

Pre-AP Algebra 2 Goal(s):

A POLYNOMIAL is a monomial or a sum of monomials.

Finding Zeros of Polynomials

Sketch the graph of the function {image} Choose the correct answer from the following. {applet}

**Get signed by your parents for 5 bonus points on the test!!

Zeros of a Polynomial Function

Polynomial Functions Equations and Graphs Characteristics

Remainder/ Factor Theorem End Behavior Zeros / Graphs Polynomials

Jeopardy Q $100 Q $100 Q $100 Q $100 Q $100 Q $200 Q $200 Q $200

Fundamental Theorem of Algebra

1) Find f(g(x)) and g(f(x) to show that f(x) and g(x) are inverses

Presentation transcript:

Remainder/ Factor Theorem End Behavior Zeros / Graphs PolynomialsExponents

Remainder/Factor Theorem 100 Main Get Answer Use the Remainder Theorem to find f(3) for f(x) = 4x 4 – 2x 3 – 10x A. -10B. -60 C. 125D. 170

Use the Remainder Theorem to find f(3) for f(x) = 4x 4 – 2x 3 – 10x A. -10B. -60 C. 125D. 170 Main Remainder/Factor Theorem 100

Main Get Answer Divide 2x 3 + 5x 2 – 7x – 1 by (2x+3) Remainder/Factor Theorem 200

Main Remainder/Factor Theorem 200 Divide 2x 3 + 5x 2 – 7x – 1 by (2x+3) x 2 + x – 5 + _14__ (2x+3)

Remainder/Factor Theorem 300 Main Get Answer Divide 3x x x + 22 by (x+4)

Remainder/Factor Theorem 300 Main 3x 2 + 4x _2__ (x+4) Divide 3x x x + 22 by (x+4)

Remainder/Factor Theorem 400 If I were to look in the dictionary under the words “greatest” and “math teacher”, whose name would I see? Main Get Answer

Remainder/Factor Theorem 400 If I were to look in the dictionary under the words “greatest” and “math teacher”, whose name would I see? KAPLAN ! Come on guys, that was the easiest 400 points in the game! Main

Remainder/Factor Theorem 500 Main Get Answer Determine if (x – 2) is a factor of: f(x) = 4x 3 – 9x 2 – 3x + 12

Remainder/Factor Theorem 500 Main No, but you must prove it with synthetic division for your points! Determine if (x – 2) is a factor of: f(x) = 4x 3 – 9x 2 – 3x + 12

End Behavior 100 Main Get Answer Describe the end behavior of f(x) = -6x x 4 – 8x As x  +, f(x)  ______ As x  -, f(x)  ______

End Behavior 100 Main Describe the end behavior of f(x) = -6x x 4 – 8x As x  +, f(x)  ______ As x  -, f(x)  ______

End Behavior 200 Main Get Answer Describe the end behavior of f(x) = 6x x 3 – 8x + 11 As x  +, f(x)  ______ As x  -, f(x)  ______

End Behavior 200 Main Describe the end behavior of f(x) = 6x x 3 – 8x + 11 As x  +, f(x)  ______ As x  -, f(x)  ______

End Behavior 300 Main Get Answer Describe the end behavior of f(x) = -x x 3 – x As x  +, f(x)  ______ As x  -, f(x)  ______ Name one zero. ________

End Behavior 300 Main Describe the end behavior of f(x) = -x x 3 – x As x  +, f(x)  ______ As x  -, f(x)  ______ Name one zero. ________ x = 0

End Behavior 400 Main Get Answer Sketch the graph. How many turning points? What are the x-intercepts? f(x) = x 3 – 4x 2 + 4x

End Behavior 400 Main Sketch the graph. How many turning points? What are the x-intercepts? f(x) = x 3 – 4x 2 + 4x  (0, 0) and (2, 0) (factor and set factors to 0— What about multiplicity?)  Think about your ends.

End Behavior 500 Main Get Answer What is your favorite subject? a) Algebra 2 b) AlgebrA 2 c) Alg. 2 d) Math – specifically Algebra 2

End Behavior 500 Main Easy choice! Of course no other subject was even a contender! What is your favorite subject? a) Algebra 2 b) AlgebrA 2 c) Alg. 2 d) Math – specifically Algebra 2

Zeros / Graphs 100 Main Get Answer Write a polynomial function of least degree that: has real coefficients, the given zeros of: 1, -1, -7 and a leading coefficient of 1. Put the polynomial in standard form.

Zeros / Graphs 100 Main Write a polynomial function of least degree that: has real coefficients, the given zeros of: 1, -1, -7 and a leading coefficient of 1. Put the polynomial in standard form. x 3 + 7x 2 – x – 7

Zeros / Graphs 200 Main Get Answer Find all of the possible rational zeros of: f (x) = 3x 5 + 2x 4 – 7x 2 – 9x + 5

Zeros / Graphs 200 Main Find all of the possible rational zeros of: f (x) = 3x 5 + 2x 4 – 7x 2 – 9x + 5       

Zeros / Graphs 300 Main Get Answer What are all the rational zeros of f(x) = x 3 − 3x 2 − 40x + 84? (You must prove it with synthetic division.)

Zeros / Graphs 300 Main What are all the rational zeros of f(x) = x 3 − 3x 2 − 40x + 84? (You must prove it with synthetic division.)            

Zeros / Graphs 400 Main Get Answer Use the graph to the right to answer the following: End Behavior: As x  + , f(x)  ______________ As x  - , f(x)  ______________ # Turning Points: _________________________ Degree of polynomial: _________________ You must give me the coordinate (if any) in the following: Absolute Min: _______ Relative Min (Name one.): __________ Absolute Max: _______ Relative Max (Name one.): _________

Zeros / Graphs 400 Main Use the graph to the right to answer the following: End Behavior: As x  + , f(x)  ______________ As x  - , f(x)  ______________ # Turning Points: _________________________ Degree of polynomial: _________________ You must give me the coordinate (if any) in the following: Absolute Min: _______ Relative Min (Name one.): __________ Absolute Max: _______ Relative Max (Name one.): _________ -- ++ 4 5 none (-4,-5) or (-1,-2) (-2, 4) or (1,4)

Zeros / Graphs 500 Main Get Answer What are all of the zeros of: f(x) = 2x 3 – 11x 2 + 8x – 15

Zeros / Graphs 500 Main What are all of the zeros of: f(x) = 2x 3 – 11x 2 + 8x – 15 Graph to find that 5 is a zero. Synthetically divide out the 5.    2x 2 – x + 3 Use quadratic formula:

Polynomials 100 Main Get Answer At most, how many roots does the following polynomial have? f(x) = 5x 4 – 2x 3 + x 2 - 7

Polynomials 100 Main At most, how many roots does the following polynomial have? f(x) = 5x 4 – 2x 3 + x 

Polynomials 200 Main Get Answer Create a polynomial with the following characteristics: quartic, leading coefficient is 9, quadratic coefficient is 4, linear coefficient is –5 and the constant term is –2.

Polynomials 200 Main Create a polynomial with the following characteristics: quartic, leading coefficient is 9, quadratic coefficient is 4, linear coefficient is –5 and the constant term is –2. 9x 4 + 4x 2 – 5x – 2

Polynomials 300 Main Get Answer If -6/11 is a zero of a polynomial function, what is a factor?

Polynomials 300 Main If -6/11 is a zero of a polynomial function, what is a factor? (11x + 6)

Polynomials 400 MainGet Answer Find (-5x x – 1) – (6x 2 + 8x – 7)

Polynomials 400 Main Find (-5x x – 1) – (6x 2 + 8x – 7) -11x 2 + 3x + 6

Polynomials 500 MainGet Answer Factor 8x

Polynomials 500 Main Factor 8x (2x + 3)(4x 2 – 6x + 9)

Exponents 100 MainGet Answer (2y -5 )(4x 0 ) Simplify.

Exponents 100 Main (2y -5 )(4x 0 ) Simplify.

Exponents 200 Main Get Answer (-2x 3 y -3 ) 2 Simplify.

Exponents 200 Main (-2x 3 y -3 ) 2 Simplify.

Exponents 300 Main Get Answer Simplify.

Exponents 300 Main Simplify.

Exponents 400 Main Get Answer (4x -2 y) -2 Simplify.

Exponents 400 Main (4x -2 y) -2 Simplify.

Exponents 500 Main Get Answer Simplify.

Exponents 500 Main Simplify.