1 POLYNOMIAL FUNCTIONS PROBLEM 4 PROBLEM 1 Standard 3, 4, 6 and 25 PROBLEM 3 PROBLEM 2 PROBLEM 8 PROBLEM 5 PROBLEM 7 PROBLEM 6 PROBLEM 9 PROBLEM 10 END.

Slides:

Advertisements

Similar presentations

SLOPE a conceptual approach

Advertisements

The Rational Zero Theorem

Operations with Complex Numbers

Roots & Zeros of Polynomials

1 THE NUMBER LINE AND NUMBER SYSTEMS Standards 6, 25 PROPERTIES OF REAL NUMBERS EXAMPLES OF PROPERTIES OF REAL NUMBERS SIMPLIFYING EXPRESSIONS APPLYING.

Rational Root Theorem.

1 Exponent Rules and Monomials Standards 3 and 4 Simplifying Monomials: Problems POLYNOMIALS Monomials and Polynomials: Adding and Multiplying Multiplying.

1 Standards 2, 5, 25 SOLVING TWO VARIABLE LINEAR INEQUALITIES INCLUDING ABSOLUTE VALUE INEQUALITIES PROBLEM 1 PROBLEM 3 PROBLEM 4 PROBLEM 2 PROBLEM 5 PROBLEM.

1 Standards 2, 25 SOLVING EQUATIONS USING DETERMINANTS PROBLEM 1 PROBLEM 3 PROBLEM 4 PROBLEM 2 SOLVING SYSTEMS OF EQUATIONS IN THREE VARIABLES PROBLEM.

Unit 4 Roots and Zeros CCSS: A. APR.3

2.5 Zeros of Polynomial Functions

Pre-Calculus 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)

1 5.6 Complex Zeros; Fundamental Theorem of Algebra In this section, we will study the following topics: Conjugate Pairs Theorem Finding a polynomial function.

1 Definition of Monomials: Problems STANDARD 10 Definition of Polynomials: Problems POLYNOMIALS Standard Form of a Polynomial: Problems Adding Polynomials:

1 MULTIPLY WITH DISTRIBUTIVE PROPERTY: Problems 1 STANDARD 10 MULTIPLY WITH DISTRIBUTIVE PROPERTY: Problems 2 POLYNOMIALS MULTIPLY WITH DISTRIBUTIVE PROPERTY:

2.7 Apply the Fundamental Theorem of Algebra day 2

DEFINITION OF A PARABOLA

SLOPE INTERCEPT FORM AND POINT-SLOPE FORM

1 Simplify Problems With Imaginary Roots Standards 5 & 6 Equation That Results In Imaginary Roots COMPLEX NUMBERS Complex Numbers In Context END SHOW Imaginary.

1 Product of a Power: problems STANDARD 10 Power of a Product: problems POLYNOMIALS: PROPERTIES OF EXPONENTS PROBLEM 1 PROBLEM 2 PROBLEM 3 PROBLEM 4 PROBLEM.

COMPOSITION AND INVERSE

The Rational Zero Theorem

Lesson 2.5 The Fundamental Theorem of Algebra. For f(x) where n > 0, there is at least one zero in the complex number system Complex → real and imaginary.

Complex Numbers Lesson 3.3.

The Fundamental Theorem of Algebra And Zeros of Polynomials

Sullivan Algebra and Trigonometry: Section 5.6 Complex Zeros; Fundamental Theorem of Algebra Objectives Utilize the Conjugate Pairs Theorem to Find the.

5.7 Apply the Fundamental Theorem of Algebra

Algebra 2 Chapter 6 Notes Polynomials and Polynomial Functions Algebra 2 Chapter 6 Notes Polynomials and Polynomial Functions.

Academy Algebra II/Trig 5.5: The Real Zeros of a Polynomial Functions HW: p.387 (14, 27, 30, 31, 37, 38, 46, 51)

MATRICES AND SYSTEMS OF EQUATIONS

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.

Adding and Subtracting Polynomials Section 0.3. Polynomial A polynomial in x is an algebraic expression of the form: The degree of the polynomial is n.

Real Zeros of a Polynomial Function Objectives: Solve Polynomial Equations. Apply Descartes Rule Find a polynomial Equation given the zeros.

Chapter 3 Polynomial and Rational Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc Zeros of Polynomial Functions.

Solving Higher Order Polynomials Unit 6 Students will solve a variety of equations and inequalities including higher order polynomials.

5.5 Theorems about Roots of Polynomial Equations P

 Evaluate a polynomial  Direct Substitution  Synthetic Substitution  Polynomial Division  Long Division  Synthetic Division  Remainder Theorem 

Lesson 2.3 Real Zeros of Polynomials. The Division Algorithm.

Do Now: Find all real zeros of the function.

Zeros of Polynomials 2.5.

Copyright © 2009 Pearson Education, Inc. CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions.

Topic: U4L5 Remainder and Factor Theorems EQ: Can I correctly apply the Remainder and Factor Theorems to help me factor higher order polynomials?

2.5 The Fundamental Theorem of Algebra Students will use the fundamental theorem of algebra to determine the number of zeros of a polynomial. Students.

Warm-Up Exercises 1. What is the degree of f (x) = 8x 6 – 4x 5 + 3x ? 2. Solve x 2 – 2x + 3 = 0 ANSWER 6 1 i 2 + _.

The Real Zeros of a Polynomial Function Obj: Apply Factor Theorem, Use Rational Zero Theorem to list roots, Apply Descartes’ Rule of Signs to determine.

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

THE FUNDAMENTAL THEOREM OF ALGEBRA. Descartes’ Rule of Signs If f(x) is a polynomial function with real coefficients, then *The number of positive real.

The Original f(x)=x 3 -9x 2 +6x+16 State the leading coefficient and the last coefficient Record all factors of both coefficients According to the Fundamental.

Lesson 76 – Introduction to Complex Numbers HL2 MATH - SANTOWSKI.

2.3 Real and Non Real Roots of a Polynomial Polynomial Identities Secondary Math 3.

7.6 Rational Zero Theorem Objectives: 1. Identify the possible rational zeros of a polynomial function. 2. Find all the rational zeros of a polynomial.

POLYNOMIALS 9/20/2015. STANDARD FORM OF A POLYNOMIAL ax + bx + cx … mx + k nn-1n-2 The degree is ‘n’

7.5 Roots and Zeros Objectives: The student will be able to…

Remainder and Factor Theorems

1 QUADRATIC FUNCTIONS PROBLEM 1 PROBLEM 2 PROBLEM 3 INTRODUCTION Standard 4, 9, 17 PROBLEM 4 PROBLEM 5 END SHOW PRESENTATION CREATED BY SIMON PEREZ. All.

Precalculus Lesson 2.5 The Fundamental Theorem of Algebra.

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.

DEFINITION OF A CIRCLE and example

SOLVING EQUATIONS USING DETERMINANTS

Finding Zeros of Polynomials

Copyright © 2014, 2010, 2007 Pearson Education, Inc.

Chapter 7.5 Roots and Zeros Standard & Honors

6-8 Roots and Zeros Given a polynomial function f(x), the following are all equivalent: c is a zero of the polynomial function f(x). x – c is a factor.

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

Presentation transcript:

1 POLYNOMIAL FUNCTIONS PROBLEM 4 PROBLEM 1 Standard 3, 4, 6 and 25 PROBLEM 3 PROBLEM 2 PROBLEM 8 PROBLEM 5 PROBLEM 7 PROBLEM 6 PROBLEM 9 PROBLEM 10 END SHOW PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

2 STANDARD 3: Students are adept at operations on polynomials, including long division. STANDARD 4: Students factor polynomials representing the difference of squares, perfect square trinomials, and the sum and difference of two cubes. STANDARD 6: Students add, subtract, multiply, and divide complex numbers STANDARD 25: Students use properties from number systems to justify steps in combining and simplifying functions. ALGEBRA II STANDARDS THIS LESSON AIMS: PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

3 ESTÁNDAR 6: Los estudiantes suman, restan, multiplican, y dividen numeros complejos. ESTÁNDAR 3: Los estudiantes hacen operaciones con polinomios incluyendo divisi[on larga. ESTÁNDAR 4: Los estudiantes factorizan diferencias de cuadrados, trinomios cuadrados perfectos, y la suma y diferencia de dos cubos. ESTÁNDAR 25: Los estudiantes usan propiedades de los sistemas numéricos para combinar y simplificar funciones. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

4 Standard 25 Evaluate: h(2)= x + 4x -2x h(2) = ( ) + 4( ) – 2( ) = 8 + 4(4) – = – = 23 Evaluate using synthetic division: h(2) = 23 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

5 Standard 25 Evaluate: h(3)= x + 3x -6x h(3) = ( ) + 3( ) – 6( ) = (9) – = – = 37 Evaluate using synthetic division: h(3) = 37 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

6 Standards 4, 25 Using synthetic division, The Remainder Theorem and the factor theorem: Two numbers that multiplied be negative twelve = (+)(-) or (-)(+) Two numbers that added be positive 4=|(+)| >| (-)| (-1)(12) -1+12=11 (-2)(6) -2+6=4 Given the following polynomial and one of its factor find the other two: h(y)= y + 2y -20y ; (y-2) y +4y (y-2)(y+6) PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

7 Standards 4, 25 Using synthetic division, The Remainder Theorem and the factor theorem: Two numbers that multiplied be negative twelve = (+)(-) or (-)(+) Two numbers that added be positive 4=|(+)| >| (-)| (-1)(5) -1+5=4 Find all the zeros of the following function if one zero is 3 h(y)= y + y -17y y + 4y -5= y – 1=0 y + 5 =0 +1 y = 1 -5 y = -5 (1,0) (-5,0) zeros: PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

Standards 3, 4, 6, 25 Find all zeros of if 2+ i is one zero of f(x) If we have a factor x – (2+ i ) then we have the conjugate as well according to the complex conjugate theorem x – (2- i ) f(x) = x – (2- i ) x – (2+ i ) because a third degree polynomial has 3 roots. (2+ i ) (2- i ) x x-x (2+ i ) + = = x -2x -x i -2x +x i i 2 2 = x – 4x + 4 –(-1) 2 F O I L = x - 4x x - x -7x x x 3 - 4x 2 + 5x - 3x 2 -12x x 2 -12x The three zeros are: +(2- i ) x – (2- i ) =0 x= 2- i +(2+ i ) x – (2+ i ) =0 x= 2+ i x+3=0 -3 x= -3 i 2 = -1 recall: Using long division to find the other factor: x+3 f(x) =x - x -7x PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

Standards 3, 4, 6, 25 Find all zeros of if 5+ i is one zero of f(x) If we have a factor x – (5+ i ) then we have the conjugate as well according to the complex conjugate theorem x – (5- i ) f(x) = x – (5- i ) x – (5+ i ) because a third degree polynomial has 3 roots. (5+ i ) (5- i ) x x-x (5+ i ) + = = x -5x -x i -5x +x i i 2 2 = x –10x +25 –(-1) 2 = x- 10x x -12x +46x x x 3 -10x 2 +26x - -2x 2 +20x x 2 +20x The three zeros are: +(5- i ) x – (5- i ) =0 x= 5- i +(5+ i ) x – (5+ i ) =0 x= 5+ i x-2=0 +2 x= 2 i 2 = -1 recall: Using long division to find the other factor: x-2 f(x) = x - 12x + 46x F O I L PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

10 Standard 25 r(x) = 5x + 4x -3x +6x -7x Using Descartes’ rule of signs, we count the number of changes in sign for the coefficients of r(x) r(x) = 5x + 4x -3x +6x -7x no yes There are four changes of sign, thus the number of positive real zeros is 4 or 2 or 0 r( ) = 5( ) + 4( ) -3( ) +6( ) -7( ) x r(-x) = -5x + 4x +3x +6x +7x Now finding r(-x) and counting the number of changes in signs for the coefficients: yes no There is only one change, thus the number of negative real zeros is only 1. State the number of positive and negative real zeros in PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

11 Standard 25 r(x) = 5x + 4x -3x +6x -7x The number of positive real zeros is 4 or 2 or 0 The number of negative real zeros is only 1. We concluded that: State the number of positive and negative real zeros in Number of Positive Real Zeros Number of Negative Real Zeros Number of Imaginary Zeros = = =5 r(x) x With the graph we verify that we have only 1 negative zero and 4 imaginary zeros. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

12 Standard 25 r(x) = 8x - 3x +2x -8x -2x Using Descartes’ rule of signs, we count the number of changes in sign for the coefficients of r(x) yes no yes There are four changes of sign, thus the number of positive real zeros is 4 or 2 or 0 r( ) = 8( ) - 3( ) +2( ) - 8( ) -2( ) x r(-x) = -8x - 3x -2x - 8x +2x Now finding r(-x) and counting the number of changes in signs for the coefficients: no yes no There is only one change, thus the number of negative real zeros is only 1. State the number of positive and negative real zeros in r(x) = 8x - 3x +2x -8x -2x PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

13 Standard 25 The number of positive real zeros is 4 or 2 or 0 The number of negative real zeros is only 1. We concluded that: State the number of positive and negative real zeros in Number of Positive Real Zeros Number of Negative Real Zeros Number of Imaginary Zeros = = =5 r(x) x With the graph we verify that we have only 1 negative zero and 2 positive real zeros and 2 imaginary zeros. r(x) = 8x - 3x +2x -8x -2x PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

14 Standard 3, 25 Write the polynomial function of least degree with integral coefficients whose zeros include 3, 2, -1. x= 3 This implies that we have zeros at the following values of x: -3 x-3=0 x – 3 x – 2 x+1 =0 xx (-3) x x (-2) + = x+1 F O I L = x- 5x x+1 f(x) x- 5x x+1x+1 X +6-5x +6x -5x 2 x 2 x x x 3 -4x 2 = x x 3 -4x 2 x= 2 -2 x-2=0 x= x+1=0 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

15 Standard 3, 25 Write the polynomial function of least degree with integral coefficients whose zeros include 5, 2-2 i x= 5 x = 2-2 i x = 2+2 i This implies that we have zeros at the following values of x: -5 x-5=0 -(2-2 i ) x -(2-2 i ) =0 -(2+2 i ) x -(2+2 i ) =0 x – (2-2 i ) x – (2+2 i ) x-5 =0 (2+2 i ) (2-2 i ) x x-x (2+2 i ) + = x-5 = x -2x -2x i -2x +2x i i 2 2 x –4x +4 –4(-1) 2 = x-5 F O I L = x- 4x x-5 f(x) x- 4x x-5x-5 X x +8x -4x 2 -5x 2 x x x 3 -9x 2 = x x 3 -9x 2 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved