Polynomials. Jackie Nicholas Jacquie Hargreaves Janet Hunter

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Polynomials. Jackie Nicholas Jacquie Hargreaves Janet Hunter"

Transcription

1 Mathematics Learning Centre Polnomials Jackie Nicholas Jacquie Hargreaves Janet Hunter c 26 Universit of Sdne

2 Mathematics Learning Centre, Universit of Sdne 1 1 Polnomials Man of the functions we will eamine will be polnomials. In this Chapter we will stud them in more detail. Definition A real polnomial, P (), of degree n is an epression of the form P () =p n n + p n 1 n 1 + p n 2 n p p 1 + p where p n,p, p 1,, p n are real and n is an integer. All polnomials are defined for all real and are continuous functions. We are familiar with the quadratic polnomial, Q() =a 2 + b + c where a. This polnomial has degree 2. The function f() = + is not a polnomial as it has a power which is not an integer and so does not satisf the definition. 1.1 Polnomial equations and their roots If, for a polnomial P (), P (k) = then we can sa 1. = k is a root of the equation P () =. 2. = k is a zero of P (). 3. k is an -intercept of the graph of P () Zeros of the quadratic polnomial The quadratic polnomial equation Q() =a 2 + b + c = has two roots that ma be: 1. real (rational or irrational) and distinct, 2. real (rational or irrational) and equal, 3. comple (not real). We can determine which one of these we have if we use the quadratic formula = b ± b 2 4ac. 2a If b 2 4ac >, there will be two real distinct roots. If b 2 4ac =, there will be two real equal roots. If b 2 4ac <, there will be comple roots. We will illustrate all of these cases with eamples, and will show the relationship between the nature and number of zeros of Q() and the -intercepts (if an) on the graph.

3 Mathematics Learning Centre, Universit of Sdne 2 1. Let Q() = We find the zeros of Q() b solving the equation Q() = = ( 1)( 3) = Therefore = 1 or 3. The roots are rational (hence real) and distinct Let Q() = Solving the equation Q() = we get, = = 4 ± Therefore = 2± 7. 2 The roots are irrational (hence real) and distinct Let Q() = Solving the equation Q() = we get, = ( 2) 2 = Therefore = 2. 3 The roots are rational (hence real) and equal. Q() = has a repeated or double root at =2. 1 Notice that the graph turns at the double root =2.

4 Mathematics Learning Centre, Universit of Sdne 3 4. Let Q() = Solving the equation Q() = we get, = = 4 ± Therefore = 2± 4. There are no real roots. In this case the roots are comple. 3 1 Notice that the graph does not intersect the -ais. That is Q() > for all real. Therefore Q is positive definite. We have given above four eamples of quadratic polnomials to illustrate the relationship between the zeros of the polnomials and their graphs. In particular we saw that: i. if the quadratic polnomial has two real distinct zeros, then the graph of the polnomial cuts the -ais at two distinct points; ii. if the quadratic polnomial has a real double (or repeated) zero, then the graph sits on the -ais; iii. if the quadratic polnomial has no real zeros, then the graph does not intersect the -ais at all. So far, we have onl considered quadratic polnomials where the coefficient of the 2 term is positive which gives us a graph which is concave up. If we consider polnomials Q() =a 2 + b + c where a< then we will have a graph which is concave down. For eample, the graph of Q() = ( ) is the reflection in the -ais of the graph of Q() = The graph of Q() = The graph of Q() = ( ).

5 Mathematics Learning Centre, Universit of Sdne Zeros of cubic polnomials A real cubic polnomial has an equation of the form P () =a 3 + b 2 + c + d where a,a, b, c and d are real. It has 3 zeros which ma be: i. 3 real distinct zeros; ii. 3 real zeros, all of which are equal (3 equal zeros); iii. 3 real zeros, 2 of which are equal; iv. 1 real zero and 2 comple zeros. We will not discuss these here. 1.2 Factorising polnomials So far for the most part, we have looked at polnomials which were alread factorised. In this section we will look at methods which will help us factorise polnomials with degree > Dividing polnomials Suppose we have two polnomials P () and A(), with the degree of P () the degree of A(), and P () is divided b A(). Then P () A() = Q()+R() A(), where Q() is a polnomial called the quotient and R() is a polnomial called the remainder, with the degree of R() < degree of A(). We can rewrite this as P () =A() Q()+R(). For eample: If P () = and A() = 2, then P () can be divided b A() as follows: The quotient is and the remainder is 21. We have = This can be written as =( 2)( )+21. Note that the degree of the polnomial 21 is.

6 Mathematics Learning Centre, Universit of Sdne The Remainder Theorem If the polnomial f() is divided b ( a) then the remainder is f(a). Proof: Following the above, we can write f() =A() Q()+R(), where A() =( a). Since the degree of A() is 1, the degree of R() is zero. That is, R() =r where r is a constant. f() = ( a)q()+r where r is a constant. f(a) = Q(a)+r = r So, if f() is divided b ( a) then the remainder is f(a). Eample Find the remainder when P () = is divided b a. +1,b Solution a. Using the Remainder Theorem: Remainder = P ( 1) = 3 ( 1) 3 1 = 27 b. Remainder = P ( 1 2 ) = 3( 1 2 )4 ( 1 2 )3 + 3( 1 2 ) 1 = = Eample When the polnomial f() is divided b 2 4, the remainder is What is the remainder when f() is divided b ( 2)? Solution Write f() =( 2 4) q()+(5 + 6). Then Remainder = f(2) = q(2)+16 = 16 A consequence of the Remainder Theorem is the Factor Theorem which we state below.

7 Mathematics Learning Centre, Universit of Sdne The Factor Theorem If = a is a zero of f(), that is f(a) =, then ( a) is a factor of f() and f() ma be written as f() =( a)q() for some polnomial q(). Also, if ( a) and ( b) are factors of f() then ( a)( b) is a factor of f() and for some polnomial Q(). f() =( a)( b) Q() Another useful fact about zeros of polnomials is given below for a polnomial of degree The cubic polnomial revisited If a (real) polnomial P () =a 3 + b 2 + c + d, where a,a, b, c and d are real, has eactl 3 real zeros α, β and γ, then P () =a( α)( β)( γ) (1) Furthermore, b epanding the right hand side of (1) and equating coefficients we get: i. ii. iii. α + β + γ = b a ; αβ + αγ + βγ = c a ; αβγ = d a. This result can be etended for polnomials of degree n. Eample Let f() = a. Factorise f(). b. Sketch the graph of = f(). c. Solve f().

8 Mathematics Learning Centre, Universit of Sdne 7 Solution a. Consider the factors of the constant term, 2. We check to see if ±1 and ±2 are solutions of the equation f() = b substitution. Since f(2) =, we know that ( 2) is a factor of f(). We use long division to determine the quotient So, f() = ( 2)(4 2 1) = ( 2)(2 1)(2 +1) b The graph of f() = c. f() when or 2. Eample Show that ( 2) and ( 3) are factors of P () = , and hence solve =. Solution P (2)= = and P (3) = = so ( 2) and ( 3) are both factors of P () and ( 2)( 3) = is also a factor of P (). Long division of P () b gives a quotient of ( + 5). So, P () = =( 2)( 3)( +5).

9 Mathematics Learning Centre, Universit of Sdne 8 Solving P () = we get ( 2)( 3)( +5)=. That is, =2or =3or = 5. Instead of using long division we could have used the facts that i. the polnomial cannot have more than three real zeros; ii. the product of the zeros must be equal to 3. Let α be the unknown root. Then 2 3 α = 3, so that α = 5. Therefore the solution of P () = = is =2or =3or = Eercises 1. When the polnomial P () is divided b ( a)( b) the quotient is Q() and the remainder is R(). a. Eplain wh R() is of the form m + c where m and c are constants. b. When a polnomial is divided b ( 2) and ( 3), the remainders are 4 and 9 respectivel. Find the remainder when the polnomial is divided b c. When P () is divided b ( a) the remainder is a 2. Also, P (b) =b 2. Find R() when P () is divided b ( a)( b). 2. a. Divide the polnomial f() = bg() = Hence write f() = g()q() + r() where q() and r() are polnomials. b. Show that f() and g() have no common zeros. (Hint: Assume that α is a common zero and show b contradiction that α does not eist.) 3. For the following polnomials, i. factorise ii. solve P () =. a. P () = b. P () = c. P () = d. P () = e. P () = Solutions 1. a. Since A() =( a)( b) is a polnomial of degree 2, the remainder R() must be a polnomial of degree < 2. So, R() is a polnomial of degree 1. That is, R() =m + c where m and c are constants. Note that if m = the remainder is a constant.

10 Mathematics Learning Centre, Universit of Sdne 9 b. Let P () =( )Q()+(m + c) =( 2)( 3)Q()+(m + c). Then and P (2) = ()( 1)Q(2) + (2m + c) = 2m + c = 4 P (3) = (1)()Q(3) + (3m + c) = 3m + c = 9 Solving simultaneousl we get that m = 5 and c = 6. R() =5 6. c. Let P () =( a)( b)q()+(m + c). Then So, the remainder is P (a) = ()(a b)q(a)+(ma + c) = am + c = a 2 and P (b) = (b a)()q(b)+(mb + c) = bm + c = b 2 Solving simultaneousl we get that m = a + b and c = ab provided a b. So, R() =(a + b) ab. 2. a =( )( ) 2 b. Let α be a common zero of f() and g(). That is, f(α) =andg(α) =. Then since f() = g()q()+ r() we have f(α) = g(α)q(α)+r(α) = ()q(α)+ r(α) since g(α) = = r(α) = since f(α) = But, from part b. r() = 2 for all values of, so we have a contradiction. Therefore, f() and g() do not have a common zero. This is an eample of a proof b contradiction. 3. a. i. P () = =( + 1)( + 2)( 4)

11 Mathematics Learning Centre, Universit of Sdne 1 ii. = 1, = 2 and = 4 are solutions of P () =. b. i. P () = = ( +2) 2 ( 5). ii. = 2 and = 5 are solutions of P () =. = 2 is a double root. c. i. P () = =(+2)( ) = (+2)( ( 1+ 5))( ( 1 5)) ii. = 2, = 1+ 5 and = 1 5 are solutions of P () =. d. i. P () = =( 2)( ) = has no real solutions. ii. = 2 is the onl real solution of P () =. e. i. P () = =( + 2)( 3)(2 1). ii. = 2, = 1 and = 3 are solutions of P () =. 2

Solving inequalities. Jackie Nicholas Jacquie Hargreaves Janet Hunter

Solving inequalities. Jackie Nicholas Jacquie Hargreaves Janet Hunter Mathematics Learning Centre Solving inequalities Jackie Nicholas Jacquie Hargreaves Janet Hunter c 6 Universit of Sdne Mathematics Learning Centre, Universit of Sdne Solving inequalities In these nots

More information

Higher. Polynomials and Quadratics 64

Higher. Polynomials and Quadratics 64 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

More information

Zero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.

Zero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P. MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called

More information

The Quadratic Function

The Quadratic Function 0 The Quadratic Function TERMINOLOGY Ais of smmetr: A line about which two parts of a graph are smmetrical. One half of the graph is a reflection of the other Coefficient: A constant multiplied b a pronumeral

More information

SAMPLE. Polynomial functions

SAMPLE. Polynomial functions Objectives C H A P T E R 4 Polnomial functions To be able to use the technique of equating coefficients. To introduce the functions of the form f () = a( + h) n + k and to sketch graphs of this form through

More information

Core Maths C2. Revision Notes

Core Maths C2. Revision Notes Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...

More information

Methods to Solve Quadratic Equations

Methods to Solve Quadratic Equations Methods to Solve Quadratic Equations We have been learning how to factor epressions. Now we will apply factoring to another skill you must learn solving quadratic equations. a b c 0 is a second-degree

More information

2.2 Absolute Value Functions

2.2 Absolute Value Functions . Absolute Value Functions 7. Absolute Value Functions There are a few was to describe what is meant b the absolute value of a real number. You ma have been taught that is the distance from the real number

More information

Zeros of Polynomial Functions. The Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra. zero in the complex number system.

Zeros of Polynomial Functions. The Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra. zero in the complex number system. _.qd /7/ 9:6 AM Page 69 Section. Zeros of Polnomial Functions 69. Zeros of Polnomial Functions What ou should learn Use the Fundamental Theorem of Algebra to determine the number of zeros of polnomial

More information

x 2 k S. S. k, k x 2 bx b 2 x b b2 4ac 2a b 2 4ac

x 2 k S. S. k, k x 2 bx b 2 x b b2 4ac 2a b 2 4ac Solving Quadratic Equations a b c 0, a 0 Methods for solving: 1. B factoring. A. First, put the equation in standard form. B. Then factor the left side C. Set each factor 0 D. Solve each equation. B square

More information

Polynomial and Rational Functions

Polynomial and Rational Functions Chapter Section.1 Quadratic Functions Polnomial and Rational Functions Objective: In this lesson ou learned how to sketch and analze graphs of quadratic functions. Course Number Instructor Date Important

More information

POLYNOMIAL FUNCTIONS

POLYNOMIAL FUNCTIONS POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a

More information

An Insight into Division Algorithm, Remainder and Factor Theorem

An Insight into Division Algorithm, Remainder and Factor Theorem An Insight into Division Algorithm, Remainder and Factor Theorem Division Algorithm Recall division of a positive integer by another positive integer For eample, 78 7, we get and remainder We confine the

More information

If (a)(b) 5 0, then a 5 0 or b 5 0.

If (a)(b) 5 0, then a 5 0 or b 5 0. chapter Algebra Ke words substitution discriminant completing the square real and distinct imaginar rational verte parabola maimum minimum surd irrational rationalising the denominator Section. Quadratic

More information

Simplification of Rational Expressions and Functions

Simplification of Rational Expressions and Functions 7.1 Simplification of Rational Epressions and Functions 7.1 OBJECTIVES 1. Simplif a rational epression 2. Identif a rational function 3. Simplif a rational function 4. Graph a rational function Our work

More information

2.2 Absolute Value Functions

2.2 Absolute Value Functions . Absolute Value Functions 7. Absolute Value Functions There are a few was to describe what is meant b the absolute value of a real number. You ma have been taught that is the distance from the real number

More information

Polynomials Past Papers Unit 2 Outcome 1

Polynomials Past Papers Unit 2 Outcome 1 PSf Polnomials Past Papers Unit 2 utcome 1 Multiple Choice Questions Each correct answer in this section is worth two marks. 1. Given p() = 2 + 6, which of the following are true? I. ( + 3) is a factor

More information

y intercept Gradient Facts Lines that have the same gradient are PARALLEL

y intercept Gradient Facts Lines that have the same gradient are PARALLEL CORE Summar Notes Linear Graphs and Equations = m + c gradient = increase in increase in intercept Gradient Facts Lines that have the same gradient are PARALLEL If lines are PERPENDICULAR then m m = or

More information

8.7 Systems of Non-Linear Equations and Inequalities

8.7 Systems of Non-Linear Equations and Inequalities 8.7 Sstems of Non-Linear Equations and Inequalities 67 8.7 Sstems of Non-Linear Equations and Inequalities In this section, we stud sstems of non-linear equations and inequalities. Unlike the sstems of

More information

STRAND F: ALGEBRA. UNIT F4 Solving Quadratic Equations: Text * * * Contents. Section. F4.1 Factorisation. F4.2 Using the Formula

STRAND F: ALGEBRA. UNIT F4 Solving Quadratic Equations: Text * * * Contents. Section. F4.1 Factorisation. F4.2 Using the Formula UNIT F4 Solving Quadratic Equations: Tet STRAND F: ALGEBRA Unit F4 Solving Quadratic Equations Tet Contents * * * Section F4. Factorisation F4. Using the Formula F4. Completing the Square UNIT F4 Solving

More information

Quadratic Functions. MathsStart. Topic 3

Quadratic Functions. MathsStart. Topic 3 MathsStart (NOTE Feb 2013: This is the old version of MathsStart. New books will be created during 2013 and 2014) Topic 3 Quadratic Functions 8 = 3 2 6 8 ( 2)( 4) ( 3) 2 1 2 4 0 (3, 1) MATHS LEARNING CENTRE

More information

Polynomial and Synthetic Division. Long Division of Polynomials. Example 1. 6x 2 7x 2 x 2) 19x 2 16x 4 6x3 12x 2 7x 2 16x 7x 2 14x. 2x 4.

Polynomial and Synthetic Division. Long Division of Polynomials. Example 1. 6x 2 7x 2 x 2) 19x 2 16x 4 6x3 12x 2 7x 2 16x 7x 2 14x. 2x 4. _.qd /7/5 9: AM Page 5 Section.. Polynomial and Synthetic Division 5 Polynomial and Synthetic Division What you should learn Use long division to divide polynomials by other polynomials. Use synthetic

More information

Maclaurin and Taylor Series

Maclaurin and Taylor Series Maclaurin and Talor Series 6.5 Introduction In this block we eamine how functions ma be epressed in terms of power series. This is an etremel useful wa of epressing a function since (as we shall see) we

More information

Zeros of Polynomial Functions

Zeros of Polynomial Functions Review: Synthetic Division Find (x 2-5x - 5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 3-5x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 3-5x 2 + x + 2. Zeros of Polynomial Functions Introduction

More information

JUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson

JUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials

More information

Graphing Quadratic Functions

Graphing Quadratic Functions A. THE STANDARD PARABOLA Graphing Quadratic Functions The graph of a quadratic function is called a parabola. The most basic graph is of the function =, as shown in Figure, and it is to this graph which

More information

13 Graphs, Equations and Inequalities

13 Graphs, Equations and Inequalities 13 Graphs, Equations and Inequalities 13.1 Linear Inequalities In this section we look at how to solve linear inequalities and illustrate their solutions using a number line. When using a number line,

More information

Chapter 4. Polynomial and Rational Functions. 4.1 Polynomial Functions and Their Graphs

Chapter 4. Polynomial and Rational Functions. 4.1 Polynomial Functions and Their Graphs Chapter 4. Polynomial and Rational Functions 4.1 Polynomial Functions and Their Graphs A polynomial function of degree n is a function of the form P = a n n + a n 1 n 1 + + a 2 2 + a 1 + a 0 Where a s

More information

PROPERTIES OF ELLIPTIC CURVES AND THEIR USE IN FACTORING LARGE NUMBERS

PROPERTIES OF ELLIPTIC CURVES AND THEIR USE IN FACTORING LARGE NUMBERS PROPERTIES OF ELLIPTIC CURVES AND THEIR USE IN FACTORING LARGE NUMBERS A ver important set of curves which has received considerabl attention in recent ears in connection with the factoring of large numbers

More information

3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes

3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general

More information

C1: Coordinate geometry of straight lines

C1: Coordinate geometry of straight lines B_Chap0_08-05.qd 5/6/04 0:4 am Page 8 CHAPTER C: Coordinate geometr of straight lines Learning objectives After studing this chapter, ou should be able to: use the language of coordinate geometr find the

More information

2.6. The Circle. Introduction. Prerequisites. Learning Outcomes

2.6. The Circle. Introduction. Prerequisites. Learning Outcomes The Circle 2.6 Introduction A circle is one of the most familiar geometrical figures and has been around a long time! In this brief Section we discuss the basic coordinate geometr of a circle - in particular

More information

Review of Essential Skills and Knowledge

Review of Essential Skills and Knowledge Review of Essential Skills and Knowledge R Eponent Laws...50 R Epanding and Simplifing Polnomial Epressions...5 R 3 Factoring Polnomial Epressions...5 R Working with Rational Epressions...55 R 5 Slope

More information

LESSON EIII.E EXPONENTS AND LOGARITHMS

LESSON EIII.E EXPONENTS AND LOGARITHMS LESSON EIII.E EXPONENTS AND LOGARITHMS LESSON EIII.E EXPONENTS AND LOGARITHMS OVERVIEW Here s what ou ll learn in this lesson: Eponential Functions a. Graphing eponential functions b. Applications of eponential

More information

SECTION 5-1 Exponential Functions

SECTION 5-1 Exponential Functions 354 5 Eponential and Logarithmic Functions Most of the functions we have considered so far have been polnomial and rational functions, with a few others involving roots or powers of polnomial or rational

More information

Zero and Negative Exponents and Scientific Notation. a a n a m n. Now, suppose that we allow m to equal n. We then have. a am m a 0 (1) a m

Zero and Negative Exponents and Scientific Notation. a a n a m n. Now, suppose that we allow m to equal n. We then have. a am m a 0 (1) a m 0. E a m p l e 666SECTION 0. OBJECTIVES. Define the zero eponent. Simplif epressions with negative eponents. Write a number in scientific notation. Solve an application of scientific notation We must have

More information

When I was 3.1 POLYNOMIAL FUNCTIONS

When I was 3.1 POLYNOMIAL FUNCTIONS 146 Chapter 3 Polnomial and Rational Functions Section 3.1 begins with basic definitions and graphical concepts and gives an overview of ke properties of polnomial functions. In Sections 3.2 and 3.3 we

More information

Chapter 2 Polynomial and Rational Functions

Chapter 2 Polynomial and Rational Functions Chapter 2 Polnomial and Rational Functions Course Number Section 2.1 Quadratic Functions and Models Instructor In this lesson ou learned how to sketch and analze graphs of functions. Date Define each term

More information

Exponential and Logarithmic Functions

Exponential and Logarithmic Functions Chapter 6 Eponential and Logarithmic Functions Section summaries Section 6.1 Composite Functions Some functions are constructed in several steps, where each of the individual steps is a function. For eample,

More information

For each learner you will need: mini-whiteboard. For each small group of learners you will need: Card set A Factors; Card set B True/false.

For each learner you will need: mini-whiteboard. For each small group of learners you will need: Card set A Factors; Card set B True/false. Level A11 of challenge: D A11 Mathematical goals Starting points Materials required Time needed Factorising cubics To enable learners to: associate x-intercepts with finding values of x such that f (x)

More information

Section 1.4: Operations on Functions

Section 1.4: Operations on Functions SECTION 14 Operations on Functions Section 14: Operations on Functions Combining Functions by Addition, Subtraction, Multiplication, Division, and Composition Combining Functions by Addition, Subtraction,

More information

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

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

More information

Graphing Nonlinear Systems

Graphing Nonlinear Systems 10.4 Graphing Nonlinear Sstems 10.4 OBJECTIVES 1. Graph a sstem of nonlinear equations 2. Find ordered pairs associated with the solution set of a nonlinear sstem 3. Graph a sstem of nonlinear inequalities

More information

CHAPTER 5 EXPONENTS AND POLYNOMIALS

CHAPTER 5 EXPONENTS AND POLYNOMIALS Chapter Five Additional Eercises 1 CHAPTER EXPONENTS AND POLYNOMIALS Section.1 The Product Rule and Power Rules for Eponents Objective 1 Use eponents. Write the epression in eponential form and evaluate.

More information

Lesson 9.1 Solving Quadratic Equations

Lesson 9.1 Solving Quadratic Equations Lesson 9.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with a. One -intercept and all nonnegative y-values. b. The verte in the third quadrant and no -intercepts. c. The verte

More information

The Graph of a Linear Equation

The Graph of a Linear Equation 4.1 The Graph of a Linear Equation 4.1 OBJECTIVES 1. Find three ordered pairs for an equation in two variables 2. Graph a line from three points 3. Graph a line b the intercept method 4. Graph a line that

More information

Section 0.2 Set notation and solving inequalities

Section 0.2 Set notation and solving inequalities Section 0.2 Set notation and solving inequalities (5/31/07) Overview: Inequalities are almost as important as equations in calculus. Man functions domains are intervals, which are defined b inequalities.

More information

More Equations and Inequalities

More Equations and Inequalities Section. Sets of Numbers and Interval Notation 9 More Equations and Inequalities 9 9. Compound Inequalities 9. Polnomial and Rational Inequalities 9. Absolute Value Equations 9. Absolute Value Inequalities

More information

MATH 4552 Cubic equations and Cardano s formulae

MATH 4552 Cubic equations and Cardano s formulae MATH 455 Cubic equations and Cardano s formulae Consider a cubic equation with the unknown z and fixed complex coefficients a, b, c, d (where a 0): (1) az 3 + bz + cz + d = 0. To solve (1), it is convenient

More information

Functions and Graphs

Functions and Graphs PSf Functions and Graphs Paper 1 Section B 1. The points A and B have coordinates (a, a 2 ) and (2b, 4b 2 ) respectivel. Determine the gradient of AB in its simplest form. 2 2. hsn.uk.net Page 1 Questions

More information

CALCULUS 1: LIMITS, AVERAGE GRADIENT AND FIRST PRINCIPLES DERIVATIVES

CALCULUS 1: LIMITS, AVERAGE GRADIENT AND FIRST PRINCIPLES DERIVATIVES 6 LESSON CALCULUS 1: LIMITS, AVERAGE GRADIENT AND FIRST PRINCIPLES DERIVATIVES Learning Outcome : Functions and Algebra Assessment Standard 1..7 (a) In this section: The limit concept and solving for limits

More information

2.1 Equations of Lines

2.1 Equations of Lines Section 2.1 Equations of Lines 1 2.1 Equations of Lines The Slope-Intercept Form Recall the formula for the slope of a line. Let s assume that the dependent variable is and the independent variable is

More information

Q (x 1, y 1 ) m = y 1 y 0

Q (x 1, y 1 ) m = y 1 y 0 . Linear Functions We now begin the stud of families of functions. Our first famil, linear functions, are old friends as we shall soon see. Recall from Geometr that two distinct points in the plane determine

More information

2-5 Rational Functions

2-5 Rational Functions -5 Rational Functions Find the domain of each function and the equations of the vertical or horizontal asymptotes, if any 1 f () = The function is undefined at the real zeros of the denominator b() = 4

More information

Imagine a cube with any side length. Imagine increasing the height by 2 cm, the. Imagine a cube. x x

Imagine a cube with any side length. Imagine increasing the height by 2 cm, the. Imagine a cube. x x OBJECTIVES Eplore functions defined b rddegree polnomials (cubic functions) Use graphs of polnomial equations to find the roots and write the equations in factored form Relate the graphs of polnomial equations

More information

C3: Functions. Learning objectives

C3: Functions. Learning objectives CHAPTER C3: Functions Learning objectives After studing this chapter ou should: be familiar with the terms one-one and man-one mappings understand the terms domain and range for a mapping understand the

More information

Section 3-3 Approximating Real Zeros of Polynomials

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

More information

Quadratic Functions and Parabolas

Quadratic Functions and Parabolas MATH 11 Quadratic Functions and Parabolas A quadratic function has the form Dr. Neal, Fall 2008 f () = a 2 + b + c where a 0. The graph of the function is a parabola that opens upward if a > 0, and opens

More information

Assessment Schedule 2013

Assessment Schedule 2013 NCEA Level Mathematics (9161) 013 page 1 of 5 Assessment Schedule 013 Mathematics with Statistics: Apply algebraic methods in solving problems (9161) Evidence Statement ONE Expected Coverage Merit Excellence

More information

3.6. The factor theorem

3.6. The factor theorem 3.6. The factor theorem Example 1. At the right we have drawn the graph of the polynomial y = x 4 9x 3 + 8x 36x + 16. Your problem is to write the polynomial in factored form. Does the geometry of the

More information

2.6. The Circle. Introduction. Prerequisites. Learning Outcomes

2.6. The Circle. Introduction. Prerequisites. Learning Outcomes The Circle 2.6 Introduction A circle is one of the most familiar geometrical figures. In this brief Section we discuss the basic coordinate geometr of a circle - in particular the basic equation representing

More information

STRAND: ALGEBRA Unit 3 Solving Equations

STRAND: ALGEBRA Unit 3 Solving Equations CMM Subject Support Strand: ALGEBRA Unit Solving Equations: Tet STRAND: ALGEBRA Unit Solving Equations TEXT Contents Section. Algebraic Fractions. Algebraic Fractions and Quadratic Equations. Algebraic

More information

THE POWER RULES. Raising an Exponential Expression to a Power

THE POWER RULES. Raising an Exponential Expression to a Power 8 (5-) Chapter 5 Eponents and Polnomials 5. THE POWER RULES In this section Raising an Eponential Epression to a Power Raising a Product to a Power Raising a Quotient to a Power Variable Eponents Summar

More information

10.2 The Unit Circle: Cosine and Sine

10.2 The Unit Circle: Cosine and Sine 0. The Unit Circle: Cosine and Sine 77 0. The Unit Circle: Cosine and Sine In Section 0.., we introduced circular motion and derived a formula which describes the linear velocit of an object moving on

More information

15.1. Exact Differential Equations. Exact First-Order Equations. Exact Differential Equations Integrating Factors

15.1. Exact Differential Equations. Exact First-Order Equations. Exact Differential Equations Integrating Factors SECTION 5. Eact First-Order Equations 09 SECTION 5. Eact First-Order Equations Eact Differential Equations Integrating Factors Eact Differential Equations In Section 5.6, ou studied applications of differential

More information

Math 100 Final Exam PRACTICE TEST Solve the equation. 1) 9x + 7(-3x - 3) = -31-2x A) 1 B) 5) -5 x A) [0, 8] D) - 1 B) (0, 8)

Math 100 Final Exam PRACTICE TEST Solve the equation. 1) 9x + 7(-3x - 3) = -31-2x A) 1 B) 5) -5 x A) [0, 8] D) - 1 B) (0, 8) Math 0 Final Eam PRACTICE TEST Solve the equation. ) 9 + 7(-3-3) = -3-7 - ) - - 3 [0, 8] (0, 8) - - - 0 8 Solve. ) 0.09 + 0.3(000 - ) = 0. {30} {300} {0} {0} - - - 0 8 [, -] - - -8 - - - 0 Solve the investment

More information

Functions. Chapter A B C D E F G H I J. The reciprocal function x

Functions. Chapter A B C D E F G H I J. The reciprocal function x Chapter 1 Functions Contents: A B C D E F G H I J Relations and functions Function notation, domain and range Composite functions, f± g Sign diagrams Inequalities (inequations) The modulus function 1 The

More information

2 Analysis of Graphs of

2 Analysis of Graphs of ch.pgs1-16 1/3/1 1:4 AM Page 1 Analsis of Graphs of Functions A FIGURE HAS rotational smmetr around an ais I if it coincides with itself b all rotations about I. Because of their complete rotational smmetr,

More information

Tim Kerins. Leaving Certificate Honours Maths - Algebra. Tim Kerins. the date

Tim Kerins. Leaving Certificate Honours Maths - Algebra. Tim Kerins. the date Leaving Certificate Honours Maths - Algebra the date Chapter 1 Algebra This is an important portion of the course. As well as generally accounting for 2 3 questions in examination it is the basis for many

More information

Solving Equations of Degree 1 (Linear Equations):

Solving Equations of Degree 1 (Linear Equations): Solving Equations of Degree 1 (Linear Equations): Definition: The equation a b0 is called an equation of degree 1. Eample: Solve for : 0. 0, Or. Dividing both sides by : giving. Easy! Check the answer!

More information

Zeros of Polynomial Functions

Zeros of Polynomial Functions Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n-1 x n-1 + + a 1 x + a 0 has integer coefficients and p/q (where p/q is reduced) is a rational zero, then p is a factor of

More information

Arithmetic Operations. The real numbers have the following properties: In particular, putting a 1 in the Distributive Law, we get

Arithmetic Operations. The real numbers have the following properties: In particular, putting a 1 in the Distributive Law, we get Review of Algebra REVIEW OF ALGEBRA Review of Algebra Here we review the basic rules and procedures of algebra that you need to know in order to be successful in calculus. Arithmetic Operations The real

More information

Core Maths C1. Revision Notes

Core Maths C1. Revision Notes Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the

More information

Graphing and transforming functions

Graphing and transforming functions Chapter 5 Graphing and transforming functions Contents: A B C D Families of functions Transformations of graphs Simple rational functions Further graphical transformations Review set 5A Review set 5B 6

More information

6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives

6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives 6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise

More information

INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1

INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1 Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.

More information

Exponential and Logarithmic Functions

Exponential and Logarithmic Functions Chapter 3 Eponential and Logarithmic Functions Section 3.1 Eponential Functions and Their Graphs Objective: In this lesson ou learned how to recognize, evaluate, and graph eponential functions. Course

More information

Analyzing the Graph of a Function

Analyzing the Graph of a Function SECTION A Summar of Curve Sketching 09 0 00 Section 0 0 00 0 Different viewing windows for the graph of f 5 7 0 Figure 5 A Summar of Curve Sketching Analze and sketch the graph of a function Analzing the

More information

Polynomials can be added or subtracted simply by adding or subtracting the corresponding terms, e.g., if

Polynomials can be added or subtracted simply by adding or subtracting the corresponding terms, e.g., if 1. Polynomials 1.1. Definitions A polynomial in x is an expression obtained by taking powers of x, multiplying them by constants, and adding them. It can be written in the form c 0 x n + c 1 x n 1 + c

More information

Solving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form

Solving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form SECTION 11.3 Solving Quadratic Equations b Graphing 11.3 OBJECTIVES 1. Find an ais of smmetr 2. Find a verte 3. Graph a parabola 4. Solve quadratic equations b graphing 5. Solve an application involving

More information

Exponential Functions

Exponential Functions CHAPTER Eponential Functions 010 Carnegie Learning, Inc. Georgia has two nuclear power plants: the Hatch plant in Appling Count, and the Vogtle plant in Burke Count. Together, these plants suppl about

More information

17.1 Connecting Intercepts and Zeros

17.1 Connecting Intercepts and Zeros Locker LESSON 7. Connecting Intercepts and Zeros Teas Math Standards The student is epected to: A.7.A Graph quadratic functions on the coordinate plane and use the graph to identif ke attributes, if possible,

More information

A Summary of Curve Sketching. Analyzing the Graph of a Function

A Summary of Curve Sketching. Analyzing the Graph of a Function 0_00.qd //0 :5 PM Page 09 SECTION. A Summar of Curve Sketching 09 0 00 Section. 0 0 00 0 Different viewing windows for the graph of f 5 7 0 Figure. 5 A Summar of Curve Sketching Analze and sketch the graph

More information

HI-RES STILL TO BE SUPPLIED

HI-RES STILL TO BE SUPPLIED 1 MRE GRAPHS AND EQUATINS HI-RES STILL T BE SUPPLIED Different-shaped curves are seen in man areas of mathematics, science, engineering and the social sciences. For eample, Galileo showed that if an object

More information

SECTION 2-5 Combining Functions

SECTION 2-5 Combining Functions 2- Combining Functions 16 91. Phsics. A stunt driver is planning to jump a motorccle from one ramp to another as illustrated in the figure. The ramps are 10 feet high, and the distance between the ramps

More information

Section V: Quadratic Equations and Functions. Module 1: Solving Quadratic Equations Using Factoring, Square Roots, Graphs, and Completing-the-Square

Section V: Quadratic Equations and Functions. Module 1: Solving Quadratic Equations Using Factoring, Square Roots, Graphs, and Completing-the-Square Haberman / Kling MTH 95 Section V: Quadratic Equations and Functions Module : Solving Quadratic Equations Using Factoring, Square Roots, Graphs, and Completing-the-Square DEFINITION: A quadratic equation

More information

I think that starting

I think that starting . Graphs of Functions 69. GRAPHS OF FUNCTIONS One can envisage that mathematical theor will go on being elaborated and etended indefinitel. How strange that the results of just the first few centuries

More information

Chapter 1 POLYNOMIAL FUNCTIONS

Chapter 1 POLYNOMIAL FUNCTIONS Chapter POLYNOMIAL FUNCTIONS Have you ever wondered how computer graphics software is able to so quickly draw the smooth, life-like faces that we see in video games and animated movies? Or how in architectural

More information

Rational Functions, Equations, and Inequalities

Rational Functions, Equations, and Inequalities Chapter 5 Rational Functions, Equations, and Inequalities GOALS You will be able to Graph the reciprocal functions of linear and quadratic functions Identif the ke characteristics of rational functions

More information

Chapter 8. Lines and Planes. By the end of this chapter, you will

Chapter 8. Lines and Planes. By the end of this chapter, you will Chapter 8 Lines and Planes In this chapter, ou will revisit our knowledge of intersecting lines in two dimensions and etend those ideas into three dimensions. You will investigate the nature of planes

More information

(2x + 1) x 3 3x2 =

(2x + 1) x 3 3x2 = BEEM13 Optimization Techniques for Economists Homework Week 2 Solutions Dieter Balkenborg Departments of Economics Universit of Eeter Eercise 1 Find the derivative of (2 + 1) 1 ln 1 3 Solution 1 d d (2

More information

Higher. Functions and Graphs. Functions and Graphs 18

Higher. Functions and Graphs. Functions and Graphs 18 hsn.uk.net Higher Mathematics UNIT UTCME Functions and Graphs Contents Functions and Graphs 8 Sets 8 Functions 9 Composite Functions 4 Inverse Functions 5 Eponential Functions 4 6 Introduction to Logarithms

More information

5.2 Inverse Functions

5.2 Inverse Functions 78 Further Topics in Functions. Inverse Functions Thinking of a function as a process like we did in Section., in this section we seek another function which might reverse that process. As in real life,

More information

GRAPHS OF RATIONAL FUNCTIONS

GRAPHS OF RATIONAL FUNCTIONS 0 (0-) Chapter 0 Polnomial and Rational Functions. f() ( 0) ( 0). f() ( 0) ( 0). f() ( 0) ( 0). f() ( 0) ( 0) 0. GRAPHS OF RATIONAL FUNCTIONS In this section Domain Horizontal and Vertical Asmptotes Oblique

More information

Course 2 Answer Key. 1.1 Rational & Irrational Numbers. Defining Real Numbers Student Logbook. The Square Root Function Student Logbook

Course 2 Answer Key. 1.1 Rational & Irrational Numbers. Defining Real Numbers Student Logbook. The Square Root Function Student Logbook Course Answer Ke. Rational & Irrational Numbers Defining Real Numbers. integers; 0. terminates; repeats 3. two; number 4. ratio; integers 5. terminating; repeating 6. rational; irrational 7. real 8. root

More information

D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review

D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review D0 APPENDIX D Precalculus Review APPENDIX D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane Just as ou can represent real numbers b

More information

a 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4)

a 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4) ROOTS OF POLYNOMIAL EQUATIONS In this unit we discuss polynomial equations. A polynomial in x of degree n, where n 0 is an integer, is an expression of the form P n (x) =a n x n + a n 1 x n 1 + + a 1 x

More information

LIMITS AND CONTINUITY

LIMITS AND CONTINUITY LIMITS AND CONTINUITY 1 The concept of it Eample 11 Let f() = 2 4 Eamine the behavior of f() as approaches 2 2 Solution Let us compute some values of f() for close to 2, as in the tables below We see from

More information

2.3 Quadratic Functions

2.3 Quadratic Functions . Quadratic Functions 9. Quadratic Functions You ma recall studing quadratic equations in Intermediate Algebra. In this section, we review those equations in the contet of our net famil of functions: the

More information

SOLVING SYSTEMS OF EQUATIONS

SOLVING SYSTEMS OF EQUATIONS SOLVING SYSTEMS OF EQUATIONS 4.. 4..4 Students have been solving equations even before Algebra. Now the focus on what a solution means, both algebraicall and graphicall. B understanding the nature of solutions,

More information