# Solving Cubic Polynomials

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Solving Cubic Polynomials 1.1 The general solution to the quadratic equation There are four steps to finding the zeroes of a quadratic polynomial. 1. First divide by the leading term, making the polynomial monic. 2. Then, given x 2 + a 1 x + a 0, substitute x = y a 1 2 (This is the depressed equation.) to obtain an equation without the linear term.. Solve then for y as a square root. (Remember to use both signs of the square root.) 4. Once this is done, recover x using the fact that x = y a 1 2. For example, let s solve 2x 2 + 7x 15 = 0. First, we divide both sides by 2 to create an equation with leading term equal to one: Then replace x by x = y a 1 2 = y 7 4 Solve for y: Then, solving back for x, we have x x 15 2 = 0. to obtain: y = 1 4 y 2 = x = 2 or 1 4 or 5. This method is equivalent to completing the square and is the steps taken in developing the muchmemorized quadratic formula. For example, if the original equation is our high school quadratic then the first step creates the equation We then write x = y b 2a so that and so ax 2 + bx + c = 0 x 2 + b a x + c a = 0. and obtain, after simplifying, y 2 b2 4ac 4a 2 = 0 y = ± b2 4ac 2a x = b 2a ± b2 4ac. 2a 1

2 The solutions to this quadratic depend heavily on the value of b 2 4ac. We give this a name (the discriminant) and a symbol ( ) and so discuss the discriminant = b 2 4ac. If the coefficients of the polynomial are integers and is a perfect square integer, we have rational roots. If the discriminant is positive, we have real roots. If the discriminant is zero, we have a single root. If the discriminant is negative, we have imaginary roots. We note for later that if the discriminant = b 2 4ac is equal to zero then we have a single root and so our polynomial is a perfect square. 1.2 The general solution to the cubic equation Every polynomial equation involves two steps to turn the polynomial into a slightly simpler polynomial. 1. First divide by the leading term, creating a monic polynomial (in which the highest power of x has coefficient one.) This does not change the roots. 2. Then, given x n +a n 1 x n 1 +a n 2 x n a 1 x+a 0, substitute x = y a n 1 to obtain an equation n without the term of degree n 1. (This is the depressed polynomial.) Since this step is reversible, solutions to the depressed equation give us solutions to the original equation. Here are the first steps in Cardano s method of solving the cubic. First, we do the two automatic steps: 1. Divide by the leading term, creating a cubic polynomial x +a 2 x 2 +a 1 x+a 0 with leading coefficient one. 2. Then substitute x = y a 2 to obtain an equation without the term of degree two. This creates an equation of the form x + P x Q = 0. Cardano would rewrite this equation in the form x + P x = Q. He then noticed (!) the following algebra identity: Given x + P x = Q, set (a b) + ab(a b) = a b. (1) ab = P and a b = Q. Solve this system by substitution and then assign x := a b. For example, we might replace a by P b equation to obtain Multiplying by b, we have (using the first equation) and then substitute into the second ( P b ) b = Q. ( P ) (b ) 2 = Qb which is a quadratic equation in b. Rewrite this equation as (b ) 2 + Qb ( P ) = 0 2

3 so that where the discriminant is b = Q 2 ± (2) := ( P ) + ( Q 2 )2. Now there are three possible solutions to equation (2). This is where Cardano struggled. The next steps sometimes involved imaginary numbers and he wasn t sure what to do with them. We will press on and finish Cardano s work by jumping ahead to Euler s day and using his understanding of complex numbers. Using Euler s formula e iθ = cos θ + i sin θ () we will create the element ω := e 2πi/ = i. Notice that ω = 1 and that (ω 2 ) = 1 so that 1, ω and ω 2 are all cube roots of 1! Using ω, we can see that any real number has three cube roots. Given one cube root, β, the others will be ωβ and ω 2 β. We use this to finish equation 2 by picking on cube root of Q/2 ±. We call that one particular solution β. The other solutions to the equation (2) are b = ωβ and b = ω 2 β. Each solution to (2) gives a value for a so that ab = P/. Let α := P β. Then are all solutions to the cubic equation. α β, αω 2 βω, and αω βω 2 Euler s example We work through an example due to Euler: We find all solutions to x 6x = 4. (4) Here P = 6 and Q = 4 and so the discriminant is = = 4 so = 2i. Therefore b = 2 ± 2i. We choose a sign and solve the cubic equation b = 2 + 2i. Write 2 + 2i in polar form as 8 (e π 4 i ). Then b = 8 e π 4 i has a particular solution Set α = 2 β β = 2 e π 4 i = 1 + i. = 2 2 e π 4 i = 2 e π 4 i = 2 e π 4 i. Or, in cartesian form, α = 1 + i. Then x = α β = ( 1 + i) (1 + i) = 2 is one solution to the cubic equation. But there are others!

4 and then Thus and If x 1 = α β is a solution then so are x 2 = αω 2 βω = 2(e π 4 i e 4π i e π 4 i e 2π i ) = 2(e 25π 12 i e 11π 12 i ) = 2(e π 12 i + e π 12 i ) x = αω βω 2 = 2(e π 4 i e 2π i e π 4 i e 4π i ) = 2(e 17π 12 i e 19π 12 i ) = 2(e 7π 12 i + e 7π 12 i ) By Euler s formula, e θi + e θi = 2 cos θ. So our answers are equivalent to x 2 = 2 2 cos( π 12 ) and x = 2 2 cos( 7π 12 ). The value of cos( π 12 ) is not immediate, but we can find it from a trig identity. Since cos(2θ) = 2 cos 2 θ 1 x 2 = 8 cos 2 θ = cos π 12 = 1 + cos 2θ = In a similar manner we can find x. Our final answers are x 2 = and x = 4 2. Another example from Euler We solve Euler s cubic: Since (a b) + ab(a b) = a b, we set x 6x = 9. (5) ab = 6 and a b = 9. so (Let P := 6; Q := 9.) Then View this as a quadratic in b so that a = 2/b; ( 2/b) b = 9 8 b 6 = 9b or b 6 + 9b + 8 = 0. Therefore either b = 8 or b = 1. Suppose b = 8 and presumably b = 2. Then a = 1 and x = a b =. (b ) 2 + 9b + 8 = 0 = (b + 8)(b + 1) = 0. (6) If b = 1 then a = 2 and x = a b =, so choosing the other factor does not give new information. We have found one solution, x =. 4

5 But what about other solutions? Set β = 2 as a solution to the cubic equation (6), so that β = 8 and let α = 1 be the corresponding choice for a. Then b = βω = 2ω is also a solution to that cubic and in this case a = αω 2 = ω 2 is the corresponding choice for a. Then a second solution is x = a b = ω 2 + 2ω = ( i) + ( 1 + i) = 1 2 ( + i). If instead we have a = αω and b = βω 2 then the final solution is x = a b = ω + 2ω 2 = ( i) + ( 1 i) = 1 2 ( i). So we have found all three solutions:, 1 2 ( + i). and 1 2 ( i). 1. Rational Solutions We can avoid the lengthy computations, above, if we are lucky enough to find a rational solution to our polynomial. For example, in the last problem, if we had merely stumbled on the root x =, we could have divided the cubic polynomial x 6x 9 by x and rewritten it as x 6x 9 = (x )(x 2 + x + ). The quadratic equation, applied to x 2 + x + would have given us the final two solutions without the extra work. So how do we find these rational solutions when they occur? If x = p q is a rational solution to the polynomial equation f(x) = 0 then qx p is a factor of the polynomial f(x) and so we can use long division to write f(x) = (qx p)g(x) where g(x) is a polynomial of smaller degree. We teach a version of this method in high school when students learn to solve quadratic equations by factoring. For example, one might solve the equation x 2 2x 8 = 0 by factoring the left-hand side into (x + 4)(x 2), obtain solutions x = 4 and x = 2 and so avoid the quadratic formula. We can try this method on polynomials of higher degree with integer coefficients. Descartes would point out, in the 1600s, that if f(x) = a n x n + a n 1 x n a 1 x + a 0 has a root x = p/q then q must divide a n and p must divide a 0 (Notice how this works on the quadratic x 2 2x 8.) With then a short list of possible rational solutions, Descartes was willing to try these solutions and exhaust the possibilities for a rational root. The modern version of this is to pull out a graphing calculator, graph the polynomial equation y = f(x) and hope that the calculator identifies a nice rational (or even integer!) root. For example, with Euler s cubic x 6x 9, we discover that x = is a root. When then divide the polynomial by x to obtain a quadratic polynomial and now we can go ahead and use the quadratic formula. This method is much faster than the general method, but it requires that we be lucky and stumble upon a root. 5

6 1.4 Quartic Polynomials After del Ferro and Tartaglia solved the general cubic equation (and the result was released to the public by Cardano), mathematicians then concentrated on the quartic equation. Cardano s student, Ferrari, around 1540, suggested a general method for solving the depressed quartic. Given the depressed quartic x 4 + px 2 + qx + r = 0 rewrite the problem so that it appears to involve a square: (x 2 + p) 2 = px 2 qx + (p 2 r). Now insert a new variable y and expand (x 2 + p + y) 2 with the hopes that the right side, (p + 2y)x 2 qx + (p 2 r + 2py + y 2 ) is a perfect square. By the quadratic formula, the right-hand side will be a perfect square when the discriminant 4(p + 2y)(p 2 r + 2py + y 2 ) q 2 is zero. So we first need to solve 4(p + 2y)(p 2 r + 2py + y 2 ) q 2 = 0 (7) for y. Since equation (7) is a cubic equation, we know how to do that! In other words, given a quartic, we make it monic, and then turn it into a depressed equation, insert y, and solve the associated cubic for y. Once this is done, we solve the quadratic equation (x 2 + p + y) 2 = (p + 2y)x 2 qx + (p 2 r + 2py + y 2 ) for x. This is lengthy and tedious, but it works in general! (Well, it works if one is willing to believe in complex numbers.) By 1545, after Cardano published Ferrari s work (with his permission, apparently!) mathematicians began to search for solutions to the general fifth-degree or quintic equation. It would take the work of Abel, Galois and others to show that in fact a general solution to the quintic is impossible. We will save that story for another time. 6

7 Exercises on cubic equations 1. For each of the equations, below, do the appropriate substitution to turn the polynomial on the left-hand side into a depressed polynomial. (a) 2x 2 5x + = 0 (b) x 6x 2 + 2x + = 0 (c) x 4 + 8x + x 12 = 0 2. Complete your work in equation part (a), above, finding the solutions to the equation by completing the square.. For parts (b) and (c) in problem 1, guess a rational solution and then use this to find all solutions to the polynomial. 4. Find a rational solution for the following polynomials. Then use this solution to find all the roots. (a) Viete and Cardano-Tartaglia examined this polynomial: x + 6x 16. (b) Viete and Cardano-Tartaglia also did this one: x 6x 162. (c) Euler s cubic: x 6x 4. 7

8 Miscellanea 1. Varadarajan, p. 47 and following. 2. 8x 6x 1 Solution. (one solution is the cosine of 20 degrees!) P = 4, Q = 1 8. Cardano and 1x 2 = x 4 + 2x + 2x + 1. (Eves 29) This is magic : Cardano added x 2 to both sides to get a perfect square. 4. Ferrari on page 29. a + b + c = 10; ac = b 2, ab = 6 leads to the quartic b 4 + 6b = 60b. This can be turned into a cubic by Ferrari and Viete s methods. 5. The quartic equation ( Ferrari on page 27 of Eves.) 6. Descarte s solution: Use it on x 4 2x 2 + 8x = 0. Or translate this by x = y + 1. (Eves p. 71) 8

### 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

### expression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method.

A polynomial of degree n (in one variable, with real coefficients) is an expression of the form: a n x n + a n 1 x n 1 + a n 2 x n 2 + + a 2 x 2 + a 1 x + a 0 where a n, a n 1, a n 2, a 2, a 1, a 0 are

### The Notebook Series. The solution of cubic and quartic equations. R.S. Johnson. Professor of Applied Mathematics

The Notebook Series The solution of cubic and quartic equations by R.S. Johnson Professor of Applied Mathematics School of Mathematics & Statistics University of Newcastle upon Tyne R.S.Johnson 006 CONTENTS

### The Method of Partial Fractions Math 121 Calculus II Spring 2015

Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method

### 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

### 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

### SOLVING POLYNOMIAL EQUATIONS

C SOLVING POLYNOMIAL EQUATIONS We will assume in this appendix that you know how to divide polynomials using long division and synthetic division. If you need to review those techniques, refer to an algebra

### 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

### Factoring Polynomials and Solving Quadratic Equations

Factoring Polynomials and Solving Quadratic Equations Math Tutorial Lab Special Topic Factoring Factoring Binomials Remember that a binomial is just a polynomial with two terms. Some examples include 2x+3

### Lagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given.

Polynomials (Ch.1) Study Guide by BS, JL, AZ, CC, SH, HL Lagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given. Sasha s method

### 3.6 The Real Zeros of a Polynomial Function

SECTION 3.6 The Real Zeros of a Polynomial Function 219 3.6 The Real Zeros of a Polynomial Function PREPARING FOR THIS SECTION Before getting started, review the following: Classification of Numbers (Appendix,

### 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

### NSM100 Introduction to Algebra Chapter 5 Notes Factoring

Section 5.1 Greatest Common Factor (GCF) and Factoring by Grouping Greatest Common Factor for a polynomial is the largest monomial that divides (is a factor of) each term of the polynomial. GCF is the

### 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

### ALGEBRA 2 CRA 2 REVIEW - Chapters 1-6 Answer Section

ALGEBRA 2 CRA 2 REVIEW - Chapters 1-6 Answer Section MULTIPLE CHOICE 1. ANS: C 2. ANS: A 3. ANS: A OBJ: 5-3.1 Using Vertex Form SHORT ANSWER 4. ANS: (x + 6)(x 2 6x + 36) OBJ: 6-4.2 Solving Equations by

### Zeros of a Polynomial Function

Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we

### SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS

(Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic

### 1.3 Algebraic Expressions

1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,

### Factoring Polynomials

Factoring Polynomials Hoste, Miller, Murieka September 12, 2011 1 Factoring In the previous section, we discussed how to determine the product of two or more terms. Consider, for instance, the equations

### FACTORISATION YEARS. A guide for teachers - Years 9 10 June 2011. The Improving Mathematics Education in Schools (TIMES) Project

9 10 YEARS The Improving Mathematics Education in Schools (TIMES) Project FACTORISATION NUMBER AND ALGEBRA Module 33 A guide for teachers - Years 9 10 June 2011 Factorisation (Number and Algebra : Module

### Exact Values of the Sine and Cosine Functions in Increments of 3 degrees

Exact Values of the Sine and Cosine Functions in Increments of 3 degrees The sine and cosine values for all angle measurements in multiples of 3 degrees can be determined exactly, represented in terms

### SOLVING POLYNOMIAL EQUATIONS BY RADICALS

SOLVING POLYNOMIAL EQUATIONS BY RADICALS Lee Si Ying 1 and Zhang De-Qi 2 1 Raffles Girls School (Secondary), 20 Anderson Road, Singapore 259978 2 Department of Mathematics, National University of Singapore,

### Polynomials and Factoring

Lesson 2 Polynomials and Factoring A polynomial function is a power function or the sum of two or more power functions, each of which has a nonnegative integer power. Because polynomial functions are built

### 1.3 Polynomials and Factoring

1.3 Polynomials and Factoring Polynomials Constant: a number, such as 5 or 27 Variable: a letter or symbol that represents a value. Term: a constant, variable, or the product or a constant and variable.

### 9. POLYNOMIALS. Example 1: The expression a(x) = x 3 4x 2 + 7x 11 is a polynomial in x. The coefficients of a(x) are the numbers 1, 4, 7, 11.

9. POLYNOMIALS 9.1. Definition of a Polynomial A polynomial is an expression of the form: a(x) = a n x n + a n-1 x n-1 +... + a 1 x + a 0. The symbol x is called an indeterminate and simply plays the role

### Mathematics. (www.tiwariacademy.com : Focus on free Education) (Chapter 5) (Complex Numbers and Quadratic Equations) (Class XI)

( : Focus on free Education) Miscellaneous Exercise on chapter 5 Question 1: Evaluate: Answer 1: 1 ( : Focus on free Education) Question 2: For any two complex numbers z1 and z2, prove that Re (z1z2) =

### Section 6.1 Factoring Expressions

Section 6.1 Factoring Expressions The first method we will discuss, in solving polynomial equations, is the method of FACTORING. Before we jump into this process, you need to have some concept of what

### ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form

ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form Goal Graph quadratic functions. VOCABULARY Quadratic function A function that can be written in the standard form y = ax 2 + bx+ c where a 0 Parabola

9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation

### FACTORING QUADRATICS 8.1.1 and 8.1.2

FACTORING QUADRATICS 8.1.1 and 8.1.2 Chapter 8 introduces students to quadratic equations. These equations can be written in the form of y = ax 2 + bx + c and, when graphed, produce a curve called a parabola.

### 1 Lecture: Integration of rational functions by decomposition

Lecture: Integration of rational functions by decomposition into partial fractions Recognize and integrate basic rational functions, except when the denominator is a power of an irreducible quadratic.

250) 960-6367 Factoring Polynomials Sometimes when we try to solve or simplify an equation or expression involving polynomials the way that it looks can hinder our progress in finding a solution. Factorization

### 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

### UNCORRECTED PAGE PROOFS

number and and algebra TopIC 17 Polynomials 17.1 Overview Why learn this? Just as number is learned in stages, so too are graphs. You have been building your knowledge of graphs and functions over time.

### Copyrighted Material. Chapter 1 DEGREE OF A CURVE

Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two

### A Different Way to Solve Quadratics Bluma s Method*

A Different Way to Solve Quadratics Bluma s Method* Abstract In this article, we introduce an approach to finding the solutions of a quadratic equation and provide two proofs of its correctness. This method

### 1.5. Factorisation. Introduction. Prerequisites. Learning Outcomes. Learning Style

Factorisation 1.5 Introduction In Block 4 we showed the way in which brackets were removed from algebraic expressions. Factorisation, which can be considered as the reverse of this process, is dealt with

### Integrals of Rational Functions

Integrals of Rational Functions Scott R. Fulton Overview A rational function has the form where p and q are polynomials. For example, r(x) = p(x) q(x) f(x) = x2 3 x 4 + 3, g(t) = t6 + 4t 2 3, 7t 5 + 3t

### March 29, 2011. 171S4.4 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

### 2.3. Finding polynomial functions. An Introduction:

2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned

### 0.4 FACTORING POLYNOMIALS

36_.qxd /3/5 :9 AM Page -9 SECTION. Factoring Polynomials -9. FACTORING POLYNOMIALS Use special products and factorization techniques to factor polynomials. Find the domains of radical expressions. Use

### Equations, Inequalities & Partial Fractions

Contents Equations, Inequalities & Partial Fractions.1 Solving Linear Equations 2.2 Solving Quadratic Equations 1. Solving Polynomial Equations 1.4 Solving Simultaneous Linear Equations 42.5 Solving Inequalities

Factoring the trinomial ax 2 + bx + c when a = 1 A trinomial in the form x 2 + bx + c can be factored to equal (x + m)(x + n) when the product of m x n equals c and the sum of m + n equals b. (Note: the

### 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

### 6.1 Add & Subtract Polynomial Expression & Functions

6.1 Add & Subtract Polynomial Expression & Functions Objectives 1. Know the meaning of the words term, monomial, binomial, trinomial, polynomial, degree, coefficient, like terms, polynomial funciton, quardrtic

### COMPLEX NUMBERS AND DIFFERENTIAL EQUATIONS

COMPLEX NUMBERS AND DIFFERENTIAL EQUATIONS BORIS HASSELBLATT CONTENTS. Introduction. Why complex numbers were introduced 3. Complex numbers, Euler s formula 3 4. Homogeneous differential equations 8 5.

### Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III

Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III Name Date Adding and Subtracting Polynomials Algebra Standard 10.0 A polynomial is a sum of one ore more monomials. Polynomial

### 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

### Algebra Practice Problems for Precalculus and Calculus

Algebra Practice Problems for Precalculus and Calculus Solve the following equations for the unknown x: 1. 5 = 7x 16 2. 2x 3 = 5 x 3. 4. 1 2 (x 3) + x = 17 + 3(4 x) 5 x = 2 x 3 Multiply the indicated polynomials

### Solving for the Roots of the Cubic Equation. Finding the solution to the roots of a polynomial equation has been a fundamental

Dallas Gosselin and Jonathan Fernandez Professor Buckmire April 18, 014 Complex Analysis Project Solving for the Roots of the Cubic Equation Finding the solution to the roots of a polynomial equation has

### Factoring Polynomials

Factoring a Polynomial Expression Factoring a polynomial is expressing the polynomial as a product of two or more factors. Simply stated, it is somewhat the reverse process of multiplying. To factor polynomials,

### 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

### Factoring Trinomials: The ac Method

6.7 Factoring Trinomials: The ac Method 6.7 OBJECTIVES 1. Use the ac test to determine whether a trinomial is factorable over the integers 2. Use the results of the ac test to factor a trinomial 3. For

### Zeros of Polynomial Functions

Zeros of Polynomial Functions Objectives: 1.Use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions 2.Find rational zeros of polynomial functions 3.Find conjugate

### Math Common Core Sampler Test

High School Algebra Core Curriculum Math Test Math Common Core Sampler Test Our High School Algebra sampler covers the twenty most common questions that we see targeted for this level. For complete tests

### Galois Theory. Richard Koch

Galois Theory Richard Koch April 2, 2015 Contents 1 Preliminaries 4 1.1 The Extension Problem; Simple Groups.................... 4 1.2 An Isomorphism Lemma............................. 5 1.3 Jordan Holder...................................

### Algebra 2: Themes for the Big Final Exam

Algebra : Themes for the Big Final Exam Final will cover the whole year, focusing on the big main ideas. Graphing: Overall: x and y intercepts, fct vs relation, fct vs inverse, x, y and origin symmetries,

### Section 5.0A Factoring Part 1

Section 5.0A Factoring Part 1 I. Work Together A. Multiply the following binomials into trinomials. (Write the final result in descending order, i.e., a + b + c ). ( 7)( + 5) ( + 7)( + ) ( + 7)( + 5) (

### Roots of Polynomials

Roots of Polynomials (Com S 477/577 Notes) Yan-Bin Jia Sep 24, 2015 A direct corollary of the fundamental theorem of algebra is that p(x) can be factorized over the complex domain into a product a n (x

### MATHEMATICS FOR ENGINEERING BASIC ALGEBRA

MATHEMATICS FOR ENGINEERING BASIC ALGEBRA TUTORIAL 3 EQUATIONS This is the one of a series of basic tutorials in mathematics aimed at beginners or anyone wanting to refresh themselves on fundamentals.

### MATH 10034 Fundamental Mathematics IV

MATH 0034 Fundamental Mathematics IV http://www.math.kent.edu/ebooks/0034/funmath4.pdf Department of Mathematical Sciences Kent State University January 2, 2009 ii Contents To the Instructor v Polynomials.

### Differentiation and Integration

This material is a supplement to Appendix G of Stewart. You should read the appendix, except the last section on complex exponentials, before this material. Differentiation and Integration Suppose we have

### Application. Outline. 3-1 Polynomial Functions 3-2 Finding Rational Zeros of. Polynomial. 3-3 Approximating Real Zeros of.

Polynomial and Rational Functions Outline 3-1 Polynomial Functions 3-2 Finding Rational Zeros of Polynomials 3-3 Approximating Real Zeros of Polynomials 3-4 Rational Functions Chapter 3 Group Activity:

### Algebra 1 If you are okay with that placement then you have no further action to take Algebra 1 Portion of the Math Placement Test

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

### Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.

Algebra 2 - Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers - {1,2,3,4,...}

### Examples of Tasks from CCSS Edition Course 3, Unit 5

Examples of Tasks from CCSS Edition Course 3, Unit 5 Getting Started The tasks below are selected with the intent of presenting key ideas and skills. Not every answer is complete, so that teachers can

### MATH 90 CHAPTER 6 Name:.

MATH 90 CHAPTER 6 Name:. 6.1 GCF and Factoring by Groups Need To Know Definitions How to factor by GCF How to factor by groups The Greatest Common Factor Factoring means to write a number as product. a

### 1.7. Partial Fractions. 1.7.1. Rational Functions and Partial Fractions. A rational function is a quotient of two polynomials: R(x) = P (x) Q(x).

.7. PRTIL FRCTIONS 3.7. Partial Fractions.7.. Rational Functions and Partial Fractions. rational function is a quotient of two polynomials: R(x) = P (x) Q(x). Here we discuss how to integrate rational

### Factoring Polynomials

Factoring Polynomials Factoring Factoring is the process of writing a polynomial as the product of two or more polynomials. The factors of 6x 2 x 2 are 2x + 1 and 3x 2. In this section, we will be factoring

### 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

### Factoring Polynomials

UNIT 11 Factoring Polynomials You can use polynomials to describe framing for art. 396 Unit 11 factoring polynomials A polynomial is an expression that has variables that represent numbers. A number can

### Unit 3: Day 2: Factoring Polynomial Expressions

Unit 3: Day : Factoring Polynomial Expressions Minds On: 0 Action: 45 Consolidate:10 Total =75 min Learning Goals: Extend knowledge of factoring to factor cubic and quartic expressions that can be factored

### Factoring Cubic Polynomials

Factoring Cubic Polynomials Robert G. Underwood 1. Introduction There are at least two ways in which using the famous Cardano formulas (1545) to factor cubic polynomials present more difficulties than

### Factoring a Difference of Two Squares. Factoring a Difference of Two Squares

284 (6 8) Chapter 6 Factoring 87. Tomato soup. The amount of metal S (in square inches) that it takes to make a can for tomato soup is a function of the radius r and height h: S 2 r 2 2 rh a) Rewrite this

### SMT 2014 Algebra Test Solutions February 15, 2014

1. Alice and Bob are painting a house. If Alice and Bob do not take any breaks, they will finish painting the house in 20 hours. If, however, Bob stops painting once the house is half-finished, then the

FACTORING QUADRATIC EQUATIONS Summary 1. Difference of squares... 1 2. Mise en évidence simple... 2 3. compounded factorization... 3 4. Exercises... 7 The goal of this section is to summarize the methods

### A. Factoring out the Greatest Common Factor.

DETAILED SOLUTIONS AND CONCEPTS - FACTORING POLYNOMIAL EXPRESSIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you!

### Algebra 1 Course Title

Algebra 1 Course Title Course- wide 1. What patterns and methods are being used? Course- wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept

### 2.5 Zeros of a Polynomial Functions

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

### College Algebra - MAT 161 Page: 1 Copyright 2009 Killoran

College Algebra - MAT 6 Page: Copyright 2009 Killoran Zeros and Roots of Polynomial Functions Finding a Root (zero or x-intercept) of a polynomial is identical to the process of factoring a polynomial.

### 4.1. COMPLEX NUMBERS

4.1. COMPLEX NUMBERS What You Should Learn Use the imaginary unit i to write complex numbers. Add, subtract, and multiply complex numbers. Use complex conjugates to write the quotient of two complex numbers

### Factoring - Solve by Factoring

6.7 Factoring - Solve by Factoring Objective: Solve quadratic equation by factoring and using the zero product rule. When solving linear equations such as 2x 5 = 21 we can solve for the variable directly

### COMPLEX NUMBERS. a bi c di a c b d i. a bi c di a c b d i For instance, 1 i 4 7i 1 4 1 7 i 5 6i

COMPLEX NUMBERS _4+i _-i FIGURE Complex numbers as points in the Arg plane i _i +i -i A complex number can be represented by an expression of the form a bi, where a b are real numbers i is a symbol with

### Introduction Assignment

PRE-CALCULUS 11 Introduction Assignment Welcome to PREC 11! This assignment will help you review some topics from a previous math course and introduce you to some of the topics that you ll be studying

### THE COMPLEX EXPONENTIAL FUNCTION

Math 307 THE COMPLEX EXPONENTIAL FUNCTION (These notes assume you are already familiar with the basic properties of complex numbers.) We make the following definition e iθ = cos θ + i sin θ. (1) This formula

### PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.

PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include

### Factoring Polynomials

Factoring Polynomials Any Any Any natural number that that that greater greater than than than 1 1can can 1 be can be be factored into into into a a a product of of of prime prime numbers. For For For

### A Systematic Approach to Factoring

A Systematic Approach to Factoring Step 1 Count the number of terms. (Remember****Knowing the number of terms will allow you to eliminate unnecessary tools.) Step 2 Is there a greatest common factor? Tool

### Factoring and Applications

Factoring and Applications What is a factor? The Greatest Common Factor (GCF) To factor a number means to write it as a product (multiplication). Therefore, in the problem 48 3, 4 and 8 are called the

### 15. Symmetric polynomials

15. Symmetric polynomials 15.1 The theorem 15.2 First examples 15.3 A variant: discriminants 1. The theorem Let S n be the group of permutations of {1,, n}, also called the symmetric group on n things.

### QUADRATIC EQUATIONS EXPECTED BACKGROUND KNOWLEDGE

MODULE - 1 Quadratic Equations 6 QUADRATIC EQUATIONS In this lesson, you will study aout quadratic equations. You will learn to identify quadratic equations from a collection of given equations and write

### Cubic Functions: Global Analysis

Chapter 14 Cubic Functions: Global Analysis The Essential Question, 231 Concavity-sign, 232 Slope-sign, 234 Extremum, 235 Height-sign, 236 0-Concavity Location, 237 0-Slope Location, 239 Extremum Location,

### 3 1. Note that all cubes solve it; therefore, there are no more

Math 13 Problem set 5 Artin 11.4.7 Factor the following polynomials into irreducible factors in Q[x]: (a) x 3 3x (b) x 3 3x + (c) x 9 6x 6 + 9x 3 3 Solution: The first two polynomials are cubics, so if

### Algebra II A Final Exam

Algebra II A Final Exam Multiple Choice Identify the choice that best completes the statement or answers the question. Evaluate the expression for the given value of the variable(s). 1. ; x = 4 a. 34 b.

### MATH 21. College Algebra 1 Lecture Notes

MATH 21 College Algebra 1 Lecture Notes MATH 21 3.6 Factoring Review College Algebra 1 Factoring and Foiling 1. (a + b) 2 = a 2 + 2ab + b 2. 2. (a b) 2 = a 2 2ab + b 2. 3. (a + b)(a b) = a 2 b 2. 4. (a

### Higher Education Math Placement

Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication

### minimal polyonomial Example

Minimal Polynomials Definition Let α be an element in GF(p e ). We call the monic polynomial of smallest degree which has coefficients in GF(p) and α as a root, the minimal polyonomial of α. Example: We

### Polynomial Degree and Finite Differences

CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial