This document provides an overview of solving polynomial equations. It defines polynomials and their key properties like degree, coefficients, and roots. It introduces several theorems for finding roots, including the Remainder Theorem, Factor Theorem, and the idea that a polynomial of degree n has n roots when counting multiplicities. Methods discussed include factoring, long division, and the quadratic formula. The document explains it is not possible to express solutions of polynomials of degree 5 or higher using radicals.
This document provides an overview of different types of numbers and their relationships. It discusses:
1) Real numbers which include rational numbers like fractions and irrational numbers like square roots. Rational numbers have repeating decimals while irrational numbers do not.
2) Complex numbers which are numbers of the form a + bi, where a and b are real numbers. They were invented to allow solutions to equations like x^2 = -1.
3) How René Descartes linked algebra and geometry by establishing a correspondence between real numbers and points on a coordinate line, allowing geometric shapes to be described with algebraic equations.
This document provides proofs of several basic limit theorems and properties from calculus. It includes:
1) Proofs of three parts of a limit theorem about combining constant multiples, sums, and products of functions with limits.
2) A proof of a basic continuity property regarding limits of composite functions.
3) Proofs of the chain rule of differentiation and that relative extrema of functions occur at critical points.
4) Proofs of two summation formulas involving sums of integers and sums of squared integers.
The proofs illustrate fundamental limit concepts and techniques like choosing appropriate δ values, using preceding results about limits, and algebraic manipulations of expressions involving limits.
This chapter introduces exponential and logarithmic functions. Exponential functions take the form f(x) = bx and model continuous exponential growth or decay. Their inverse functions are logarithmic functions of the form x = logb y. Key points covered include:
- Exponential growth occurs when b > 1 and decay when 0 < b < 1.
- Logarithmic functions have a domain of positive real numbers and range of all real numbers. Their graphs are reflections of exponential graphs in the line y = x.
- Important properties are established, such as logb(bn) = n, logb1 = 0, and the relationship between exponential and logarithmic forms of an equation.
The document is a presentation on polynomials. It defines a polynomial as an expression that can contain constants, variables, and exponents, but cannot contain division by a variable. It discusses the key characteristics of polynomials including their degree, standard form, zeros, factoring, and algebraic identities. Examples are provided to illustrate different types of polynomials like monomials, binomials, trinomials, and how to add, subtract, multiply and divide polynomials.
The document discusses polynomial division algorithms. It introduces long division and synthetic division as methods for dividing polynomials. Long division is analogous to dividing numbers, while synthetic division is simpler but only applies when dividing a polynomial by a monomial. The key points are:
- Long division allows dividing any polynomial P(x) by any polynomial D(x) to obtain a quotient Q(x) and remainder R(x) such that P(x) = Q(x)D(x) + R(x) and the degree of R(x) is less than the degree of D(x).
- Synthetic division is more efficient than long division when dividing a polynomial by a monomial of the form (
The document discusses the remainder theorem for polynomials. It defines the division algorithm for polynomials which divides a polynomial P(x) by (x-c) to get a unique quotient polynomial Q(x) and remainder R. The remainder theorem then states that the remainder R is equal to the value of P(c). The document proves the theorem and provides examples of using it to find the remainder when one polynomial is divided by another. It also provides exercises for students to find remainders using the theorem.
english mathematics dictionary
kamus bahassa inggris untuk matematika
oleh neneng
Nurwaningsih
(06081281520066)
Nurwaningsih30@gmail.com
PROGRAM STUDI PENDIDIKAN MATEMATIKA
FAKULTAS KEGURUAN DAN ILMU PENDIDIKAN
UNIVERSITAS SRIWIJAYA
INDRALAYA
2017
semoga bermanfaat
The document discusses solving literal equations by isolating the variable of interest on one side of the equation. It provides examples of solving equations for various variables by adding, subtracting, multiplying, or dividing both sides of the equation by the same quantity. The goal is to isolate the variable being solved for so it stands alone on one side of the equal sign. Steps include clearing fractions, moving all other terms to the other side of the equation, and then dividing both sides by the coefficient of the variable being solved for.
This document provides an overview of different types of numbers and their relationships. It discusses:
1) Real numbers which include rational numbers like fractions and irrational numbers like square roots. Rational numbers have repeating decimals while irrational numbers do not.
2) Complex numbers which are numbers of the form a + bi, where a and b are real numbers. They were invented to allow solutions to equations like x^2 = -1.
3) How René Descartes linked algebra and geometry by establishing a correspondence between real numbers and points on a coordinate line, allowing geometric shapes to be described with algebraic equations.
This document provides proofs of several basic limit theorems and properties from calculus. It includes:
1) Proofs of three parts of a limit theorem about combining constant multiples, sums, and products of functions with limits.
2) A proof of a basic continuity property regarding limits of composite functions.
3) Proofs of the chain rule of differentiation and that relative extrema of functions occur at critical points.
4) Proofs of two summation formulas involving sums of integers and sums of squared integers.
The proofs illustrate fundamental limit concepts and techniques like choosing appropriate δ values, using preceding results about limits, and algebraic manipulations of expressions involving limits.
This chapter introduces exponential and logarithmic functions. Exponential functions take the form f(x) = bx and model continuous exponential growth or decay. Their inverse functions are logarithmic functions of the form x = logb y. Key points covered include:
- Exponential growth occurs when b > 1 and decay when 0 < b < 1.
- Logarithmic functions have a domain of positive real numbers and range of all real numbers. Their graphs are reflections of exponential graphs in the line y = x.
- Important properties are established, such as logb(bn) = n, logb1 = 0, and the relationship between exponential and logarithmic forms of an equation.
The document is a presentation on polynomials. It defines a polynomial as an expression that can contain constants, variables, and exponents, but cannot contain division by a variable. It discusses the key characteristics of polynomials including their degree, standard form, zeros, factoring, and algebraic identities. Examples are provided to illustrate different types of polynomials like monomials, binomials, trinomials, and how to add, subtract, multiply and divide polynomials.
The document discusses polynomial division algorithms. It introduces long division and synthetic division as methods for dividing polynomials. Long division is analogous to dividing numbers, while synthetic division is simpler but only applies when dividing a polynomial by a monomial. The key points are:
- Long division allows dividing any polynomial P(x) by any polynomial D(x) to obtain a quotient Q(x) and remainder R(x) such that P(x) = Q(x)D(x) + R(x) and the degree of R(x) is less than the degree of D(x).
- Synthetic division is more efficient than long division when dividing a polynomial by a monomial of the form (
The document discusses the remainder theorem for polynomials. It defines the division algorithm for polynomials which divides a polynomial P(x) by (x-c) to get a unique quotient polynomial Q(x) and remainder R. The remainder theorem then states that the remainder R is equal to the value of P(c). The document proves the theorem and provides examples of using it to find the remainder when one polynomial is divided by another. It also provides exercises for students to find remainders using the theorem.
english mathematics dictionary
kamus bahassa inggris untuk matematika
oleh neneng
Nurwaningsih
(06081281520066)
Nurwaningsih30@gmail.com
PROGRAM STUDI PENDIDIKAN MATEMATIKA
FAKULTAS KEGURUAN DAN ILMU PENDIDIKAN
UNIVERSITAS SRIWIJAYA
INDRALAYA
2017
semoga bermanfaat
The document discusses solving literal equations by isolating the variable of interest on one side of the equation. It provides examples of solving equations for various variables by adding, subtracting, multiplying, or dividing both sides of the equation by the same quantity. The goal is to isolate the variable being solved for so it stands alone on one side of the equal sign. Steps include clearing fractions, moving all other terms to the other side of the equation, and then dividing both sides by the coefficient of the variable being solved for.
The document discusses factorable polynomials and how to graph them. It defines a factorable polynomial as one that can be written as the product of linear factors using real numbers. For large values of x, the leading term of a polynomial dominates so the graph resembles that of the leading term. To graph a factorable polynomial, one first graphs the individual factors like x^n and then combines them, which gives smooth curves tending to the graphs of the leading terms for large x.
The document discusses functions and their graphical representations. It defines key terms like domain, range, and one-to-one and many-to-one mappings. It then focuses on quadratic functions, showing that their graphs take characteristic U-shaped or inverted U-shaped forms. The document also examines inequalities involving quadratic expressions and how to determine the range of values satisfying such inequalities by analyzing the graph of the quadratic function.
The document describes aspects of polynomials including:
- Definitions of polynomials and their components
- Algebraic operations on polynomials like addition, subtraction, and multiplication
- Dividing polynomials using long division and synthetic (Horner's) division methods
- The remainder theorem stating the remainder of dividing a polynomial by (x - k) is the value of the polynomial at k
- The factorization theorem relating the factors of a polynomial to its roots
- Techniques for factorizing polynomials based on the sums of coefficients
The document contains examples and explanations of key polynomial concepts over multiple pages.
Factor Theorem and Remainder Theorem. Mathematics10 Project under Mrs. Marissa De Ocampo. Prepared by Danielle Diva, Ronalie Mejos, Rafael Vallejos and Mark Lenon Dacir of 10- Einstein. CNSTHS.
The remainder theorem states that when a polynomial f(x) is divided by (x - c), the remainder is equal to f(c). This is based on the concept of synthetic division, where a polynomial is divided into a quotient and remainder. Some examples are worked out to demonstrate evaluating a polynomial f(x) at a value c to find the remainder when divided by (x - c), in accordance with the remainder theorem.
This document contains an unsolved mathematics paper from 2010 for the IIT JEE entrance exam. It has 4 sections - multiple choice questions with single answers, multiple choice questions with multiple possible answers, paragraph style questions, and questions requiring integer answers. The questions cover topics like complex numbers, matrices, geometry, trigonometry, and hyperbolas. An accurate summary is provided in 3 sentences or less.
The document discusses evaluating the formula log[(2x+1)/(sin1/3(x)+1)] at x=0 and x=10 degrees using a scientific calculator. It explains that the answer is 0 at x=0 and approximately 1.13 at x=10 degrees. It then describes the keyboard of a typical scientific calculator, noting the number, operation, yx, sin, log and formula keys. The rest of the document provides examples and definitions of algebraic, trigonometric and exponential-log formulas.
The document discusses the factor theorem and how to determine if a polynomial is a factor of another polynomial. It provides examples of using the factor theorem to show that (x + 1) is a factor of 2x^3 + 5x^2 - 3 and that (x - 2) is a factor of x^4 + x^3 - x^2 - x - 18. It also gives an example of finding a polynomial function given its zeros as -2, 1, -1. The document provides exercises for using the factor theorem to determine unknown values in polynomials.
This document discusses algebraic expressions and polynomial expressions. It provides examples of algebraic expressions and defines them as formulas constructed with variables and numbers using basic arithmetic operations. Polynomials are defined as expressions of the form anxn + an-1xn-1 + ... + a1x + a0, where the ai's are numbers. The document gives examples of factoring polynomials and evaluating polynomial expressions at given values. It also discusses using factoring to find the roots of polynomial equations.
The document discusses arithmetic sequences and provides examples to illustrate how to determine if a sequence is arithmetic, derive the specific formula for an arithmetic sequence from the general formula, and use the specific formula to calculate future terms. It defines an arithmetic sequence as one where the terms follow a linear formula of an = d*n + c. Examples show how to identify the common difference d between terms and plug into the general formula along with the first term a1 to derive the specific formula for different sequences.
The document discusses various transformations that can be performed on graphs of functions, including vertical translations, stretches and compressions, and vertical reflections. Vertical translations move the graph up or down by adding or subtracting a constant value to the output. Stretches and compressions multiply the output by a constant value greater than or less than 1, respectively. Reflecting the output about the x-axis vertically reflects the entire graph. These transformations can be represented by modifying the original function definition.
This document defines polynomials and discusses their key properties. It begins by defining a polynomial as an algebraic expression with two or more terms where the power of each variable is a positive integer. The degree of a polynomial is defined as the highest power of the variable. Polynomials are then classified based on their degree as constant, linear, quadratic, cubic, etc. The document also discusses the zeros or roots of a polynomial, which are the values that make the polynomial equal to zero. It shows how the zeros relate to the coefficients of the polynomial and can be found using factoring or solving techniques. Examples are provided to illustrate dividing polynomials using the division algorithm.
This document discusses factorials and the binomial theorem. It begins by defining factorials and providing examples of simplifying expressions with factorials. It then explains the binomial theorem, which gives a formula for expanding binomial expressions as binomial series. Specifically, it shows that the coefficients of terms in the binomial expansion can be determined using Pascal's triangle and factorials. It provides examples of using the binomial theorem to expand binomial expressions and find specific terms. In the examples, it demonstrates expanding binomials, finding coefficients, and determining terms with given exponents.
This document contains a presentation on polynomials. It defines what a polynomial is and discusses how the degree of a polynomial relates to the number of zeros. It provides an example of a cubic polynomial and discusses the relationship between the zeros and coefficients of a cubic polynomial. Specifically, it states that the sum of the zeros equals the negative of the coefficient of x^2, the sum of the products of the zeros equals the coefficient of x, and the product of the zeros equals the negative of the constant term. The document then provides two example questions - one asking to find the zeros of a quadratic polynomial and verify the relationship between the zeros and coefficients, and another asking to find a quadratic polynomial with given zeros.
The document discusses linear equations and how to graph them. It explains that linear equations relate the x-coordinate and y-coordinate of points in a straight line. To graph a linear equation, one finds ordered pairs that satisfy the equation by choosing values for x and solving for y, then plots the points. An example demonstrates graphing the linear equation y = 2x - 5 by making a table of x and y values and plotting the line.
t5 graphs of trig functions and inverse trig functionsmath260
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive functioning. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms.
The document contains questions related to trigonometric functions, sets, relations and functions, complex numbers, and sequences and series. Some questions ask students to prove trigonometric identities, find sets operations, determine if relations are functions, solve complex equations, and evaluate infinite geometric series. The document provides hints for many questions and includes the answers for some questions.
The document discusses methods for graphing quadratic equations. It explains that the graphs of quadratic equations are called parabolas, which are symmetric around a center line with a highest or lowest point called the vertex. It provides an example of graphing the equation y = x^2 - 4x - 12 by first finding the vertex, then making a table of symmetric x and y values around the vertex and plotting the points. The document also describes how to find the x-intercepts and y-intercepts of a parabola and provides an alternate two-step method for graphing a parabola.
The document discusses several topics in algebra including:
1. Indices laws including am x an = am + n, am ÷ an = am - n, and (am)n = amn. Negative and fractional indices are also discussed.
2. Logarithms including the definition that logarithm of 'x' to base 'a' is the power to which 'a' must be raised to give 'x'. Change of base formula is also provided.
3. Series including the definition of finite and infinite series. Notation of sigma notation ∑ is introduced to represent the sum of terms.
Guidelines for Modelling Groundwater Surface Water Interaction in eWater SourceeWater
One of the key challenges in modelling GW-SW interactions is the significant time-scale
differences between surface water and groundwater processes. Because groundwater
movement can be orders of magnitude slower than surface water movement, the
responses of groundwater systems to hydrological and management drivers such as
climate variability, land use change, and groundwater extraction can be very damped and
lagged. Hence, a key requirement in modelling GW-SW interactions in river system
models is to account for these time lags.
The modelling of GW-SW interactions in river system models is still very much in its
infancy, not just in Australia, but also throughout the world. As such, there is no consensus
on implementation of this functionality in river system models, and hence the little
discussion in the literature so far on what constitutes Best Practice Modelling in this
domain.
This document provides an introduction to PHP by summarizing its history and key features. PHP was created in 1994 by Rasmus Lerdorf as a set of Common Gateway Interface scripts for tracking visits to his online resume. It has since evolved into a full-featured programming language used widely by major companies like Google, Facebook, and Bank of America. The document outlines PHP's core syntax like variables, constants, includes, and flow control structures. It also discusses databases, MVC patterns, classes, and tools that employers seek like contributions to open source projects.
This document discusses distance, circles, and quadratic equations in three parts:
1) It derives the formula for finding the distance between two points in a plane as the square root of the sum of the squares of the differences of their x- and y-coordinates.
2) It derives the midpoint formula for finding the midpoint between two points as the average of their x-coordinates and the average of their y-coordinates.
3) It discusses the standard equation of a circle, gives methods for finding the center and radius from different forms of the circle equation, and notes degenerate cases where the equation does not represent a circle.
The document discusses factorable polynomials and how to graph them. It defines a factorable polynomial as one that can be written as the product of linear factors using real numbers. For large values of x, the leading term of a polynomial dominates so the graph resembles that of the leading term. To graph a factorable polynomial, one first graphs the individual factors like x^n and then combines them, which gives smooth curves tending to the graphs of the leading terms for large x.
The document discusses functions and their graphical representations. It defines key terms like domain, range, and one-to-one and many-to-one mappings. It then focuses on quadratic functions, showing that their graphs take characteristic U-shaped or inverted U-shaped forms. The document also examines inequalities involving quadratic expressions and how to determine the range of values satisfying such inequalities by analyzing the graph of the quadratic function.
The document describes aspects of polynomials including:
- Definitions of polynomials and their components
- Algebraic operations on polynomials like addition, subtraction, and multiplication
- Dividing polynomials using long division and synthetic (Horner's) division methods
- The remainder theorem stating the remainder of dividing a polynomial by (x - k) is the value of the polynomial at k
- The factorization theorem relating the factors of a polynomial to its roots
- Techniques for factorizing polynomials based on the sums of coefficients
The document contains examples and explanations of key polynomial concepts over multiple pages.
Factor Theorem and Remainder Theorem. Mathematics10 Project under Mrs. Marissa De Ocampo. Prepared by Danielle Diva, Ronalie Mejos, Rafael Vallejos and Mark Lenon Dacir of 10- Einstein. CNSTHS.
The remainder theorem states that when a polynomial f(x) is divided by (x - c), the remainder is equal to f(c). This is based on the concept of synthetic division, where a polynomial is divided into a quotient and remainder. Some examples are worked out to demonstrate evaluating a polynomial f(x) at a value c to find the remainder when divided by (x - c), in accordance with the remainder theorem.
This document contains an unsolved mathematics paper from 2010 for the IIT JEE entrance exam. It has 4 sections - multiple choice questions with single answers, multiple choice questions with multiple possible answers, paragraph style questions, and questions requiring integer answers. The questions cover topics like complex numbers, matrices, geometry, trigonometry, and hyperbolas. An accurate summary is provided in 3 sentences or less.
The document discusses evaluating the formula log[(2x+1)/(sin1/3(x)+1)] at x=0 and x=10 degrees using a scientific calculator. It explains that the answer is 0 at x=0 and approximately 1.13 at x=10 degrees. It then describes the keyboard of a typical scientific calculator, noting the number, operation, yx, sin, log and formula keys. The rest of the document provides examples and definitions of algebraic, trigonometric and exponential-log formulas.
The document discusses the factor theorem and how to determine if a polynomial is a factor of another polynomial. It provides examples of using the factor theorem to show that (x + 1) is a factor of 2x^3 + 5x^2 - 3 and that (x - 2) is a factor of x^4 + x^3 - x^2 - x - 18. It also gives an example of finding a polynomial function given its zeros as -2, 1, -1. The document provides exercises for using the factor theorem to determine unknown values in polynomials.
This document discusses algebraic expressions and polynomial expressions. It provides examples of algebraic expressions and defines them as formulas constructed with variables and numbers using basic arithmetic operations. Polynomials are defined as expressions of the form anxn + an-1xn-1 + ... + a1x + a0, where the ai's are numbers. The document gives examples of factoring polynomials and evaluating polynomial expressions at given values. It also discusses using factoring to find the roots of polynomial equations.
The document discusses arithmetic sequences and provides examples to illustrate how to determine if a sequence is arithmetic, derive the specific formula for an arithmetic sequence from the general formula, and use the specific formula to calculate future terms. It defines an arithmetic sequence as one where the terms follow a linear formula of an = d*n + c. Examples show how to identify the common difference d between terms and plug into the general formula along with the first term a1 to derive the specific formula for different sequences.
The document discusses various transformations that can be performed on graphs of functions, including vertical translations, stretches and compressions, and vertical reflections. Vertical translations move the graph up or down by adding or subtracting a constant value to the output. Stretches and compressions multiply the output by a constant value greater than or less than 1, respectively. Reflecting the output about the x-axis vertically reflects the entire graph. These transformations can be represented by modifying the original function definition.
This document defines polynomials and discusses their key properties. It begins by defining a polynomial as an algebraic expression with two or more terms where the power of each variable is a positive integer. The degree of a polynomial is defined as the highest power of the variable. Polynomials are then classified based on their degree as constant, linear, quadratic, cubic, etc. The document also discusses the zeros or roots of a polynomial, which are the values that make the polynomial equal to zero. It shows how the zeros relate to the coefficients of the polynomial and can be found using factoring or solving techniques. Examples are provided to illustrate dividing polynomials using the division algorithm.
This document discusses factorials and the binomial theorem. It begins by defining factorials and providing examples of simplifying expressions with factorials. It then explains the binomial theorem, which gives a formula for expanding binomial expressions as binomial series. Specifically, it shows that the coefficients of terms in the binomial expansion can be determined using Pascal's triangle and factorials. It provides examples of using the binomial theorem to expand binomial expressions and find specific terms. In the examples, it demonstrates expanding binomials, finding coefficients, and determining terms with given exponents.
This document contains a presentation on polynomials. It defines what a polynomial is and discusses how the degree of a polynomial relates to the number of zeros. It provides an example of a cubic polynomial and discusses the relationship between the zeros and coefficients of a cubic polynomial. Specifically, it states that the sum of the zeros equals the negative of the coefficient of x^2, the sum of the products of the zeros equals the coefficient of x, and the product of the zeros equals the negative of the constant term. The document then provides two example questions - one asking to find the zeros of a quadratic polynomial and verify the relationship between the zeros and coefficients, and another asking to find a quadratic polynomial with given zeros.
The document discusses linear equations and how to graph them. It explains that linear equations relate the x-coordinate and y-coordinate of points in a straight line. To graph a linear equation, one finds ordered pairs that satisfy the equation by choosing values for x and solving for y, then plots the points. An example demonstrates graphing the linear equation y = 2x - 5 by making a table of x and y values and plotting the line.
t5 graphs of trig functions and inverse trig functionsmath260
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive functioning. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms.
The document contains questions related to trigonometric functions, sets, relations and functions, complex numbers, and sequences and series. Some questions ask students to prove trigonometric identities, find sets operations, determine if relations are functions, solve complex equations, and evaluate infinite geometric series. The document provides hints for many questions and includes the answers for some questions.
The document discusses methods for graphing quadratic equations. It explains that the graphs of quadratic equations are called parabolas, which are symmetric around a center line with a highest or lowest point called the vertex. It provides an example of graphing the equation y = x^2 - 4x - 12 by first finding the vertex, then making a table of symmetric x and y values around the vertex and plotting the points. The document also describes how to find the x-intercepts and y-intercepts of a parabola and provides an alternate two-step method for graphing a parabola.
The document discusses several topics in algebra including:
1. Indices laws including am x an = am + n, am ÷ an = am - n, and (am)n = amn. Negative and fractional indices are also discussed.
2. Logarithms including the definition that logarithm of 'x' to base 'a' is the power to which 'a' must be raised to give 'x'. Change of base formula is also provided.
3. Series including the definition of finite and infinite series. Notation of sigma notation ∑ is introduced to represent the sum of terms.
Guidelines for Modelling Groundwater Surface Water Interaction in eWater SourceeWater
One of the key challenges in modelling GW-SW interactions is the significant time-scale
differences between surface water and groundwater processes. Because groundwater
movement can be orders of magnitude slower than surface water movement, the
responses of groundwater systems to hydrological and management drivers such as
climate variability, land use change, and groundwater extraction can be very damped and
lagged. Hence, a key requirement in modelling GW-SW interactions in river system
models is to account for these time lags.
The modelling of GW-SW interactions in river system models is still very much in its
infancy, not just in Australia, but also throughout the world. As such, there is no consensus
on implementation of this functionality in river system models, and hence the little
discussion in the literature so far on what constitutes Best Practice Modelling in this
domain.
This document provides an introduction to PHP by summarizing its history and key features. PHP was created in 1994 by Rasmus Lerdorf as a set of Common Gateway Interface scripts for tracking visits to his online resume. It has since evolved into a full-featured programming language used widely by major companies like Google, Facebook, and Bank of America. The document outlines PHP's core syntax like variables, constants, includes, and flow control structures. It also discusses databases, MVC patterns, classes, and tools that employers seek like contributions to open source projects.
This document discusses distance, circles, and quadratic equations in three parts:
1) It derives the formula for finding the distance between two points in a plane as the square root of the sum of the squares of the differences of their x- and y-coordinates.
2) It derives the midpoint formula for finding the midpoint between two points as the average of their x-coordinates and the average of their y-coordinates.
3) It discusses the standard equation of a circle, gives methods for finding the center and radius from different forms of the circle equation, and notes degenerate cases where the equation does not represent a circle.
Survey Grade LiDAR Technologies for Transportation EngineeringQuantum Spatial
This presentation was given during the 2013 Annual Civil Engineering Conference by Tim Stagg of AeroMetric. It covers system/sensor configurations, application advantages/disadvantages, analysis from sensor data, feature extraction/deliverables, and client pains in relation to survey grade LiDAR technologies for transportation engineering.
WE1.L10 - GRACE Applications to Regional Hydrology and Water Resourcesgrssieee
This document summarizes the applications of NASA's GRACE mission for monitoring regional hydrology and water resources. GRACE uses two satellites to measure small changes in Earth's gravity field caused by the redistribution of water on land and oceans. GRACE data has been used to monitor seasonal water storage changes, depleting groundwater aquifers, declining glaciers and ice sheets, and rising sea levels. Ensuring continuity of GRACE measurements is important for long-term climate monitoring, and NASA has proposed a GRACE Follow-On mission to launch in 2016 to fill the gap until next-generation gravity missions.
The document discusses multimedia content on the web including images, audio, animated content, and using the Windows Media Player. It explains how browsers handle different media types using plug-ins and helper applications. It also describes common image formats, compressed audio formats, streaming audio, creating animations with JavaScript, Flash and video. Finally, it outlines the features and functionality of the Windows Media Player.
The document discusses various types of computer input devices such as keyboards, mice, touchscreens, scanners, cameras, and biometric devices. It describes how these devices work and are used for entering data, images, video, and instructions into computers. Examples of recommended input configurations are provided for different types of users including home users, small office/home office users, mobile users, and power users.
Chapter 4 Form Factors & Power SuppliesPatty Ramsey
The document discusses computer form factors, power supplies, and electrical troubleshooting. It covers different form factors like ATX, microATX and BTX that specify motherboard and case dimensions. Power supplies convert alternating current to direct current needed by components. Electrical issues can be caused by static electricity, electromagnetic interference, power surges or inadequate power supplies. Troubleshooting involves checking for loose connections, overheating, defective fans or capacitors, and replacing the power supply if needed.
The document discusses different computer components related to power and electricity, including form factors, power supplies, cases, and how electricity is measured. It covers topics like ATX, MicroATX, and BTX form factors; desktop and tower cases; voltages, amps, ohms, and watts; AC and DC power; surge protection; UPS systems; and how computers meet Energy Star standards through power management features. The document provides information to help understand and troubleshoot electrical issues in personal computers.
This chapter discusses synchronous communication technologies including Internet Relay Chat (IRC), instant messaging, video conferencing, and Voice over Internet Protocol (VoIP). It defines these technologies, explains how they work, and provides examples of using IRC clients, Windows Live Messenger, Skype, and other synchronous communication software.
This document defines and provides examples of absolute value, including:
- Absolute value strips away the minus sign if a number is negative and leaves it unchanged if nonnegative.
- The absolute value of a product is the product of the absolute values.
- Absolute value has a geometric interpretation as distance on a number line.
- Important inequalities involving absolute value, like |x-a|<k, have solution sets that consist of values within k units of a.
- The triangle inequality states the absolute value of a sum is less than or equal to the sum of the absolute values.
The document discusses emoji and how they are used internationally to exchange ideas. It explains that emoji are part of Unicode so all devices can understand the same pictures. It notes that while Unicode provides code names for emoji in English, there is no source for official names in other languages. The document proposes creating an Emoji International Name Finder project to collect emoji names in different languages and share them online to help people understand emoji in languages other than English.
Discussion of the science, collection and availability of lidar, specifically topobathymetric lidar. Use of NOAA/USGS Interagency Elevation Inventory leveraged
The document discusses new features in PHP 5.3 including new syntax, functions, extensions, and changes from previous versions. Some key changes include new array, date, and PHP core functions; the addition of namespaces, lambda functions, and closures; and removal of support for certain Windows versions and extensions. Backwards compatibility issues are addressed for magic methods, reserved words, and deprecations.
The document provides instructions on how to execute C# programs and includes examples of simple C# programs. It discusses installing the .NET framework, setting the PATH variable, using commands like "csc" and "dotnet" to compile and run programs. Example programs shown include adding two numbers, displaying command line arguments, using the Math class, and handling exceptions. Custom exception classes are also demonstrated.
This document is about Unix commands for bioinformaticians. It discusses Unix folders and files, processes, and redirection. It provides examples of commands for listing, moving, copying, reading and editing files. It also demonstrates running processes, controlling processes, and redirecting inputs/outputs. The goal is to introduce basic Unix skills like navigating the filesystem, working with files, and running programs needed for bioinformatics tasks.
This chapter discusses accessing information resources on the web, including the difference between the surface web and deep web. It covers various search tools like search engines, subject directories, and meta search engines. Boolean logic and search syntax are explained to refine queries. Advanced search features and evaluating results are also summarized. Methods to define search questions and formulate strategies are provided to efficiently find relevant information online.
This document provides an overview of trigonometric functions and identities. It defines angles and their measurement in degrees and radians. It discusses trigonometric functions for right triangles, extending the definitions to angles in rectangular coordinate systems. Examples are provided to illustrate evaluating trigonometric functions of various angles. Key relationships between arc length, angle, radius, and area are also summarized.
The document provides an introduction to PHP, covering topics such as what PHP is, PHP files, why use PHP, PHP requests, the PHP development environment, PHP syntax, comments, mixing PHP with HTML, variables, operators, control structures like if/else statements, loops, functions, arrays, cookies, and sessions. PHP is introduced as a server-side scripting language used to build dynamic web applications. Key aspects explained include PHP files containing HTML, text, and scripts which are executed on the server and returned to the browser as HTML.
- Polynomials are expressions constructed from variables and constants with non-negative whole number exponents.
- The degree of a polynomial is the highest exponent among its terms. Zeroes are values that make the polynomial equal to zero.
- There is a relationship between the number of zeroes a polynomial can have and its degree. Linear polynomials have at most 1 zero, quadratics have at most 2 zeros, and cubics have at most 3 zeros.
- The coefficients of a polynomial are related to its zeroes through formulas involving the sum and product of the zeroes.
The document explains the Remainder Theorem in multiple ways using different examples and proofs. It states that the Remainder Theorem provides a test to determine if a polynomial f(x) is divisible by a polynomial of the form x-c. It proves that the remainder obtained when dividing f(x) by x-c is equal to the value of f(x) when x is substituted with c. It provides multiple examples working through applying the Remainder Theorem to determine if various polynomials are divisible.
The document discusses the Remainder Theorem, which states that the remainder obtained when dividing a polynomial f(x) by a linear polynomial x-c is equal to the value of f(x) when x is substituted with c. It provides examples to show how to use the Remainder Theorem to determine whether a given polynomial is divisible by x-c. The document also contains practice exercises for readers to apply the Remainder Theorem.
This PowerPoint presentation covers polynomials, including:
- Definitions of polynomials, monomials, binomials, trinomials, and the degree of a polynomial.
- The geometric meaning of zeros of polynomials - linear polynomials have one zero, quadratics have up to two zeros, and cubics have up to three.
- The relationship between the zeros and coefficients of a quadratic polynomial - the sum of the zeros equals the negative of the coefficient of x divided by the coefficient of x^2, and the product of the zeros equals the constant term divided by the coefficient of x^2.
- The division algorithm for polynomials - any polynomial p(x) can be divided by a non-zero polynomial
The document defines different types of polynomials and their key properties. It discusses linear, quadratic, and cubic polynomials, and defines them based on their highest degree term. It also covers the degree of a polynomial, zeros of polynomials, and the relationship between the zeros and coefficients of quadratic and cubic polynomials. Finally, it discusses the division algorithm for polynomials.
This document discusses algebraic fractions and polynomials. It covers dividing polynomials by monomials and other polynomials. The key steps of polynomial long division and Ruffini's rule for polynomial division are explained. Finding the quotient, remainder, and whether a polynomial is divisible are discussed. Finding the roots of polynomials and using the remainder theorem are also covered. Various techniques for factorizing polynomials are presented, including taking out common factors, using identities, the fundamental theorem of algebra, and Ruffini's rule.
- The order of a root of a polynomial is the number of times the root repeats.
- The polynomial x5 + 2x4 + x3 has two roots, x = 0 with order 3 and x = -1 with order 2.
- In general, polynomials of the form k(x - c1)m(x - c2)m...(x - cn)m have roots x = c1 with order m1, x = c2 with order m2, and so on.
Quotient of polynomial using (Synthetic Division) and Zeros of Polynomial Fu...magnesium121
This document discusses using synthetic division to find the quotient and remainder when dividing polynomials. It provides the step-by-step process for synthetic division, including examples of dividing polynomials where the divisor is of the form x + c and ax + c. The key steps are writing the coefficients of the dividend as the first row, taking the negative of the constant term of the divisor as the multiplier, bringing down terms and multiplying and adding sequentially until the last term. The numbers in the third row give the quotient polynomial. The document also discusses using factoring and setting each factor equal to 0 to find the zeros of a polynomial function.
- The order of a root of a polynomial is the number of times the root repeats.
- The polynomial x5 + 2x4 + x3 has two roots, x = 0 with order 3 and x = -1 with order 2.
- In general, polynomials of the form k(x - c1)m(x - c2)m...(x - cn)m have roots x = c1 with order m1, x = c2 with order m2, and so on.
This document discusses using synthetic division and the remainder theorem to find the value of polynomial functions at given points. It provides examples of using both synthetic division and the remainder theorem to find the value of polynomials like P(x) = 2x^3 - 8x^2 + 19x - 12 at x = 3. The key points are that the remainder R obtained from synthetic division gives the value of the polynomial function at the given point, f(c), and that if R = 0, then x - c is a factor of the polynomial. Exercises are provided to have students practice finding polynomial values using these methods.
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This module discusses polynomial functions of degree greater than two. The key points are:
1. The graph of a third-degree polynomial has both a minimum and maximum point, while higher degree polynomials have one less turning point than their degree.
2. Methods like finding upper and lower bounds and Descartes' Rule of Signs can help determine properties of the graph like zeros.
3. Odd degree polynomials increase on the far left and right if the leading term is positive, and decrease if negative. Even degree polynomials increase on the far left and decrease on the far right, or vice versa.
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- Terms for integers, fractions, real and complex numbers, exponents, and basic arithmetic operations.
- Algebraic expressions, indices, matrices, inequalities, polynomial equations, and congruences.
- The use of definite and indefinite articles for theorems, conjectures, and mathematical concepts.
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1. November 4, 2004 15:00 k34-appb Sheet number 1 Page number 15 cyan magenta yellow black
A15
a p p e n d i x b
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 book.
A BRIEF REVIEW OF POLYNOMIALS
Recall that if n is a nonnegative integer, then a polynomial of degree n is a function that
can be written in the following forms, depending on whether you want the powers of x in
ascending or descending order:
c0 + c1x + c2x2
+ · · · + cnxn
(cn = 0)
cnxn
+ cn−1xn−1
+ · · · + c1x + c0 (cn = 0)
The numbers c0, c1, . . . , cn are called the coefficients of the polynomial. The coefficient cn
(which multiplies the highest power of x) is called the leading coefficient, the term cnxn
is
called the leading term, and the coefficient c0 is called the constant term. Polynomials of
degree 1, 2, 3, 4, and 5 are called linear, quadratic, cubic, quartic, and quintic, respectively.
For simplicity, general polynomials of low degree are often written without subscripts on
the coefficients:
p(x) = a Constant polynomial
p(x) = ax + b (a = 0) Linear polynomial
p(x) = ax2
+ bx + c (a = 0) Quadratic polynomial
p(x) = ax3
+ bx2
+ cx + d (a = 0) Cubic polynomial
When you attempt to factor a polynomial completely, one of three things can happen:
• You may be able to decompose the polynomial into distinct linear factors using only real
numbers; for example,
x3
+ x2
− 2x = x(x2
+ x − 2) = x(x − 1)(x + 2)
• You may be able to decompose the polynomial into linear factors using only real numbers,
but some of the factors may be repeated; for example,
x6
− 3x4
+ 2x3
= x3
(x3
− 3x + 2) = x3
(x − 1)2
(x + 2) (1)
• You may be able to decompose the polynomial into linear and quadratic factors using
only real numbers, but you may not be able to decompose the quadratic factors into
linear factors using only real numbers (such quadratic factors are said to be irreducible
over the real numbers); for example,
x4
− 1 = (x2
− 1)(x2
+ 1) = (x − 1)(x + 1)(x2
+ 1)
= (x − 1)(x + 1)(x − i)(x + i)
Here, the factor x2
+ 1 is irreducible over the real numbers.
2. November 4, 2004 15:00 k34-appb Sheet number 2 Page number 16 cyan magenta yellow black
A16 Appendix B: Solving Polynomial Equations
In general, if p(x) is a polynomial of degree n with leading coefficient a, and if complex
numbers are allowed, then p(x) can be factored as
p(x) = a(x − r1)(x − r2) · · · (x − rn) (2)
where r1, r2, . . . , rn are called the zeros of p(x) or the roots of the equation p(x) = 0, and
(2) is called the complete linear factorization of p(x). If some of the factors in (2) are
repeated, then they can be combined; for example, if the first k factors are distinct and the
rest are repetitions of the first k, then (2) can be expressed in the form
p(x) = a(x − r1)m1
(x − r2)m2
· · · (x − rk)mk
(3)
wherer1, r2, . . . , rk arethedistinct rootsofp(x) = 0. Theexponentsm1, m2, . . . , mk tellus
how many times the various factors occur in the complete linear factorization; for example,
in (3) the factor (x − r1) occurs m1 times, the factor (x − r2) occurs m2 times, and so forth.
Some techniques for factoring polynomials are discussed later in this appendix. In general,
if a factor (x − r) occurs m times in the complete linear factorization of a polynomial, then
we say that r is a root or zero of multiplicity m, and if (x − r) has no repetitions (i.e., r has
multiplicity 1), then we say that r is a simple root or zero. For example, it follows from (1)
that the equation x6
− 3x4
+ 2x3
= 0 can be expressed as
x3
(x − 1)2
(x + 2) = 0 (4)
so this equation has three distinct roots—a root x = 0 of multiplicity 3, a root x = 1 of mul-
tiplicity 2, and a simple root x = −2.
Note that in (3) the multiplicities of the roots must add up to n, since p(x) has degree n;
that is,
m1 + m2 + · · · + mk = n
For example, in (4) the multiplicities add up to 6, which is the same as the degree of the
polynomial.
It follows from (2) that a polynomial of degree n can have at most n distinct roots; if all
of the roots are simple, then there will be exactly n, but if some are repeated, then there will
be fewer than n. However, when counting the roots of a polynomial, it is standard practice
to count multiplicities, since that convention allows us to say that a polynomial of degree n
has n roots. For example, from (1) the six roots of the polynomial p(x) = x6
− 3x4
+ 2x3
are
r = 0, 0, 0, 1, 1, −2
In summary, we have the following important theorem.
B.1 theorem. If complex roots are allowed, and if roots are counted according to
their multiplicities, then a polynomial of degree n has exactly n roots.
THE REMAINDER THEOREM
When two positive integers are divided, the numerator can be expressed as the quotient plus
the remainder over the divisor, where the remainder is less than the divisor. For example,
17
5
= 3 + 2
5
If we multiply this equation through by 5, we obtain
17 = 5 · 3 + 2
which states that the numerator is the divisor times the quotient plus the remainder.
3. November 4, 2004 15:00 k34-appb Sheet number 3 Page number 17 cyan magenta yellow black
Appendix B: Solving Polynomial Equations A17
The following theorem, which we state without proof, is an analogous result for division
of polynomials.
B.2 theorem. If p(x) and s(x) are polynomials, and if s(x) is not the zero poly-
nomial, then p(x) can be expressed as
p(x) = s(x)q(x) + r(x)
where q(x) and r(x) are the quotient and remainder that result when p(x) is divided by
s(x), and either r(x) is the zero polynomial or the degree of r(x) is less than the degree
of s(x).
In the special case where p(x) is divided by a first-degree polynomial of the form x − c,
the remainder must be some constant r, since it is either zero or has degree less than 1.
Thus, Theorem B.2 implies that
p(x) = (x − c)q(x) + r
and this in turn implies that p(c) = r. In summary, we have the following theorem.
B.3 theorem (Remainder Theorem). If a polynomial p(x) is divided by x − c,
then the remainder is p(c).
Example 1 According to the Remainder Theorem, the remainder on dividing
p(x) = 2x3
+ 3x2
− 4x − 3
by x + 4 should be
p(−4) = 2(−4)3
+ 3(−4)2
− 4(−4) − 3 = −67
Show that this is so.
Solution. By long division
2x2
− 5x + 16
x + 4|2x3
+ 3x2
− 4x − 3
2x3
+ 8x2
−5x2
− 4x
−5x2
− 20x
16x − 3
16x + 64
−67
which shows that the remainder is −67.
Alternative Solution. Because we are dividing by an expression of the form x − c
(where c = −4), we can use synthetic division rather than long division. The computations
are −4| 2 3 −4 −3
−8 20 −64
2 −5 16 −67
which again shows that the remainder is −67.
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A18 Appendix B: Solving Polynomial Equations
THE FACTOR THEOREM
To factor a polynomial p(x) is to write it as a product of lower-degree polynomials, called
factors of p(x). For s(x) to be a factor of p(x) there must be no remainder when p(x) is
divided by s(x). For example, if p(x) can be factored as
p(x) = s(x)q(x) (5)
then
p(x)
s(x)
= q(x) (6)
so dividing p(x) by s(x) produces a quotient q(x) with no remainder. Conversely, (6)
implies (5), so s(x) is a factor of p(x) if there is no remainder when p(x) is divided by
s(x).
In the special case where x − c is a factor of p(x), the polynomial p(x) can be expressed
as
p(x) = (x − c)q(x)
which implies that p(c) = 0. Conversely, if p(c) = 0, then the RemainderTheorem implies
that x − c is a factor of p(x), since the remainder is 0 when p(x) is divided by x − c. These
results are summarized in the following theorem.
B.4 theorem (Factor Theorem). A polynomial p(x) has a factor x − c if and only
if p(c) = 0.
It follows from this theorem that the statements below say the same thing in different
ways:
• x − c is a factor of p(x).
• p(c) = 0.
• c is a zero of p(x).
• c is a root of the equation p(x) = 0.
• c is a solution of the equation p(x) = 0.
• c is an x-intercept of y = p(x).
Example 2 Confirm that x − 1 is a factor of
p(x) = x3
− 3x2
− 13x + 15
by dividing x − 1 into p(x) and checking that the remainder is zero.
Solution. By long division
x2
− 2x − 15
x − 1|x3
− 3x2
− 13x + 15
x3
− x2
−2x2
− 13x
−2x2
+ 2x
−15x + 15
−15x + 15
0
which shows that the remainder is zero.
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Appendix B: Solving Polynomial Equations A19
Alternative Solution. Because we are dividing by an expression of the form x − c, we
can use synthetic division rather than long division. The computations are
1| 1 −3 −13 15
1 −2 −15
1 −2 −15 0
which again confirms that the remainder is zero.
USING ONE FACTOR TO FIND OTHER FACTORS
If x − c is a factor of p(x), and if q(x) = p(x)/(x − c), then
p(x) = (x − c)q(x) (7)
so that additional linear factors of p(x) can be obtained by factoring the quotient q(x).
Example 3 Factor
p(x) = x3
− 3x2
− 13x + 15 (8)
completely into linear factors.
Solution. We showed in Example 2 that x − 1 is a factor of p(x) and we also showed
that p(x)/(x − 1) = x2
− 2x − 15. Thus,
x3
− 3x2
− 13x + 15 = (x − 1)(x2
− 2x − 15)
Factoring x2
− 2x − 15 by inspection yields
x3
− 3x2
− 13x + 15 = (x − 1)(x − 5)(x + 3)
which is the complete linear factorization of p(x).
METHODS FOR FINDING ROOTS
A general quadratic equation ax2
+ bx + c = 0 can be solved by using the quadratic for-
mula to express the solutions of the equation in terms of the coefficients. Versions of this
formula were known since Babylonian times, and by the seventeenth century formulas had
been obtained for solving general cubic and quartic equations. However, attempts to find
formulas for the solutions of general fifth-degree equations and higher proved fruitless.
The reason for this became clear in 1829 when the French mathematician Evariste Galois
(1811–1832) proved that it is impossible to express the solutions of a general fifth-degree
equation or higher in terms of its coefficients using algebraic operations.
Today, we have powerful computer programs for finding the zeros of specific polynomi-
als. For example, it takes only seconds for a computer algebra system, such as Mathematica,
Maple, or Derive, to show that the zeros of the polynomial
p(x) = 10x4
− 23x3
− 10x2
+ 29x + 6 (9)
are
x = −1, x = −1
5
, x = 3
2
, and x = 2 (10)
The algorithms that these programs use to find the integer and rational zeros of a polynomial,
if any, are based on the following theorem, which is proved in advanced algebra courses.
6. November 4, 2004 15:00 k34-appb Sheet number 6 Page number 20 cyan magenta yellow black
A20 Appendix B: Solving Polynomial Equations
B.5 theorem. Suppose that
p(x) = cnxn
+ cn−1xn−1
+ · · · + c1x + c0
is a polynomial with integer coefficients.
(a) If r is an integer zero of p(x), then r must be a divisor of the constant term c0.
(b) If r = a/b is a rational zero of p(x) in which all common factors of a and b have
been canceled, then a must be a divisor of the constant term c0, and b must be a
divisor of the leading coefficient cn.
For example, in (9) the constant term is 6 (which has divisors ±1, ±2, ±3, and ±6) and
the leading coefficient is 10 (which has divisors ±1, ±2, ±5, and ±10). Thus, the only
possible integer zeros of p(x) are
±1, ±2, ±3, ±6
and the only possible noninteger rational zeros are
±1
2
, ±1
5
, ± 1
10
, ±2
5
, ±3
2
, ±3
5
, ± 3
10
, ±6
5
Using a computer, it is a simple matter to evaluate p(x) at each of the numbers in these lists
to show that its only rational zeros are the numbers in (10).
Example 4 Solve the equation x3
+ 3x2
− 7x − 21 = 0.
Solution. The solutions of the equation are the zeros of the polynomial
p(x) = x3
+ 3x2
− 7x − 21
We will look for integer zeros first. All such zeros must divide the constant term, so the
only possibilities are ±1, ±3, ±7, and ±21. Substituting these values into p(x) (or using
the method of Exercise 6) shows that x = −3 is an integer zero. This tells us that x + 3 is
a factor of p(x) and that p(x) can be written as
x3
+ 3x2
− 7x − 21 = (x + 3)q(x)
where q(x) is the quotient that results when x3
+ 3x2
− 7x − 21 is divided by x + 3. We
leave it for you to perform the division and show that q(x) = x2
− 7; hence,
x3
+ 3x2
− 7x − 21 = (x + 3)(x2
− 7) = (x + 3)(x +
√
7)(x −
√
7)
which tells us that the solutions of the given equation are x = 3, x =
√
7 ≈ 2.65, and
x = −
√
7 ≈ −2.65.
EXERCISE SET B C CAS
1–2 Findthequotientq(x)andtheremainder r(x)thatresult
when p(x) is divided by s(x).
1. (a) p(x) = x4
+ 3x3
− 5x + 10; s(x) = x2
− x + 2
(b) p(x) = 6x4
+ 10x2
+ 5; s(x) = 3x2
− 1
(c) p(x) = x5
+ x3
+ 1; s(x) = x2
+ x
2. (a) p(x) = 2x4
− 3x3
+ 5x2
+ 2x + 7; s(x) = x2
− x + 1
(b) p(x) = 2x5
+ 5x4
− 4x3
+ 8x2
+ 1; s(x) = 2x2
− x + 1
(c) p(x) = 5x6
+ 4x2
+ 5; s(x) = x3
+ 1
3–4 Use synthetic division to find the quotient q(x) and the
remainder r that result when p(x) is divided by s(x).
3. (a) p(x) = 3x3
− 4x − 1; s(x) = x − 2
(b) p(x) = x4
− 5x2
+ 4; s(x) = x + 5
(c) p(x) = x5
− 1; s(x) = x − 1
4. (a) p(x) = 2x3
− x2
− 2x + 1; s(x) = x − 1
(b) p(x) = 2x4
+ 3x3
− 17x2
− 27x − 9; s(x) = x + 4
(c) p(x) = x7
+ 1; s(x) = x − 1
7. November 4, 2004 15:00 k34-appb Sheet number 7 Page number 21 cyan magenta yellow black
Appendix B: Solving Polynomial Equations A21
5. Let p(x) = 2x4
+ x3
− 3x2
+ x − 4. Use synthetic divi-
sion and the Remainder Theorem to find p(0), p(1), p(−3),
and p(7).
6. Let p(x) be the polynomial in Example 4. Use synthetic
division and the Remainder Theorem to evaluate p(x) at
x = ±1, ±3, ±7, and ±21.
7. Let p(x) = x3
+ 4x2
+ x − 6. Find a polynomial q(x) and
a constant r such that
(a) p(x) = (x − 2)q(x) + r
(b) p(x) = (x + 1)q(x) + r.
8. Let p(x) = x5
− 1. Find a polynomial q(x) and a constant
r such that
(a) p(x) = (x + 1)q(x) + r
(b) p(x) = (x − 1)q(x) + r.
9. In each part, make a list of all possible candidates for the
rational zeros of p(x).
(a) p(x) = x7
+ 3x3
− x + 24
(b) p(x) = 3x4
− 2x2
+ 7x − 10
(c) p(x) = x35
− 17
10. Find all integer zeros of
p(x) = x6
+ 5x5
− 16x4
− 15x3
− 12x2
− 38x − 21
11–15 Factor the polynomials completely.
11. p(x) = x3
− 2x2
− x + 2
12. p(x) = 3x3
+ x2
− 12x − 4
13. p(x) = x4
+ 10x3
+ 36x2
+ 54x + 27
14. p(x) = 2x4
+ x3
+ 3x2
+ 3x − 9
15. p(x) = x5
+ 4x4
− 4x3
− 34x2
− 45x − 18
16.C For each of the factorizations that you obtained in Exercises
11–15, check your answer using a CAS.
17–21 Find all real solutions of the equations.
17. x3
+ 3x2
+ 4x + 12 = 0
18. 2x3
− 5x2
− 10x + 3 = 0
19. 3x4
+ 14x3
+ 14x2
− 8x − 8 = 0
20. 2x4
− x3
− 14x2
− 5x + 6 = 0
21. x5
− 2x4
− 6x3
+ 5x2
+ 8x + 12 = 0
22.C For each of the equations you solved in Exercises 17–21,
check your answer using a CAS.
23. Find all values of k for which x − 1 is a factor of the poly-
nomial p(x) = k2
x3
− 7kx + 10.
24. Is x + 3 a factor of x7
+ 2187? Justify your answer.
25.C A 3-cm-thick slice is cut from a cube, leaving a volume of
196 cm3
. Use a CAS to find the length of a side of the
original cube.
26. (a) Show that there is no positive rational number that ex-
ceeds its cube by 1.
(b) Does there exist a real number that exceeds its cube by
1? Justify your answer.
27. Use the Factor Theorem to show each of the following.
(a) x − y is a factor of xn
− yn
for all positive integer val-
ues of n.
(b) x + y is a factor of xn
− yn
for all positive even integer
values of n.
(c) x + y is a factor of xn
+ yn
for all positive odd integer
values of n.