Section A.6. Solving Equations. Math Precalculus I. Solving Equations Section A.6
|
|
- Emil Carroll
- 6 years ago
- Views:
Transcription
1 Section A.6 Solving Equations Math Precalculus I
2 A.6 Solving Equations Simplify 1 2x 6 x + 2 x 2 9
3 A.6 Solving Equations Simplify 1 2x 6 x + 2 x 2 9 Factor: 2x 6 = 2(x 3) x 2 9 = (x 3)(x + 3) LCD = 2(x 3)(x + 3)
4 A.6 Solving Equations Simplify 1 2x 6 x + 2 x 2 9 LCD = 2(x 3)(x + 3), so 1 2x 6 x + 2 x 2 9 = 1 2(x 3) x + 3 x + 3 x + 2 (x 3)(x + 3) 2 2 (x + 3) (2x + 4) = 2(x 3)(x + 3) x 1 = 2(x 3)(x + 3)
5 Solving Equations Equation: Two expressions set equal to each other.
6 Solving Equations Equation: Two expressions set equal to each other. To solve means to find the values of the variables that make the equation a true statement.
7 Solving Equations Equation: Two expressions set equal to each other. To solve means to find the values of the variables that make the equation a true statement. A solution must be in the domain of the original equation.
8 Three types of solutions
9 Three types of solutions Conditional: True for some values of x but not others. x + 2 = 3 is a conditional equation because 1 is a solution but other numbers are not.
10 Three types of solutions Conditional: True for some values of x but not others. x + 2 = 3 is a conditional equation because 1 is a solution but other numbers are not. Contradiction: An equation that is false for all values of x. There is no solution. x + 1 = x is a contradiction. If you take a number and add 1 to it, you don t get back the original number.
11 Three types of solutions Conditional: True for some values of x but not others. x + 2 = 3 is a conditional equation because 1 is a solution but other numbers are not. Contradiction: An equation that is false for all values of x. There is no solution. x + 1 = x is a contradiction. If you take a number and add 1 to it, you don t get back the original number. Identity: An equation that is true for all values of x. The solution is all real numbers. x + 1 = 1 + x is an identity. It is an example of the commutative property of addition.
12 2 3 (2x + 3) = 1 (3 x) 1 2
13 2 3 (2x + 3) = 1 (3 x)
14 5(x 4) (3 x) = 2(x + 5) + 4x
15 5(x 4) (3 x) = 2(x + 5) + 4x
16 5(x 4) (3 x) = 2(x + 5) + 4x No solution!
17 6x 2 5 = 13x
18 6x 2 5 = 13x
19 6x 2 5 = 13x We can check that both of these work
20 Factoring 3x x = 6x 3
21 Factoring 3x x = 6x
22 Absolute Values 4 3x 4 = 20
23 Absolute Values 4 3x 4 = 20 Rule for absolute values If y = b, then y = b or y = b. Use this to split the equation into 2 different equations without an absolute value.
24 Absolute Values 4 3x 4 =
25 5 x 2 + 3x 2 3 = 7
26 5 x 2 + 3x 2 3 =
27 Square Roots (x 2) 2 = 9
28 Square Roots (x 2) 2 = 9 Rule for square roots If y 2 = b, then y = b or y = b. Use this to split the equation into 2 different equations.
29 Square Roots (x 2) 2 =
30 Completing the Square x 2 + 6x = 4
31 Completing the Square x 2 + 6x = 4 Completing the square Since (x + a) 2 = x 2 + 2ax + a 2, let b = 2a so that b 2 quadratic equation like = a. In a x 2 + bx = c we can solve by adding ( b 2) 2 to both sides.
32 Completing the Square 2x 2 3x = 1
33 Completing the Square 2x 2 3x =
34 Quadratic Equation The standard form for a quadratic equation is ax 2 + bx + c = 0 where a, b, c are real numbers and a 0. We can use completing the square to solve this equation to get the quadratic formula: x 1,2 = b ± b 2 4ac 2a
35 Quadratic Equation 2x 2 3x = 1
36 Quadratic Equation 2x 2 3x = 1 Same answer as we found completing the square
37 How about the cubic equation ax 3 + bx 2 + cx + d = 0? The solution is:
38 x 1 = b 3a a a 2 [ ] 2b 3 9abc + 27a 2 d + (2b 3 9abc + 27a 2 d) 2 4(b 2 3ac) 3 [ ] 2b 3 9abc + 27a 2 d (2b 3 9abc + 27a 2 d) 2 4(b 2 3ac) 3 x 2 = b 3a 1 + i a 2 1 i a 2 [ ] 2b 3 9abc + 27a 2 d + (2b 3 9abc + 27a 2 d) 2 4(b 2 3ac) 3 [ ] 2b 3 9abc + 27a 2 d (2b 3 9abc + 27a 2 d) 2 4(b 2 3ac) 3 x 3 = b 3a 1 i a i a 2 [ ] 2b 3 9abc + 27a 2 d + (2b 3 9abc + 27a 2 d) 2 4(b 2 3ac) 3 [ ] 2b 3 9abc + 27a 2 d (2b 3 9abc + 27a 2 d) 2 4(b 2 3ac) 3
39 Gerolamo Cardano
40 Cubic Equations No magical formula. Hopefully we can factor. x 3 1 = 0
41 Cubic Equations No magical formula. Hopefully we can factor. x 3 1 =
42 Solve: ( ) ( ) 2x 2 x x 2 2x = (2x 1)x
43 Solve: ( ) ( ) 2x 2 x x 2 2x = (2x 1)x
44 Read section A.8 before next lecture.
Roots and Coefficients Polynomials Preliminary Maths Extension 1
Preliminary Maths Extension Question If, and are the roots of x 5x x 0, find the following. (d) (e) Question If p, q and r are the roots of x x x 4 0, evaluate the following. pq r pq qr rp p q q r r p
More information1 Quadratic Functions
Unit 1 Quadratic Functions Lecture Notes Introductory Algebra Page 1 of 8 1 Quadratic Functions In this unit we will learn many of the algebraic techniques used to work with the quadratic function fx)
More information( ) and D( x) have been written out in
PART E: REPEATED (OR POWERS OF) LINEAR FACTORS Example (Section 7.4: Partial Fractions) 7.27 Find the PFD for x 2 ( x + 2). 3 Solution Step 1: The expression is proper, because 2, the degree of N ( x),
More information9.5. Polynomial and Rational Inequalities. Objectives. Solve quadratic inequalities. Solve polynomial inequalities of degree 3 or greater.
Chapter 9 Section 5 9.5 Polynomial and Rational Inequalities Objectives 1 3 Solve quadratic inequalities. Solve polynomial inequalities of degree 3 or greater. Solve rational inequalities. Objective 1
More informationMath 096--Quadratic Formula page 1
Math 096--Quadratic Formula page 1 A Quadratic Formula. Use the quadratic formula to solve quadratic equations ax + bx + c = 0 when the equations can t be factored. To use the quadratic formula, the equation
More informationWhen using interval notation use instead of open circles, and use instead of solid dots.
P.1 Real Numbers PreCalculus P.1 REAL NUMBERS Learning Targets for P1 1. Describe an interval on the number line using inequalities. Describe an interval on the number line using interval notation (closed
More informationTropical Polynomials
1 Tropical Arithmetic Tropical Polynomials Los Angeles Math Circle, May 15, 2016 Bryant Mathews, Azusa Pacific University In tropical arithmetic, we define new addition and multiplication operations on
More informationLecture 27. Quadratic Formula
Lecture 7 Quadratic Formula Goal: to solve any quadratic equation Quadratic formula 3 Plan Completing the square 5 Completing the square 6 Proving quadratic formula 7 Proving quadratic formula 8 Proving
More informationIntegration of Rational Functions by Partial Fractions
Title Integration of Rational Functions by MATH 1700 MATH 1700 1 / 11 Readings Readings Readings: Section 7.4 MATH 1700 2 / 11 Rational functions A rational function is one of the form where P and Q are
More informationA Sampling of Symmetry and Solvability. Sarah Witherspoon Professor of Mathematics Texas A&M University
A Sampling of Symmetry and Solvability Sarah Witherspoon Professor of Mathematics Texas A&M University Polynomial Equations Examples: x 9 2x 4 5 = 0, x 2 y 3 3y 2 = 0, x 2 + y 2 + z 2 = 1 Solving polynomial
More informationSection 8.3 Partial Fraction Decomposition
Section 8.6 Lecture Notes Page 1 of 10 Section 8.3 Partial Fraction Decomposition Partial fraction decomposition involves decomposing a rational function, or reversing the process of combining two or more
More informationMath 113 Winter 2013 Prof. Church Midterm Solutions
Math 113 Winter 2013 Prof. Church Midterm Solutions Name: Student ID: Signature: Question 1 (20 points). Let V be a finite-dimensional vector space, and let T L(V, W ). Assume that v 1,..., v n is a basis
More informationIntegration of Rational Functions by Partial Fractions
Title Integration of Rational Functions by Partial Fractions MATH 1700 December 6, 2016 MATH 1700 Partial Fractions December 6, 2016 1 / 11 Readings Readings Readings: Section 7.4 MATH 1700 Partial Fractions
More informationCHAPTER 2 POLYNOMIALS KEY POINTS
CHAPTER POLYNOMIALS KEY POINTS 1. Polynomials of degrees 1, and 3 are called linear, quadratic and cubic polynomials respectively.. A quadratic polynomial in x with real coefficient is of the form a x
More informationMath 231E, Lecture 25. Integral Test and Estimating Sums
Math 23E, Lecture 25. Integral Test and Estimating Sums Integral Test The definition of determining whether the sum n= a n converges is:. Compute the partial sums s n = a k, k= 2. Check that s n is a convergent
More informationQuestion 1: The graphs of y = p(x) are given in following figure, for some Polynomials p(x). Find the number of zeroes of p(x), in each case.
Class X - NCERT Maths EXERCISE NO:.1 Question 1: The graphs of y = p(x) are given in following figure, for some Polynomials p(x). Find the number of zeroes of p(x), in each case. (i) (ii) (iii) (iv) (v)
More informationB.3 Solving Equations Algebraically and Graphically
B.3 Solving Equations Algebraically and Graphically 1 Equations and Solutions of Equations An equation in x is a statement that two algebraic expressions are equal. To solve an equation in x means to find
More informationPartial Fraction Decomposition
Partial Fraction Decomposition As algebra students we have learned how to add and subtract fractions such as the one show below, but we probably have not been taught how to break the answer back apart
More informationPartial Fraction Decomposition Honors Precalculus Mr. Velazquez Rm. 254
Partial Fraction Decomposition Honors Precalculus Mr. Velazquez Rm. 254 Adding and Subtracting Rational Expressions Recall that we can use multiplication and common denominators to write a sum or difference
More informationUse the Rational Zero Theorem to list all the possible rational zeros of the following polynomials. (1-2) 4 3 2
Name: Math 114 Activity 1(Due by EOC Apr. 17) Dear Instructor or Tutor, These problems are designed to let my students show me what they have learned and what they are capable of doing on their own. Please
More informationEquations in Quadratic Form
Equations in Quadratic Form MATH 101 College Algebra J. Robert Buchanan Department of Mathematics Summer 2012 Objectives In this lesson we will learn to: make substitutions that allow equations to be written
More informationUpdated: January 16, 2016 Calculus II 7.4. Math 230. Calculus II. Brian Veitch Fall 2015 Northern Illinois University
Math 30 Calculus II Brian Veitch Fall 015 Northern Illinois University Integration of Rational Functions by Partial Fractions From algebra, we learned how to find common denominators so we can do something
More informationFactors, Zeros, and Roots
Factors, Zeros, and Roots Solving polynomials that have a degree greater than those solved in previous courses is going to require the use of skills that were developed when we previously solved quadratics.
More informationMath123 Lecture 1. Dr. Robert C. Busby. Lecturer: Office: Korman 266 Phone :
Lecturer: Math1 Lecture 1 Dr. Robert C. Busby Office: Korman 66 Phone : 15-895-1957 Email: rbusby@mcs.drexel.edu Course Web Site: http://www.mcs.drexel.edu/classes/calculus/math1_spring0/ (Links are case
More informationSection September 6, If n = 3, 4, 5,..., the polynomial is called a cubic, quartic, quintic, etc.
Section 2.1-2.2 September 6, 2017 1 Polynomials Definition. A polynomial is an expression of the form a n x n + a n 1 x n 1 + + a 1 x + a 0 where each a 0, a 1,, a n are real numbers, a n 0, and n is a
More informationChapter 1A -- Real Numbers. iff. Math Symbols: Sets of Numbers
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1A! Page 1 Chapter 1A -- Real Numbers Math Symbols: iff or Example: Let A = {2, 4, 6, 8, 10, 12, 14, 16,...} and let B = {3, 6, 9, 12, 15, 18,
More informationName: Partners: PreCalculus. Review 5 Version A
Name: Partners: PreCalculus Date: Review 5 Version A [A] Circle whether each statement is true or false. 1. 3 log 3 5x = 5x 2. log 2 16 x+3 = 4x + 3 3. ln x 6 + ln x 5 = ln x 30 4. If ln x = 4, then e
More informationPolynomial Functions
Polynomial Functions Equations and Graphs Characteristics The Factor Theorem The Remainder Theorem http://www.purplemath.com/modules/polyends5.htm 1 A cross-section of a honeycomb has a pattern with one
More informationAppendix 1. Cardano s Method
Appendix 1 Cardano s Method A1.1. Introduction This appendix gives the mathematical method to solve the roots of a polynomial of degree three, called a cubic equation. Some results in this section can
More informationPARTIAL FRACTION DECOMPOSITION. Mr. Velazquez Honors Precalculus
PARTIAL FRACTION DECOMPOSITION Mr. Velazquez Honors Precalculus ADDING AND SUBTRACTING RATIONAL EXPRESSIONS Recall that we can use multiplication and common denominators to write a sum or difference of
More informationMATH 250 REVIEW TOPIC 3 Partial Fraction Decomposition and Irreducible Quadratics. B. Decomposition with Irreducible Quadratics
Math 250 Partial Fraction Decomposition Topic 3 Page MATH 250 REVIEW TOPIC 3 Partial Fraction Decomposition and Irreducible Quadratics I. Decomposition with Linear Factors Practice Problems II. A. Irreducible
More informationKing Fahd University of Petroleum and Minerals Prep-Year Math Program Math Term 161 Recitation (R1, R2)
Math 001 - Term 161 Recitation (R1, R) Question 1: How many rational and irrational numbers are possible between 0 and 1? (a) 1 (b) Finite (c) 0 (d) Infinite (e) Question : A will contain how many elements
More informationPARTIAL FRACTIONS AND POLYNOMIAL LONG DIVISION. The basic aim of this note is to describe how to break rational functions into pieces.
PARTIAL FRACTIONS AND POLYNOMIAL LONG DIVISION NOAH WHITE The basic aim of this note is to describe how to break rational functions into pieces. For example 2x + 3 = + x 3 x +. The point is that we don
More informationGraphs of polynomials. Sue Gordon and Jackie Nicholas
Mathematics Learning Centre Graphs of polynomials Sue Gordon and Jackie Nicholas c 2004 University of Sydney Mathematics Learning Centre, University of Sydney 1 1 Graphs of Polynomials Polynomials are
More informationMATH10001 Mathematical Workshop Graph Fitting Project part 2
MATH10001 Mathematical Workshop Graph Fitting Project part 2 Polynomial models Modelling a set of data with a polynomial curve can be convenient because polynomial functions are particularly easy to differentiate
More informationA quadratic expression is a mathematical expression that can be written in the form 2
118 CHAPTER Algebra.6 FACTORING AND THE QUADRATIC EQUATION Textbook Reference Section 5. CLAST OBJECTIVES Factor a quadratic expression Find the roots of a quadratic equation A quadratic expression is
More informationNumerical Algorithms. IE 496 Lecture 20
Numerical Algorithms IE 496 Lecture 20 Reading for This Lecture Primary Miller and Boxer, Pages 124-128 Forsythe and Mohler, Sections 1 and 2 Numerical Algorithms Numerical Analysis So far, we have looked
More informationTo solve a radical equation, you must take both sides of an equation to a power.
Topic 5 1 Radical Equations A radical equation is an equation with at least one radical expression. There are four types we will cover: x 35 3 4x x 1x 7 3 3 3 x 5 x 1 To solve a radical equation, you must
More informationSolving Quadratic Equations
Solving Quadratic Equations MATH 101 College Algebra J. Robert Buchanan Department of Mathematics Summer 2012 Objectives In this lesson we will learn to: solve quadratic equations by factoring, solve quadratic
More informationSolving Quadratic Equations
Concepts: Solving Quadratic Equations, Completing the Square, The Quadratic Formula, Sketching Quadratics Solving Quadratic Equations Completing the Square ax + bx + c = a x + ba ) x + c Factor so the
More informationOCR Maths FP1. Topic Questions from Papers. Roots of Polynomial Equations
OCR Maths FP1 Topic Questions from Papers Roots of Polynomial Equations PhysicsAndMathsTutor.com 18 (a) The quadratic equation x 2 2x + 4 = 0hasroots and. (i) Write down the values of + and. [2] (ii) Show
More informationLecture 26. Quadratic Equations
Lecture 26 Quadratic Equations Quadratic polynomials....................................................... 2 Quadratic polynomials....................................................... 3 Quadratic equations
More informationWhat if the characteristic equation has a double root?
MA 360 Lecture 17 - Summary of Recurrence Relations Friday, November 30, 018. Objectives: Prove basic facts about basic recurrence relations. Last time, we looked at the relational formula for a sequence
More informationWhat if the characteristic equation has complex roots?
MA 360 Lecture 18 - Summary of Recurrence Relations (cont. and Binomial Stuff Thursday, November 13, 01. Objectives: Examples of Recurrence relation solutions, Pascal s triangle. A quadratic equation What
More informationMA 1128: Lecture 19 4/20/2018. Quadratic Formula Solving Equations with Graphs
MA 1128: Lecture 19 4/20/2018 Quadratic Formula Solving Equations with Graphs 1 Completing-the-Square Formula One thing you may have noticed when you were completing the square was that you followed the
More informationCore Mathematics 1 Quadratics
Regent College Maths Department Core Mathematics 1 Quadratics Quadratics September 011 C1 Note Quadratic functions and their graphs. The graph of y ax bx c. (i) a 0 (ii) a 0 The turning point can be determined
More informationChapter 1: Precalculus Review
: Precalculus Review Math 115 17 January 2018 Overview 1 Important Notation 2 Exponents 3 Polynomials 4 Rational Functions 5 Cartesian Coordinates 6 Lines Notation Intervals: Interval Notation (a, b) (a,
More informationax 2 + bx + c = 0 What about any cubic equation?
(Trying to) solve Higher order polynomial euations. Some history: The uadratic formula (Dating back to antiuity) allows us to solve any uadratic euation. What about any cubic euation? ax 2 + bx + c = 0
More informationCM2104: Computational Mathematics General Maths: 2. Algebra - Factorisation
CM204: Computational Mathematics General Maths: 2. Algebra - Factorisation Prof. David Marshall School of Computer Science & Informatics Factorisation Factorisation is a way of simplifying algebraic expressions.
More informationNotes on Linear Algebra I. # 1
Notes on Linear Algebra I. # 1 Oussama Moutaoikil Contents 1 Introduction 1 2 On Vector Spaces 5 2.1 Vectors................................... 5 2.2 Vector Spaces................................ 7 2.3
More informationBefore this course is over we will see the need to split up a fraction in a couple of ways, one using multiplication and the other using addition.
CH MORE FRACTIONS Introduction I n this chapter we tie up some loose ends. First, we split a single fraction into two fractions, followed by performing our standard math operations on positive and negative
More informationSecondary Math 3 Honors - Polynomial and Polynomial Functions Test Review
Name: Class: Date: Secondary Math 3 Honors - Polynomial and Polynomial Functions Test Review 1 Write 3x 2 ( 2x 2 5x 3 ) in standard form State whether the function is even, odd, or neither Show your work
More informationWhat students need to know for PRE-CALCULUS Students expecting to take Pre-Calculus should demonstrate the ability to:
What students need to know for PRE-CALCULUS 2014-2015 Students expecting to take Pre-Calculus should demonstrate the ability to: General: keep an organized notebook take good notes complete homework every
More informationPARTIAL FRACTIONS AND POLYNOMIAL LONG DIVISION. The basic aim of this note is to describe how to break rational functions into pieces.
PARTIAL FRACTIONS AND POLYNOMIAL LONG DIVISION NOAH WHITE The basic aim of this note is to describe how to break rational functions into pieces. For example 2x + 3 1 = 1 + 1 x 1 3 x + 1. The point is that
More informationMath 2142 Homework 5 Part 1 Solutions
Math 2142 Homework 5 Part 1 Solutions Problem 1. For the following homogeneous second order differential equations, give the general solution and the particular solution satisfying the given initial conditions.
More informationUnraveling the Complexities of the Cubic Formula. Bruce Bordwell & Mark Omodt Anoka-Ramsey Community College Coon Rapids, Minnesota
Unraveling the Complexities of the Cubic Formula Bruce Bordell & Mark Omodt Anoka-Ramsey Community College Coon Rapids, Minnesota A little history Quadratic Equations Cubic Equations del Ferro (1465 156)
More informationHonors Advanced Algebra Unit 3: Polynomial Functions October 28, 2016 Task 10: Factors, Zeros, and Roots: Oh My!
Honors Advanced Algebra Name Unit 3: Polynomial Functions October 8, 016 Task 10: Factors, Zeros, and Roots: Oh My! MGSE9 1.A.APR. Know and apply the Remainder Theorem: For a polynomial p(x) and a number
More informationMathematics AQA Advanced Subsidiary GCE Core 1 (MPC1) January 2010
Link to past paper on AQA website: http://store.aqa.org.uk/qual/gce/pdf/aqa-mpc1-w-qp-jan10.pdf These solutions are for your personal use only. DO NOT photocopy or pass on to third parties. If you are
More informationBefore this course is over we will see the need to split up a fraction in a couple of ways, one using multiplication and the other using addition.
CH 0 MORE FRACTIONS Introduction I n this chapter we tie up some loose ends. First, we split a single fraction into two fractions, followed by performing our standard math operations on positive and negative
More informationPolynomial Form. Factored Form. Perfect Squares
We ve seen how to solve quadratic equations (ax 2 + bx + c = 0) by factoring and by extracting square roots, but what if neither of those methods are an option? What do we do with a quadratic equation
More informationComplex numbers, the exponential function, and factorization over C
Complex numbers, the exponential function, and factorization over C 1 Complex Numbers Recall that for every non-zero real number x, its square x 2 = x x is always positive. Consequently, R does not contain
More informationQuadratic and Rational Inequalities
Quadratic and Rational Inequalities Definition of a Quadratic Inequality A quadratic inequality is any inequality that can be put in one of the forms ax 2 + bx + c < 0 ax 2 + bx + c > 0 ax 2 + bx + c
More information( ) c. m = 0, 1 2, 3 4
G Linear Functions Probably the most important concept from precalculus that is required for differential calculus is that of linear functions The formulas you need to know backwards and forwards are:
More informationToday. Couple of more induction proofs. Stable Marriage.
Today. Couple of more induction proofs. Stable Marriage. Strengthening: need to... Theorem: For all n 1, n i=1 1 2. (S i 2 n = n i=1 1.) i 2 Base: P(1). 1 2. Ind Step: k i=1 1 2. i 2 k+1 i=1 1 i 2 = k
More informationEquations. Rational Equations. Example. 2 x. a b c 2a. Examine each denominator to find values that would cause the denominator to equal zero
Solving Other Types of Equations Rational Equations Examine each denominator to find values that would cause the denominator to equal zero Multiply each term by the LCD or If two terms cross-multiply Solve,
More informationHow to Do Word Problems. Solving Linear Equations
Solving Linear Equations Properties of Equality Property Name Mathematics Operation Addition Property If A = B, then A+C = B +C Subtraction Property If A = B, then A C = B C Multiplication Property If
More informationAPPENDIX : PARTIAL FRACTIONS
APPENDIX : PARTIAL FRACTIONS Appendix : Partial Fractions Given the expression x 2 and asked to find its integral, x + you can use work from Section. to give x 2 =ln( x 2) ln( x + )+c x + = ln k x 2 x+
More informationFormative Assignment PART A
MHF4U_2011: Advanced Functions, Grade 12, University Preparation Unit 2: Advanced Polynomial and Rational Functions Activity 2: Families of polynomial functions Formative Assignment PART A For each of
More informationClass IX Chapter 2 Polynomials Maths
NCRTSOLUTIONS.BLOGSPOT.COM Class IX Chapter 2 Polynomials Maths Exercise 2.1 Question 1: Which of the following expressions are polynomials in one variable and which are No. It can be observed that the
More informationFactor each expression. Remember, always find the GCF first. Then if applicable use the x-box method and also look for difference of squares.
NOTES 11: RATIONAL EXPRESSIONS AND EQUATIONS Name: Date: Period: Mrs. Nguyen s Initial: LESSON 11.1 SIMPLIFYING RATIONAL EXPRESSIONS Lesson Preview Review Factoring Skills and Simplifying Fractions Factor
More informationSUMMER REVIEW PACKET. Name:
Wylie East HIGH SCHOOL SUMMER REVIEW PACKET For students entering Regular PRECALCULUS Name: Welcome to Pre-Calculus. The following packet needs to be finished and ready to be turned the first week of the
More informationAlgebra 2 Summer Math Answer Section
Algebra 2 Summer Math Answer Section 1. ANS: A PTS: 1 DIF: Level B REF: MALG0064 STA: SC.HSCS.MTH.00.AL1.A1.I.C.4 TOP: Lesson 1.1 Evaluate Expressions KEY: word volume cube area solid 2. ANS: C PTS: 1
More informationChapter 8 ~ Quadratic Functions and Equations In this chapter you will study... You can use these skills...
Chapter 8 ~ Quadratic Functions and Equations In this chapter you will study... identifying and graphing quadratic functions transforming quadratic equations solving quadratic equations using factoring
More informationSecondary Math 2H Unit 3 Notes: Factoring and Solving Quadratics
Secondary Math H Unit 3 Notes: Factoring and Solving Quadratics 3.1 Factoring out the Greatest Common Factor (GCF) Factoring: The reverse of multiplying. It means figuring out what you would multiply together
More information4.5 Integration of Rational Functions by Partial Fractions
4.5 Integration of Rational Functions by Partial Fractions From algebra, we learned how to find common denominators so we can do something like this, 2 x + 1 + 3 x 3 = 2(x 3) (x + 1)(x 3) + 3(x + 1) (x
More informationDesert Mountain H. S. Math Department Summer Work Packet
Course #44-443 Desert Mountain H. S. Math Department Summer Work Packet Honors/AP/IB level math courses at Desert Mountain are for students who are enthusiastic learners of mathematics and whose work ethic
More informationTwitter: @Owen134866 www.mathsfreeresourcelibrary.com Prior Knowledge Check 1) Factorise each polynomial: a) x 2 6x + 5 b) x 2 16 c) 9x 2 25 2) Simplify the following algebraic fractions fully: a) x 2
More informationLinear equations are equations involving only polynomials of degree one.
Chapter 2A Solving Equations Solving Linear Equations Linear equations are equations involving only polynomials of degree one. Examples include 2t +1 = 7 and 25x +16 = 9x 4 A solution is a value or a set
More informationMath Lecture 3 Notes
Math 1010 - Lecture 3 Notes Dylan Zwick Fall 2009 1 Operations with Real Numbers In our last lecture we covered some basic operations with real numbers like addition, subtraction and multiplication. This
More informationPhysicsAndMathsTutor.com
Question Answer Marks Guidance x x x mult throughout by (x + )(x ) or combining fractions and mult up oe (can retain denominator throughout). Condone a single computational error provided that there xx)
More informationSolving a quartic equation by the method of radicals
Solving a quartic equation by the method of radicals Peter Haggstrom www.gotohaggstrom.com mathsatbondibeach@gmail.com September 18, 01 1 Introduction This paper deals with the solution of quartic equations
More informationIntegration by partial fractions
Roberto s Notes on Integral Calculus Chapter : Integration methods Section 15 Integration by partial fractions with non-repeated quadratic factors What you need to know already: How to use the integration
More informationLecture 6 & 7. Shuanglin Shao. September 16th and 18th, 2013
Lecture 6 & 7 Shuanglin Shao September 16th and 18th, 2013 1 Elementary matrices 2 Equivalence Theorem 3 A method of inverting matrices Def An n n matrice is called an elementary matrix if it can be obtained
More informationMULTIPLYING TRINOMIALS
Name: Date: 1 Math 2 Variable Manipulation Part 4 Polynomials B MULTIPLYING TRINOMIALS Multiplying trinomials is the same process as multiplying binomials except for there are more terms to multiply than
More informatione. some other answer 6. The graph of the parabola given below has an axis of symmetry of: a. y = 5 b. x = 3 c. y = 3 d. x = 5 e. Some other answer.
Intermediate Algebra Solutions Review Problems Final Exam MTH 099 December, 006 1. True or False: (a + b) = a + b. True or False: x + y = x + y. True or False: The parabola given by the equation y = x
More informationMath 54 HW 4 solutions
Math 54 HW 4 solutions 2.2. Section 2.2 (a) False: Recall that performing a series of elementary row operations A is equivalent to multiplying A by a series of elementary matrices. Suppose that E,...,
More informationThere are two main properties that we use when solving linear equations. Property #1: Additive Property of Equality
Chapter 1.1: Solving Linear and Literal Equations Linear Equations Linear equations are equations of the form ax + b = c, where a, b and c are constants, and a zero. A hint that an equation is linear is
More informationConcept Category 4. Quadratic Equations
Concept Category 4 Quadratic Equations 1 Solving Quadratic Equations by the Square Root Property Square Root Property We previously have used factoring to solve quadratic equations. This chapter will introduce
More informationQuadratics - Quadratic Formula
9.4 Quadratics - Quadratic Formula Objective: Solve quadratic equations by using the quadratic formula. The general from of a quadratic is ax + bx + c = 0. We will now solve this formula for x by completing
More informationb for the linear system x 1 + x 2 + a 2 x 3 = a x 1 + x 3 = 3 x 1 + x 2 + 9x 3 = 3 ] 1 1 a 2 a
Practice Exercises for Exam Exam will be on Monday, September 8, 7. The syllabus for Exam consists of Sections One.I, One.III, Two.I, and Two.II. You should know the main definitions, results and computational
More informationSolutions to Chapter Review Questions, Chapter 0
Instructor s Solutions Manual, Chapter 0 Review Question 1 Solutions to Chapter Review Questions, Chapter 0 1. Explain how the points on the real line correspond to the set of real numbers. solution Start
More informationWarm-Up. Use long division to divide 5 into
Warm-Up Use long division to divide 5 into 3462. 692 5 3462-30 46-45 12-10 2 Warm-Up Use long division to divide 5 into 3462. Divisor 692 5 3462-30 46-45 12-10 2 Quotient Dividend Remainder Warm-Up Use
More informationTenth Maths Polynomials
Tenth Maths Polynomials Polynomials are algebraic expressions constructed using constants and variables. Coefficients operate on variables, which can be raised to various powers of non-negative integer
More informationA polynomial expression is the addition or subtraction of many algebraic terms with positive integer powers.
LEAVING CERT Honours Maths notes on Algebra. A polynomial expression is the addition or subtraction of many algebraic terms with positive integer powers. The degree is the highest power of x. 3x 2 + 2x
More informationStudents expecting to take Advanced Qualitative Reasoning should demonstrate the ability to:
What students need to know for AQR Students expecting to take Advanced Qualitative Reasoning should demonstrate the ability to: General: keep an organized notebook take good notes complete homework every
More informationMath-2A Lesson 13-3 (Analyzing Functions, Systems of Equations and Inequalities) Which functions are symmetric about the y-axis?
Math-A Lesson 13-3 (Analyzing Functions, Systems of Equations and Inequalities) Which functions are symmetric about the y-axis? f ( x) x x x x x x 3 3 ( x) x We call functions that are symmetric about
More informationMATH 425-Spring 2010 HOMEWORK ASSIGNMENTS
MATH 425-Spring 2010 HOMEWORK ASSIGNMENTS Instructor: Shmuel Friedland Department of Mathematics, Statistics and Computer Science email: friedlan@uic.edu Last update April 18, 2010 1 HOMEWORK ASSIGNMENT
More informationPrecalculus Notes: Unit P Prerequisite Skills
Syllabus Objective Note: Because this unit contains all prerequisite skills that were taught in courses prior to precalculus, there will not be any syllabus objectives listed. Teaching this unit within
More informationPOLYNOMIALS CHAPTER 2. (A) Main Concepts and Results
CHAPTER POLYNOMIALS (A) Main Concepts and Results Meaning of a Polynomial Degree of a polynomial Coefficients Monomials, Binomials etc. Constant, Linear, Quadratic Polynomials etc. Value of a polynomial
More informationAnswers to Sample Exam Problems
Math Answers to Sample Exam Problems () Find the absolute value, reciprocal, opposite of a if a = 9; a = ; Absolute value: 9 = 9; = ; Reciprocal: 9 ; ; Opposite: 9; () Commutative law; Associative law;
More information