October 28, Complex Numbers.notebook. Discriminant
|
|
- Hollie Hancock
- 5 years ago
- Views:
Transcription
1 OBJECTIVE Students will be able to utilize complex numbers to simplify roots of negative numbers. Students will be able to plot complex numbers on a complex coordinate plane. Students will be able to add and subtract complex numbers. Students will be able to multiply complex numbers. Students will be able to divide complex numbers using complex conjugates. Students will be able to find the absolute value of complex numbers. Discriminant Value greater than 0 and a perfect square greater than 0 and NOT a perfect square equal to 0 less than 0 Discriminant Type and Number of Roots rational roots irrational roots 1 rational root complex roots Complex roots result when the discriminant is negative. This will be our focus for today. Oct 4 10:58 AM Oct 4 11:06 AM Two Complex Roots Ax + Bx + C Discriminant is the expression b 4ac. The discriminant's value tells in advance how many answers and what type of answers to the quadratic formula there will be. x 5x + 7 = 0 ( 5) 4(1)(7) 5 4(1)(7) Two complex roots. Oct 4 11:37 AM Oct 4 11:07 AM You've been told that you can't take the square root of a negative number. 5 That's because you had no numbers which were negative after you'd squared them (so you couldn't "go backwards" by taking the square root). Every number was positive after you squared it. (+ 5) = + 5 ( 5) = + 5 So +5 is +5 and 5 or ±5 So you couldn't very well square root a negative and expect to come up with anything sensible. *Source Purplemath.com Now, however, you can take the square root of a negative number, but it involves using a new number to do it. This new number was invented (discovered?) around the time of the Reformation. At that time, nobody believed that any "real world" use would be found for this new number, other than easing the computations involved in solving certain equations, so the new number was viewed as being a pretend number invented for convenience sake. (But then, when you think about it, aren't all numbers inventions? It's not like numbers grow on trees! They live in our heads. We made them all up! Why not invent a new one, as long as it works okay with what we already have?) This new number was called "i", standing for "imaginary", because "everybody knew" that i wasn't "real". *Source Purplemath.com Oct 4 11:53 AM Oct 4 11:57 AM 1
2 i (imaginary number) = -1 i = ( -1 ) = - 1 i = -1 5 = 1 * 5 = i * 5 = ±5i 11 = 1 * 11 = i * 11 = ±11i So, let's see how this imaginary number is utilized. 48 = 1 * 48 = i * 16 * 3 = ±4i 3 Oct 4 1:01 PM Oct 4 1:15 PM Ax + Bx + C Discriminant is the expression b 4ac. The discriminant's value tells in advance how many answers and what type of answers to the quadratic formula there will be. x 5x + 7 = 0 ( 5) 4(1)(7) 5 4(1)(7) Two complex roots. Now let's solve for those roots. x 5x + 7 = 0 x = - (-5) ± (-5) - 4(1)(7) (1) x = - (-5) ± 5-8 (1) x = - (-5) ± - 3 (1) x = + 5 ± - 3 x = + 5 ± -1 * 3 x = + 5 ± i 3 Oct 4 1:35 PM Oct 4 1:36 PM 4x 3x + 3 = 0 Operations with Perform Operations with just as you would with a variable or monomial expression with a variable. Add imaginary numbers i + 3i. i + 3i = ( + 3)i = 5i Subtract imaginary numbers 16i 5i. 16i 5i = (16 5)i = 11i Multiply imaginary numbers (3i)(4i). (3i)(4i) = (3 4)(ii) = (1)(i ) = (1)( 1) = 1 Multiply imaginary numbers (i)(i)( 3i). (i)(i)( 3i) = ( 3)(i i i) = ( 6)(i i)=( 6)( 1 i) = ( 6)( i) = 6i Oct 4 1:46 PM Oct 4 1:38 PM
3 Pattern of Powers with i 1 i - 1 i 3 i i i 4 1 i 5 i 4 * i 1 i i 6 i 4 * i - 1 i 7 i 4 * i 3 i i 8 i 4 * i 4 1 Pattern of Powers with Simplify i 17 i 17 = i = i = i 1 = i Simplify i 10 i 10 = i 4 30 = i = i 0 = 1 Simplify i 64,00 i 64,00 = i 64,000 + = i 4 16,000 + = i = 1 Oct 4 1:45 PM Oct 4 1:58 PM Pattern of Powers with Simplify i 8 Simplify i 38 Simplify i 103 Simplify i 8,001 Complex Numbers A complex number is a number in the form of a + bi where a and b are real numbers. a is the real portion of the complex number. bi is the imaginary portion of the complex number. If b 0, then a + bi is an imaginary number i If a = 0 and b 0, then a + bi is a pure imaginary number i 6i Oct 4 1:15 PM Oct 4 1:30 PM Perform just as you would simplify like and unlike terms. Combine like terms. Leave unlike terms alone. Add Complex numbers (4 + i ) + (5 + 3i ) = 9 i + 3i = 5i 9 + 5i Subtract imaginary numbers (4 + i ) (5 + 3i ) (4 + i ) (5 + 3i ) (4 + i ) + ( 5 3i ) 4 5 = 1 i 3i = i 1 i Simplify (7 + 6i) + ( + 4i) Simplify (3 + i) + (9 + 8i) Simplify (1 5i) + (4 3i) Simplify (9 + 7i) (6 i) Simplify (5 i) (3 8i) Oct 4 1:06 PM Oct 4 1:18 PM 3
4 Multiply Complex numbers 9i (3 + i) 9i (3) = 7i 9i (i) = 18i 7i + 18( 1) 7i i Write in standard form. Multiply Complex numbers (4 + i ) (5 + 3i ) First (4)(5) = 0 Outer (4)(3i) = 1i Inner (i)(5) = 10i Last (i)(3i) = 6i 0 + 1i + 10i + 6i 0 + i + 6i 0 + i + 6( 1) Remember: i = i i (3 + i ) (6 + 8i ) (7 + 9i ) (5 + 4i ) Oct 4 1:3 PM Oct 4 3:01 PM Complex conjugates are to complex numbers that when multiplied together will result in a product that is a real number. (4 + 3i) (4 3i) First (4)(4) = 16 Outer (4)( 3i) = 1i Inner (4)(3i) = 1i Last (3i)( 3i) = 9i 16 1i + 1i 9i 16 9 i 16 9( 1) Remember: i = 1 Complex numbers must have real number denominators i i 3 i Allowed NOT Allowed Oct 4 :3 PM Oct 4 1:49 PM Dividing Complex numbers (4 + i ) (5 + 3i ) (4 + i ) * (5 3i ) (5 + 3i ) * (5 3i ) Multiply top and bottom by the complex conjugate of the denominator. First + Outer + Inner + Last (4)(5) + (4)( 3i) + (i )(5) + (i )( 3i ) (5)(5) + (5)( 3i ) + ( 3i )(5) + ( 3i )( 3i ) First + Outer + Inner + Last 0 i 6i 0 i 6( 1) 5 9i Remember: i = 1 5 9( 1) 0 i i i 17 ( + 3i ) (6 + 5i ) Oct 4 :4 PM Oct 4 :58 PM 4
5 (7 + 4i ) ( + i ) Oct 4 :59 PM Oct 4 3:06 PM (4 + 3i) 3i 4 (-5 + 4i) Oct 4 3:08 PM Oct 4 3:4 PM (- - 6i) The Absolute Value of a Complex Number is the distance the Complex Number is from the origin (0,0) on a coordinate plane. z = a + b where z is the complex number a + bi Oct 4 3:5 PM Oct 4 3:8 PM 5
6 Review DISTANCE FORMULA (x - x 1 ) + (y - y 1 ) Given A (, 5) B ( 7, 4), find AB. (-7-4) + ( - -5) (-11) + (7) (11) + (49) 170 Review DISTANCE FORMULA (x - x 1 ) + (y - y 1 ) Given C (4, ) D (3, 7), find CD. Oct 4 3:38 PM Oct 4 3:50 PM (x - x 1 ) + (y - y 1 ) complex number (5 + 3i) so (5, 3) origin (0, 0) (5-0) + (3-0) (5) + (3) z = a + b Remember Find the absolute value of the complex number ( + 7i). Oct 4 3:5 PM Oct 4 4:04 PM Find 4 + 9i. Oct 4 4:05 PM 6
Solving Quadratic Equations by Formula
Algebra Unit: 05 Lesson: 0 Complex Numbers All the quadratic equations solved to this point have had two real solutions or roots. In some cases, solutions involved a double root, but there were always
More informationCHAPTER EIGHT: SOLVING QUADRATIC EQUATIONS Review April 9 Test April 17 The most important equations at this level of mathematics are quadratic
CHAPTER EIGHT: SOLVING QUADRATIC EQUATIONS Review April 9 Test April 17 The most important equations at this level of mathematics are quadratic equations. They can be solved using a graph, a perfect square,
More informationUNIT 5 QUADRATIC FUNCTIONS Lesson 2: Creating and Solving Quadratic Equations in One Variable Instruction
Prerequisite Skills This lesson requires the use of the following skills: simplifying radicals working with complex numbers Introduction You can determine how far a ladder will extend from the base of
More informationMath 1302 Notes 2. How many solutions? What type of solution in the real number system? What kind of equation is it?
Math 1302 Notes 2 We know that x 2 + 4 = 0 has How many solutions? What type of solution in the real number system? What kind of equation is it? What happens if we enlarge our current system? Remember
More informationComplex Numbers: Definition: A complex number is a number of the form: z = a + bi where a, b are real numbers and i is a symbol with the property: i
Complex Numbers: Definition: A complex number is a number of the form: z = a + bi where a, b are real numbers and i is a symbol with the property: i 2 = 1 Sometimes we like to think of i = 1 We can treat
More informationInstructor Quick Check: Question Block 11
Instructor Quick Check: Question Block 11 How to Administer the Quick Check: The Quick Check consists of two parts: an Instructor portion which includes solutions and a Student portion with problems for
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 informationA2 HW Imaginary Numbers
Name: A2 HW Imaginary Numbers Rewrite the following in terms of i and in simplest form: 1) 100 2) 289 3) 15 4) 4 81 5) 5 12 6) -8 72 Rewrite the following as a radical: 7) 12i 8) 20i Solve for x in simplest
More informationCHAPTER 3: Quadratic Functions and Equations; Inequalities
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 3: Quadratic Functions and Equations; Inequalities 3.1 The Complex Numbers 3.2 Quadratic Equations, Functions, Zeros, and
More informationChapter 6 Complex Numbers
Chapter 6 Complex Numbers Lesson 1: Imaginary Numbers Lesson 2: Complex Numbers Lesson 3: Quadratic Formula Lesson 4: Discriminant This assignment is a teacher-modified version of Algebra 2 Common Core
More information5.1 Monomials. Algebra 2
. Monomials Algebra Goal : A..: Add, subtract, multiply, and simplify polynomials and rational expressions (e.g., multiply (x ) ( x + ); simplify 9x x. x Goal : Write numbers in scientific notation. Scientific
More informationSeptember 12, Math Analysis Ch 1 Review Solutions. #1. 8x + 10 = 4x 30 4x 4x 4x + 10 = x = x = 10.
#1. 8x + 10 = 4x 30 4x 4x 4x + 10 = 30 10 10 4x = 40 4 4 x = 10 Sep 5 7:00 AM 1 #. 4 3(x + ) = 5x 7(4 x) 4 3x 6 = 5x 8 + 7x CLT 3x = 1x 8 +3x +3x = 15x 8 +8 +8 6 = 15x 15 15 x = 6 15 Sep 5 7:00 AM #3.
More informationSOLUTIONS FOR PROBLEMS 1-30
. Answer: 5 Evaluate x x + 9 for x SOLUTIONS FOR PROBLEMS - 0 When substituting x in x be sure to do the exponent before the multiplication by to get (). + 9 5 + When multiplying ( ) so that ( 7) ( ).
More informationRoots are: Solving Quadratics. Graph: y = 2x 2 2 y = x 2 x 12 y = x 2 + 6x + 9 y = x 2 + 6x + 3. real, rational. real, rational. real, rational, equal
Solving Quadratics Graph: y = 2x 2 2 y = x 2 x 12 y = x 2 + 6x + 9 y = x 2 + 6x + 3 Roots are: real, rational real, rational real, rational, equal real, irrational 1 To find the roots algebraically, make
More informationMultiplication of Polynomials
Summary 391 Chapter 5 SUMMARY Section 5.1 A polynomial in x is defined by a finite sum of terms of the form ax n, where a is a real number and n is a whole number. a is the coefficient of the term. n is
More informationQuadratic Formula: - another method for solving quadratic equations (ax 2 + bx + c = 0)
In the previous lesson we showed how to solve quadratic equations that were not factorable and were not perfect squares by making perfect square trinomials using a process called completing the square.
More informationMath 3 Variable Manipulation Part 4 Polynomials B COMPLEX NUMBERS A Complex Number is a combination of a Real Number and an Imaginary Number:
Math 3 Variable Manipulation Part 4 Polynomials B COMPLEX NUMBERS A Complex Number is a combination of a Real Number and an Imaginary Number: 1 Examples: 1 + i 39 + 3i 0.8.i + πi + i/ A Complex Number
More informationLesson 3.5 Exercises, pages
Lesson 3.5 Exercises, pages 232 238 A 4. Calculate the value of the discriminant for each quadratic equation. a) 5x 2-9x + 4 = 0 b) 3x 2 + 7x - 2 = 0 In b 2 4ac, substitute: In b 2 4ac, substitute: a 5,
More informationUnit 2-1: Factoring and Solving Quadratics. 0. I can add, subtract and multiply polynomial expressions
CP Algebra Unit -1: Factoring and Solving Quadratics NOTE PACKET Name: Period Learning Targets: 0. I can add, subtract and multiply polynomial expressions 1. I can factor using GCF.. I can factor by grouping.
More informationComplex Numbers. The Imaginary Unit i
292 Chapter 2 Polynomial and Rational Functions SECTION 2.1 Complex Numbers Objectives Add and subtract complex numbers. Multiply complex numbers. Divide complex numbers. Perform operations with square
More informationPre-Calculus Summer Packet
2013-2014 Pre-Calculus Summer Packet 1. Complete the attached summer packet, which is due on Friday, September 6, 2013. 2. The material will be reviewed in class on Friday, September 6 and Monday, September
More informationSkills Practice Skills Practice for Lesson 4.1
Skills Practice Skills Practice for Lesson.1 Name Date Thinking About Numbers Counting Numbers, Whole Numbers, Integers, Rational and Irrational Numbers Vocabulary Define each term in your own words. 1.
More informationMath 2 Variable Manipulation Part 3 COMPLEX NUMBERS A Complex Number is a combination of a Real Number and an Imaginary Number:
Math 2 Variable Manipulation Part 3 COMPLEX NUMBERS A Complex Number is a combination of a Real Number and an Imaginary Number: 1 Examples: 1 + i 39 + 3i 0.8 2.2i 2 + πi 2 + i/2 A Complex Number is just
More informationMathematical Focus 1 Complex numbers adhere to certain arithmetic properties for which they and their complex conjugates are defined.
Situation: Complex Roots in Conjugate Pairs Prepared at University of Georgia Center for Proficiency in Teaching Mathematics June 30, 2013 Sarah Major Prompt: A teacher in a high school Algebra class has
More information3.4. ZEROS OF POLYNOMIAL FUNCTIONS
3.4. ZEROS OF POLYNOMIAL FUNCTIONS What You Should Learn Use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions. Find rational zeros of polynomial functions. Find
More informationChapter 4: Quadratic Functions and Factoring 4.1 Graphing Quadratic Functions in Stand
Chapter 4: Quadratic Functions and Factoring 4.1 Graphing Quadratic Functions in Stand VOCAB: a quadratic function in standard form is written y = ax 2 + bx + c, where a 0 A quadratic Function creates
More information6: Polynomials and Polynomial Functions
6: Polynomials and Polynomial Functions 6-1: Polynomial Functions Okay you know what a variable is A term is a product of constants and powers of variables (for example: x ; 5xy ) For now, let's restrict
More informationLesson 21 Not So Dramatic Quadratics
STUDENT MANUAL ALGEBRA II / LESSON 21 Lesson 21 Not So Dramatic Quadratics Quadratic equations are probably one of the most popular types of equations that you ll see in algebra. A quadratic equation has
More information4-1 Graphing Quadratic Functions
4-1 Graphing Quadratic Functions Quadratic Function in standard form: f() a b c The graph of a quadratic function is a. y intercept Ais of symmetry -coordinate of verte coordinate of verte 1) f ( ) 4 a=
More informationFinding the Equation of a Graph. I can give the equation of a curve given just the roots.
National 5 W 7th August Finding the Equation of a Parabola Starter Sketch the graph of y = x - 8x + 15. On your sketch clearly identify the roots, axis of symmetry, turning point and y intercept. Today
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 information5.3. Polynomials and Polynomial Functions
5.3 Polynomials and Polynomial Functions Polynomial Vocabulary Term a number or a product of a number and variables raised to powers Coefficient numerical factor of a term Constant term which is only a
More informationCHAPTER 1 QUADRATIC FUNCTIONS AND FACTORING
CHAPTER 1 QUADRATIC FUNCTIONS AND FACTORING Big IDEAS: 1) Graphing and writing quadratic functions in several forms ) Solving quadratic equations using a variety of methods 3) Performing operations with
More informationB. Complex number have a Real part and an Imaginary part. 1. written as a + bi some Examples: 2+3i; 7+0i; 0+5i
Section 11.8 Complex Numbers I. The Complex Number system A. The number i = -1 1. 9 and 24 B. Complex number have a Real part and an Imaginary part II. Powers of i 1. written as a + bi some Examples: 2+3i;
More informationMultiplying a Monomial times a Monomial. To multiply a monomial term times a monomial term with radicals you use the following rule A B C D = A C B D
Section 7 4A: Multiplying Radical Expressions Multiplying a Monomial times a Monomial To multiply a monomial term times a monomial term with radicals you use the following rule A B C D = A C B D In other
More informationSection 3.6 Complex Zeros
04 Chapter Section 6 Complex Zeros When finding the zeros of polynomials, at some point you're faced with the problem x = While there are clearly no real numbers that are solutions to this equation, leaving
More informationSection 1.3 Review of Complex Numbers
1 Section 1. Review of Complex Numbers Objective 1: Imaginary and Complex Numbers In Science and Engineering, such quantities like the 5 occur all the time. So, we need to develop a number system that
More informationMidterm 3 Review. Terms. Formulas and Rules to Use. Math 1010, Fall 2011 Instructor: Marina Gresham. Odd Root ( n x where n is odd) Exponent
Math 1010, Fall 2011 Instructor: Marina Gresham Terms Midterm 3 Review Exponent Polynomial - Monomial - Binomial - Trinomial - Standard Form - Degree - Leading Coefficient - Constant Term Difference of
More informationSection 4.3: Quadratic Formula
Objective: Solve quadratic equations using the quadratic formula. In this section we will develop a formula to solve any quadratic equation ab c 0 where a b and c are real numbers and a 0. Solve for this
More informationUnit 5 Solving Quadratic Equations
SM Name: Period: Unit 5 Solving Quadratic Equations 5.1 Solving Quadratic Equations by Factoring Quadratic Equation: Any equation that can be written in the form a b c + + = 0, where a 0. Zero Product
More informationLesson #33 Solving Incomplete Quadratics
Lesson # Solving Incomplete Quadratics A.A.4 Know and apply the technique of completing the square ~ 1 ~ We can also set up any quadratic to solve it in this way by completing the square, the technique
More informationMath 1 Variable Manipulation Part 6 Polynomials
Name: Math 1 Variable Manipulation Part 6 Polynomials Date: 1 VOCABULARY Constant: A term that does not have a variable is called a constant. Example: the number 5 is a constant because it does not have
More informationAssociative Property. The word "associative" comes from "associate" or "group; the Associative Property is the rule that refers to grouping.
Associative Property The word "associative" comes from "associate" or "group; the Associative Property is the rule that refers to grouping. For addition, the rule is "a + (b + c) = (a + b) + c; in numbers,
More informationDegree of a polynomial
Variable Algebra Term Polynomial Monomial Binomial Trinomial Degree of a term Degree of a polynomial Linear A generalization of arithmetic. Letters called variables are used to denote numbers, which are
More informationReview 1. 1 Relations and Functions. Review Problems
Review 1 1 Relations and Functions Objectives Relations; represent a relation by coordinate pairs, mappings and equations; functions; evaluate a function; domain and range; operations of functions. Skills
More informationSect Complex Numbers
161 Sect 10.8 - Complex Numbers Concept #1 Imaginary Numbers In the beginning of this chapter, we saw that the was undefined in the real numbers since there is no real number whose square is equal to a
More informationImaginary numbers and real numbers make up the set of complex numbers. Complex numbers are written in the form: a + bi. Real part ~ -- Imaginary part
C Complex Numbers Imaginary numbers and real numbers make up the set of Complex numbers are written in the form: a + bi Real part ~ -- Imaginary part Rules for Simplifying Complex Numbers 1. Identify the
More informationMath-2 Lesson 2-4. Radicals
Math- Lesson - Radicals = What number is equivalent to the square root of? Square both sides of the equation ( ) ( ) = = = is an equivalent statement to = 1.7 1.71 1.70 1.701 1.7008... There is no equivalent
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 informationMath 2 Variable Manipulation Part 3 Polynomials A
Math 2 Variable Manipulation Part 3 Polynomials A 1 MATH 1 REVIEW: VOCABULARY Constant: A term that does not have a variable is called a constant. Example: the number 5 is a constant because it does not
More informationAlgebra I Unit Report Summary
Algebra I Unit Report Summary No. Objective Code NCTM Standards Objective Title Real Numbers and Variables Unit - ( Ascend Default unit) 1. A01_01_01 H-A-B.1 Word Phrases As Algebraic Expressions 2. A01_01_02
More informationSection 1.1. Chapter 1. Quadratics. Parabolas. Example. Example. ( ) = ax 2 + bx + c -2-1
Chapter 1 Quadratic Functions and Factoring Section 1.1 Graph Quadratic Functions in Standard Form Quadratics The polynomial form of a quadratic function is: f x The graph of a quadratic function is a
More informationAlg. 1 Radical Notes
Alg. 1 Radical Notes Evaluating Square Roots and Cube Roots (Day 1) Objective: SWBAT find the square root and cube roots of monomials Perfect Squares: Perfect Cubes: 1 =11 1 = 11 =1111 11 1 =111 1 1 =
More informationCP Algebra 2. Unit 2-1 Factoring and Solving Quadratics
CP Algebra Unit -1 Factoring and Solving Quadratics Name: Period: 1 Unit -1 Factoring and Solving Quadratics Learning Targets: 1. I can factor using GCF.. I can factor by grouping. Factoring Quadratic
More informationSolving Quadratic Equations Review
Math III Unit 2: Polynomials Notes 2-1 Quadratic Equations Solving Quadratic Equations Review Name: Date: Period: Some quadratic equations can be solved by. Others can be solved just by using. ANY quadratic
More informationPerform the following operations. 1) (2x + 3) + (4x 5) 2) 2(x + 3) 3) 2x (x 4) 4) (2x + 3)(3x 5) 5) (x 4)(x 2 3x + 5)
2/24 week Add subtract polynomials 13.1 Multiplying Polynomials 13.2 Radicals 13.6 Completing the square 13.7 Real numbers 15.1 and 15.2 Complex numbers 15.3 and 15.4 Perform the following operations 1)
More informationDay 3 (2-1) Daily Objective: I can transform quadratic functions. I can describe the effects of changes in the coefficients of y = a(x h) 2 +k.
Day 1 (1-1) I can apply transformations to points and sets of points. I can interpret transformations of real-world data. E.2.b Use transformations to draw the graph of a relation and determine a relation
More informationStudy Guide for Math 095
Study Guide for Math 095 David G. Radcliffe November 7, 1994 1 The Real Number System Writing a fraction in lowest terms. 1. Find the largest number that will divide into both the numerator and the denominator.
More informationP.6 Complex Numbers. -6, 5i, 25, -7i, 5 2 i + 2 3, i, 5-3i, i. DEFINITION Complex Number. Operations with Complex Numbers
SECTION P.6 Complex Numbers 49 P.6 Complex Numbers What you ll learn about Complex Numbers Operations with Complex Numbers Complex Conjugates and Division Complex Solutions of Quadratic Equations... and
More information2.5 Operations With Complex Numbers in Rectangular Form
2.5 Operations With Complex Numbers in Rectangular Form The computer-generated image shown is called a fractal. Fractals are used in many ways, such as making realistic computer images for movies and squeezing
More informationSection 5.5 Complex Numbers
Name: Period: Section 5.5 Comple Numbers Objective(s): Perform operations with comple numbers. Essential Question: Tell whether the statement is true or false, and justify your answer. Every comple number
More informationRadicals: To simplify means that 1) no radicand has a perfect square factor and 2) there is no radical in the denominator (rationalize).
Summer Review Packet for Students Entering Prealculus Radicals: To simplify means that 1) no radicand has a perfect square factor and ) there is no radical in the denominator (rationalize). Recall the
More informationNUMBERS( A group of digits, denoting a number, is called a numeral. Every digit in a numeral has two values:
NUMBERS( A number is a mathematical object used to count and measure. A notational symbol that represents a number is called a numeral but in common use, the word number can mean the abstract object, the
More informationQUADRATIC EQUATIONS. + 6 = 0 This is a quadratic equation written in standard form. x x = 0 (standard form with c=0). 2 = 9
QUADRATIC EQUATIONS A quadratic equation is always written in the form of: a + b + c = where a The form a + b + c = is called the standard form of a quadratic equation. Eamples: 5 + 6 = This is a quadratic
More informationQuestion Bank Quadratic Equations
Question Bank Quadratic Equations 1. Solve the following equations for x : (i) 8x + 15 6x (ii) x (x + 5) 5 Solution. (i) Given 8x + 15 6x 8x 6x + 15 0 [Putting in the form as ax + bx + c 0] 8x 0x 6x +
More informationIn a quadratic expression the highest power term is a square. E.g. x x 2 2x 5x 2 + x - 3
A. Quadratic expressions B. The difference of two squares In a quadratic expression the highest power term is a square. E.g. x + 3x x 5x + x - 3 If a quadratic expression has no x term and both terms are
More informationBeginning Algebra. 1. Review of Pre-Algebra 1.1 Review of Integers 1.2 Review of Fractions
1. Review of Pre-Algebra 1.1 Review of Integers 1.2 Review of Fractions Beginning Algebra 1.3 Review of Decimal Numbers and Square Roots 1.4 Review of Percents 1.5 Real Number System 1.6 Translations:
More information30 Wyner Math Academy I Fall 2015
30 Wyner Math Academy I Fall 2015 CHAPTER FOUR: QUADRATICS AND FACTORING Review November 9 Test November 16 The most common functions in math at this level are quadratic functions, whose graphs are parabolas.
More informationHerndon High School Geometry Honors Summer Assignment
Welcome to Geometry! This summer packet is for all students enrolled in Geometry Honors at Herndon High School for Fall 07. The packet contains prerequisite skills that you will need to be successful in
More informationUnit 5 Test: 9.1 Quadratic Graphs and Their Properties
Unit 5 Test: 9.1 Quadratic Graphs and Their Properties Quadratic Equation: (Also called PARABOLAS) 1. of the STANDARD form y = ax 2 + bx + c 2. a, b, c are all real numbers and a 0 3. Always have an x
More informationChapter 1 Notes: Quadratic Functions
19 Chapter 1 Notes: Quadratic Functions (Textbook Lessons 1.1 1.2) Graphing Quadratic Function A function defined by an equation of the form, The graph is a U-shape called a. Standard Form Vertex Form
More informationPENNSYLVANIA. Page 1 of 5
Know: Understand: Do: CC.2.1.HS.F.7 -- Essential Apply concepts of complex numbers in polynomial identities and quadratic equations to solve problems. CC.2.2.HS.D.4 -- Essential Understand the relationship
More informationALGEBRA 1B GOALS. 1. The student should be able to use mathematical properties to simplify algebraic expressions.
GOALS 1. The student should be able to use mathematical properties to simplify algebraic expressions. 2. The student should be able to add, subtract, multiply, divide, and compare real numbers. 3. The
More informationSECTION Types of Real Numbers The natural numbers, positive integers, or counting numbers, are
SECTION.-.3. Types of Real Numbers The natural numbers, positive integers, or counting numbers, are The negative integers are N = {, 2, 3,...}. {..., 4, 3, 2, } The integers are the positive integers,
More information2.1 Notes: Simplifying Radicals (Real and Imaginary)
Chapter 2 Calendar Name: Day Date Assignment (Due the next class meeting) Friday Tuesday Wednesday Thursday Friday Monday Tuesday Wednesday Thursday Friday Monday Tuesday Wednesday Thursday 8/30/13 (A)
More informationMATH 190 KHAN ACADEMY VIDEOS
MATH 10 KHAN ACADEMY VIDEOS MATTHEW AUTH 11 Order of operations 1 The Real Numbers (11) Example 11 Worked example: Order of operations (PEMDAS) 7 2 + (7 + 3 (5 2)) 4 2 12 Rational + Irrational Example
More informationA Add, subtract, multiply, and simplify polynomials and rational expressions.
ED 337 Paul Garrett Selected Response Assessment 10 th Grade Mathematics Examination Algebra II Clear Purpose: The purpose of this selected response is to ensure an understanding of expressions, manipulation
More informationCourse Name: MAT 135 Spring 2017 Master Course Code: N/A. ALEKS Course: Intermediate Algebra Instructor: Master Templates
Course Name: MAT 135 Spring 2017 Master Course Code: N/A ALEKS Course: Intermediate Algebra Instructor: Master Templates Course Dates: Begin: 01/15/2017 End: 05/31/2017 Course Content: 279 Topics (207
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 informationR1: Sets A set is a collection of objects sets are written using set brackets each object in onset is called an element or member
Chapter R Review of basic concepts * R1: Sets A set is a collection of objects sets are written using set brackets each object in onset is called an element or member Ex: Write the set of counting numbers
More information5-9. Complex Numbers. Key Concept. Square Root of a Negative Real Number. Key Concept. Complex Numbers VOCABULARY TEKS FOCUS ESSENTIAL UNDERSTANDING
TEKS FOCUS 5-9 Complex Numbers VOCABULARY TEKS (7)(A) Add, subtract, and multiply complex TEKS (1)(F) Analyze mathematical relationships to connect and communicate mathematical ideas. Additional TEKS (1)(D),
More information7x 5 x 2 x + 2. = 7x 5. (x + 1)(x 2). 4 x
Advanced Integration Techniques: Partial Fractions The method of partial fractions can occasionally make it possible to find the integral of a quotient of rational functions. Partial fractions gives us
More informationGrade 7/8 Math Circles Winter March 20/21/22 Types of Numbers
Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 7/8 Math Circles Winter 2018 - March 20/21/22 Types of Numbers Introduction Today, we take our number
More informationALGEBRA 2 Summer Review Assignments Graphing
ALGEBRA 2 Summer Review Assignments Graphing To be prepared for algebra two, and all subsequent math courses, you need to be able to accurately and efficiently find the slope of any line, be able to write
More informationThe Quadratic Formula, the Discriminant, and Solving Quadratic Equations and Inequalities
CHAPTER The Quadratic Formula, the Discriminant, and Solving Quadratic Equations and Inequalities 009 Carnegie Learning, Inc. The Chinese invented rockets over 700 years ago. Since then rockets have been
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/TRIGONMETRY TOPIC REVIEW QUARTER 2 POWERS OF I
ALGEBRA /TRIGONMETRY TOPIC REVIEW QUARTER Imaginary Unit: i = i i i i 0 = = i = = i Imaginary numbers appear when you have a negative number under a radical. POWERS OF I Higher powers if i: If you have
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 informationCambium Learning Voyager
Product Sample For questions or more information, contact: Cambium Learning Voyager 17855 Dallas Parkway, Ste. 400 Dallas, TX 7587 1 888 399 1995 www.voyagerlearning.com 9CHAPTER Objective 5 Solve quadratic
More information3.3 Real Zeros of Polynomial Functions
71_00.qxp 12/27/06 1:25 PM Page 276 276 Chapter Polynomial and Rational Functions. Real Zeros of Polynomial Functions Long Division of Polynomials Consider the graph of f x 6x 19x 2 16x 4. Notice in Figure.2
More informationNAME DATE PERIOD. A negative exponent is the result of repeated division. Extending the pattern below shows that 4 1 = 1 4 or 1. Example: 6 4 = 1 6 4
Lesson 4.1 Reteach Powers and Exponents A number that is expressed using an exponent is called a power. The base is the number that is multiplied. The exponent tells how many times the base is used as
More information83. 31x + 2x + 9 = 3. Review Exercises. 85. Divide using synthetic division: 86. Divide: 90. Rationalize the denominator: Complex Numbers
718 CHAPTER 10 Radicals, Radical Functions, and Rational Exponents 76. Now that I know how to solve radical equations, I can use models that are radical functions to determine the value of the independent
More informationS4 (4.3) Quadratic Functions.notebook February 06, 2018
Daily Practice 2.11.2017 Q1. Multiply out and simplify 3g - 5(2g + 4) Q2. Simplify Q3. Write with a rational denominator Today we will be learning about quadratic functions and their graphs. Q4. State
More informationQuadratic Functions. Key Terms. Slide 1 / 200. Slide 2 / 200. Slide 3 / 200. Table of Contents
Slide 1 / 200 Quadratic Functions Table of Contents Key Terms Identify Quadratic Functions Explain Characteristics of Quadratic Functions Solve Quadratic Equations by Graphing Solve Quadratic Equations
More informationQuadratic Functions. Key Terms. Slide 2 / 200. Slide 1 / 200. Slide 3 / 200. Slide 4 / 200. Slide 6 / 200. Slide 5 / 200.
Slide 1 / 200 Quadratic Functions Slide 2 / 200 Table of Contents Key Terms Identify Quadratic Functions Explain Characteristics of Quadratic Functions Solve Quadratic Equations by Graphing Solve Quadratic
More informationSlide 1 / 200. Quadratic Functions
Slide 1 / 200 Quadratic Functions Key Terms Slide 2 / 200 Table of Contents Identify Quadratic Functions Explain Characteristics of Quadratic Functions Solve Quadratic Equations by Graphing Solve Quadratic
More informationP3.C8.COMPLEX NUMBERS
Recall: Within the real number system, we can solve equation of the form and b 2 4ac 0. ax 2 + bx + c =0, where a, b, c R What is R? They are real numbers on the number line e.g: 2, 4, π, 3.167, 2 3 Therefore,
More informationSolving Quadratic & Higher Degree Equations
Chapter 9 Solving Quadratic & Higher Degree Equations Sec 1. Zero Product Property Back in the third grade students were taught when they multiplied a number by zero, the product would be zero. In algebra,
More informationChapter R - Basic Algebra Operations (94 topics, no due date)
Course Name: Math 00024 Course Code: N/A ALEKS Course: College Algebra Instructor: Master Templates Course Dates: Begin: 08/15/2014 End: 08/15/2015 Course Content: 207 topics Textbook: Barnett/Ziegler/Byleen/Sobecki:
More information