Imaginary numbers and real numbers make up the set of complex numbers. Complex numbers are written in the form: a + bi. Real part ~ -- Imaginary part

Size: px
Start display at page:

Download "Imaginary numbers and real numbers make up the set of complex numbers. Complex numbers are written in the form: a + bi. Real part ~ -- Imaginary part"

Transcription

1 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 real part of the number. Identify what will become the imaginary part. 2. Simplify the imaginary part of the complex number. 3. Rewrite the complex number in the form a + bi. Rewrite the complex number. v in the form a + bi Step 1 Identify the real part of the number. Identify what will become the imaginary part. Step 2 Simplify the imaginary part of the complex number. Step 3 Rewrite the complex number in the form a + bi. Practice A Rewrite each complex number 1. V-18-2 Identify the real part of the number. Identify what will become the imaginary part. Simplify the imaginary part of the complex number. Rewrite the complex number in the form a + bi. 2. ~+4 3. v v v-81 = imaginary part 3 = real part v-81 = vcf x V8T V-81 = 9i 3 + 9i in the form a + bi. = imaginary part = real part V-18 = vcf x v-18 = v v v Algebra ~ Saddleback Educational Publishing 2006 (888)

2 C Adding Complex Numbers Imaginary numbers and real numbers make up the set of Complex numbers are written in the form: Real part a + bi ~ Imaginary part You can apply what you know about operations with real numbers to any questions with Rules for Adding Complex Numbers 1. For each complex number (in the form a + bi) identify the real part and the imaginary part. 2. Group the real parts together; group the imaginary parts together. 3. Simplify. Express the sum in terms of a + bi. Add. (3 + 4;) + (-2 + 6;) Step 1 For each complex number (in the form a + bi) identify the real part and the imaginary part. Step 2 Group the real parts together; group the imaginary parts together. Step 3 Simplify. Express the sum in terms of a + bi. Real parts = +3 and -2 Imaginary parts = +4i and +6i (3-2)+(4i+6i) 1 + loi Practice :B Add. 1. (-5-4i) + (3 + 6i) For each complex number (in the form a + bi) identify the real part and the imaginary part. Group the real parts together; group the imaginary parts together. Simplify. Express the sum in terms of a + bi. 2. (4 + 4i) + (3 - i) 3. (-7+2i)+(6-6i) 4. (12-3i)+(-12-6i) Real parts = -5 and Imaginary parts = -4i and (-5 + ) + (-4i+ ) (8 + 6i) + (8-6i) 6. (12-3i) + (-9 + i) 7. (4 + v-16) + (2 + v-25) Saddleback Educational Publishing 2006 (888)

3 C Subtracting Complex Numbers You can apply what you know about operations with real numbers to any operation with Rules for Subtracting Complex Numbers 1. For the complex number to the right of the minus sign, change the sign in front of the real part of the number complex and in front of the imaginary part. Change the minus sign (between the two complex numbers) to a plus. 2. For each complex number, identify the real part and the imaginary part. 3. Group the real parts together; group the imaginary parts together. Separate the real part from the imaginary part with a plus sign. 4. Simplify. Express the difference in terms of a + bi. Subtract. (3 + 4;) - (-4 + 2;) Step 1 For the complex number to the right of the minus sign, change the sign in front of the real part of the number complex and in front of the imaginary part. Change the minus sign (between the two complex numbers) to a plus. Step 2 For each complex number, identify the real part and the imaginary part. Step 3 Group the real parts together; group the imaginary parts together. Separate the real part from the imaginary part with a plus sign. Step 4 Simplify. Express the difference in terms of a + bi. (3 + 4i) - (-4 + 2i) = (3 + 4i) + (4-2i) Real parts: 3 and 4 Imaginary parts: 4i and -2i (3+4)+(4i-2i) (3 + 4) + (4i - 2i) = 7 + 2i Practice C!. Subtract. 1. (9 + 4i) - (2 + 5i) Step 1 (9 + 4i) - (2 + 5i) 9+4i+ Step 2 Real parts: 9 and ; imaginary parts: 4i and Step 3 (9 ) + (4i Step 4 2. (10 + 2i) - (4 + i) (2 + 6i) 3. (5-3i) - (-3-2i) 5. (6 + 3i) - 3i 64 Algebra ; Saddleback Educational Publishing 2006 (888)

4 C Multiplying Complex Numbers You can apply what you know about operations with real numbers to any operation with Rules for Multiplying Complex Numbers For two imaginary numbers: 1. Multiply the whole numbers; multiply i by i, if applicable. 2. Remember, i x i = P = -1. For two complex numbers: 1. Use the FOIL method. 2. Remember, i x i = P = Simplify by combining like terms. Multiply. (5 + 7;)(-2 + 6;) Step 1 Use the FOIL method. Step 2 Remember, i xi = P = -1. Step 3 Simplify by combining like terms. Practice J) Multiply. 1. (3+6i)(4-8i) Use the FOIL method. Remember, i x i = i2 = -1. (5 + 7i)(-2 + 6i) = i + (-l4i) + 42P (30i+ (-14i)) + 42(-1) (30i + (-14i)) - 42 = i - 42 = i (3 + 6i)(4-8i) = 12-24i i+ + = 12-24i+ + Simplify by combining like terms = 2. (5 + 6i)(3-4i) 3. (3 + 2i)(5 + 3i) 4. (7 + 3i)(4-2i) 5. (9 + 4i)(3 + 4i) Saddleback Educational Publishing 2006 (888) vwvw.sdlback.com 65

5 C Dividing Complex Numbers You can apply what you know about operations with real numbers to any operation with Rules for Dividing Complex Numbers 1. Multiply numerator and denominator by the complex conjugate of the denominator. Use the same complex number, but with the opposite operation sign between the real and imaginary parts. 2. Follow the rules for multiplying two 3. Simplify. Express the quotient in a + bi form. Solve I2/ Step 1 Multiply numerator and denominator by the complex conjugate of the denominator. Use the same complex number, but with the opposite operation sign between the real and imaginary parts. Step 2 Follow the rules for multiplying two Step 3 Simplify. Express the quotient in a + biform. Practice E:. Solve. 1 5-i i Multiply numerator and denominator by the complex conjugate of the denominator. Use the same complex number, but with the opposite operation sign between the real and imaginary parts. Follow the rules for multiplying two The complex conjugate of 3 - i is 3 + i i X 3 + i 3-i 3+i 2 + 2i X 3 + i = 6 + 2i + 6i + 2i2 3-i 3+i 9-3i+3i-l" i + 6i + 2i2 = 4 + 8i = -±- +!i = 1+ii 9-3i+3i-i The complex conjugate of i is 5-i 2+4ix--- --x 5-i -2 +4i Simplify. Express the quotient a + biform. in =-----= (2 + 4i) -7- (6 + i) 3. (3 + 3i) -7- (2 - i) 4. (-2 + i) -7- (3-2i) 5. (2 + 5i) -7- (8 + i) 66 Saddle back Educational Publishing 2006 (888)

6 C Absolute Value and Complex Numbers You can apply what you know about real numbers, including absolute value, to complex numbers. The absolute value of a complex number is its distance from the origin on the complex number plane. Rules for Finding the Absolute Value of a Complex Number 1. Write the complex number in the form I a + bi I. 2. Put the real number and the coefficient of the imaginary number into the formula Va2 + b2. 3. Simplify. The absolute value is always a positive number. Find 14-3;1. Step 1 Write the complex number in the form 1 a + bi I. 14-3il Step 2 Put the real number and the coefficient of the imaginary number into the formula "';~a2-+-b-2. Step 3 Simplify. The absolute value is always a positive number. Practice F Find the absolute value il Write the complex number in the form I a + bi I. Put the real number and the coefficient of the imaginary number into the formula "';-a-2-+-b ij 14-6il =V + 2 Simplify. The absolute value is always a positive number il il il il 6. j5i i + 21 Saddleback Educational Publishing 2006 (888)

7 C Finding a Complex Solution to a Simple Quadratic Equation You can use the concept of imaginary numbers to help find the solution to certain quadratic equations. The solution of some quadratic equations is a complex number. Rules for Finding Complex Solutions to Simple Quadratic Equations 1. Isolate the term with the i2 variable on one side of the equation. 2. Simplify the equation so you have i2 on one side. 3. Find the square root of each side. The result will be x = ± Solve. 5x = 0 Step 1 Isolate the term with the i2 variable on one side of the equation. Step 2 Simplify the equation so you have i2 on one side. Step 3 Find the square root of each side. The result will be x = ± complex number. Practice Go Solve. 1. 4i = 0 Isolate the term with the i2 variable on one side of the equation. Simplify the equation so you have i2 on one side. Find the square root of each side. The result will be x = ± complex number. 5i = i2 = i = i2 = -25 R =v-25 =v-1 x 25 x= ±i x 5 = ±5i 4i2+32 =0 4i2= 4i2-o- = -o- i2= R=v =v x=± 2. i = i2+48=0 4. i = i2 + 5 = Saddleback Educational Publishing 200G (888)

8 C Finding a Complex Solution to a Quadratic Equation You can apply what you know about operations with real numbers to any operation with You can use the concept of imaginary numbers to help find the solution to certain quadratic equations. The solution of some quadratic equations is a complex number. Rules for Finding Complex Solutions to Quadratic Equations 1. Write the quadratic equation in standard form (ax2+ bx + c = 0). 2. Determine the values for a, b, and c. 3. Plug a, b, and c into the quadratic formula. 4. If the number under the square root sign is negative, apply the concept of imaginary numbers to simplify. Solve. x2 = -4x - 29 Step 1 Write the quadratic equation in standard form (axl + bx + c = 0). Step 2 Determine the values for a, b, and c. Step 3 Plug a, b, and c into the quadratic formula. xl + 4x+ 29 = 0 a = 1, b = 4, c = 29 -b ± ~ -4 ±,/42-4(1)(29) x = 2a 2(1) Step 4 If the number under the square root sign is negative, apply the concept of imaginary numbers to simplify. Practice H Solve. 1. xl = 2x - 26 Write the quadratic equation in standard form (axl + bx + c = 0). Determine the values for a, b, and c. Plug a, b, and c into the quadratic formula. If the number under the square root sign is negative, apply the concept of imaginary numbers to simplify. 2. 6xl = -4x xl-4x=-10 xl - 2x+ 26 = 0 a=,b=,c= -b±~ x= 2a = x=---- ±V- ± 4. 4xl=-16x xl = -10x+ 14 Saddle back Educational Publishing 2006 (888)

When you square a binomial, you can apply the FOIL method to find the product. You can also apply the following rules as a short cut.

When you square a binomial, you can apply the FOIL method to find the product. You can also apply the following rules as a short cut. Squaring a Binomial When you square a binomial, you can apply the FOIL method to find the product. You can also apply the following rules as a short cut. Solve. (x 3) 2 Step 1 Square the first term. Rules

More information

Chapter 1.6. Perform Operations with Complex Numbers

Chapter 1.6. Perform Operations with Complex Numbers Chapter 1.6 Perform Operations with Complex Numbers EXAMPLE Warm-Up 1 Exercises Solve a quadratic equation Solve 2x 2 + 11 = 37. 2x 2 + 11 = 37 2x 2 = 48 Write original equation. Subtract 11 from each

More information

5-9. Complex Numbers. Key Concept. Square Root of a Negative Real Number. Key Concept. Complex Numbers VOCABULARY TEKS FOCUS ESSENTIAL UNDERSTANDING

5-9. 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 information

Solving Quadratic Equations by Formula

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 information

Equations. Rational Equations. Example. 2 x. a b c 2a. Examine each denominator to find values that would cause the denominator to equal zero

Equations. 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 information

5.3. Polynomials and Polynomial Functions

5.3. 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 information

Section 1.3 Review of Complex Numbers

Section 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 information

3-3 Complex Numbers. Simplify. SOLUTION: 2. SOLUTION: 3. (4i)( 3i) SOLUTION: 4. SOLUTION: 5. SOLUTION: esolutions Manual - Powered by Cognero Page 1

3-3 Complex Numbers. Simplify. SOLUTION: 2. SOLUTION: 3. (4i)( 3i) SOLUTION: 4. SOLUTION: 5. SOLUTION: esolutions Manual - Powered by Cognero Page 1 1. Simplify. 2. 3. (4i)( 3i) 4. 5. esolutions Manual - Powered by Cognero Page 1 6. 7. Solve each equation. 8. Find the values of a and b that make each equation true. 9. 3a + (4b + 2)i = 9 6i Set the

More information

CHAPTER 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 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 information

Warm-Up. Simplify the following terms:

Warm-Up. Simplify the following terms: Warm-Up Simplify the following terms: 81 40 20 i 3 i 16 i 82 TEST Our Ch. 9 Test will be on 5/29/14 Complex Number Operations Learning Targets Adding Complex Numbers Multiplying Complex Numbers Rules for

More information

A quadratic expression is a mathematical expression that can be written in the form 2

A 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 information

CHAPTER 3: Quadratic Functions and Equations; Inequalities

CHAPTER 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 information

Review 1. 1 Relations and Functions. Review Problems

Review 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 information

Basic Equation Solving Strategies

Basic Equation Solving Strategies Basic Equation Solving Strategies Case 1: The variable appears only once in the equation. (Use work backwards method.) 1 1. Simplify both sides of the equation if possible.. Apply the order of operations

More information

Solving Quadratic Equations

Solving 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 information

Section 3.6 Complex Zeros

Section 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 information

Math 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: 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 information

Complex Numbers. Copyright Cengage Learning. All rights reserved.

Complex Numbers. Copyright Cengage Learning. All rights reserved. 4 Complex Numbers Copyright Cengage Learning. All rights reserved. 4.1 Complex Numbers Copyright Cengage Learning. All rights reserved. Objectives Use the imaginary unit i to write complex numbers. Add,

More information

Solving Equations Quick Reference

Solving Equations Quick Reference Solving Equations Quick Reference Integer Rules Addition: If the signs are the same, add the numbers and keep the sign. If the signs are different, subtract the numbers and keep the sign of the number

More information

Polynomial and Rational Functions. Copyright Cengage Learning. All rights reserved.

Polynomial and Rational Functions. Copyright Cengage Learning. All rights reserved. 2 Polynomial and Rational Functions Copyright Cengage Learning. All rights reserved. 2.4 Complex Numbers Copyright Cengage Learning. All rights reserved. What You Should Learn Use the imaginary unit i

More information

Secondary Math 2H Unit 3 Notes: Factoring and Solving Quadratics

Secondary 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 information

1 Quadratic Functions

1 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

Solving Linear Equations

Solving Linear Equations Solving Linear Equations Golden Rule of Algebra: Do unto one side of the equal sign as you will do to the other Whatever you do on one side of the equal sign, you MUST do the same exact thing on the other

More information

Radicals: To simplify means that 1) no radicand has a perfect square factor and 2) there is no radical in the denominator (rationalize).

Radicals: 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 information

September 12, Math Analysis Ch 1 Review Solutions. #1. 8x + 10 = 4x 30 4x 4x 4x + 10 = x = x = 10.

September 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 information

Fundamental Theorem of Algebra (NEW): A polynomial function of degree n > 0 has n complex zeros. Some of these zeros may be repeated.

Fundamental Theorem of Algebra (NEW): A polynomial function of degree n > 0 has n complex zeros. Some of these zeros may be repeated. .5 and.6 Comple Numbers, Comple Zeros and the Fundamental Theorem of Algebra Pre Calculus.5 COMPLEX NUMBERS 1. Understand that - 1 is an imaginary number denoted by the letter i.. Evaluate the square root

More information

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

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 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 information

Sect Complex Numbers

Sect 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 information

Mixed Review Write an equation for each problem. Then solve the equation. 1. The difference between 70 and a number is 2. A number minus 13 is 1.

Mixed Review Write an equation for each problem. Then solve the equation. 1. The difference between 70 and a number is 2. A number minus 13 is 1. 1 of 12 2/23/2009 4:28 PM Name Mixed Review Write an equation for each problem. Then solve the equation. 1. The difference between 70 and a number is 2. A number minus 13 is 1. 66. 3. A number multiplied

More information

UNIT 5 QUADRATIC FUNCTIONS Lesson 2: Creating and Solving Quadratic Equations in One Variable Instruction

UNIT 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 information

Study Guide for Math 095

Study 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 information

Complex Numbers. Essential Question What are the subsets of the set of complex numbers? Integers. Whole Numbers. Natural Numbers

Complex Numbers. Essential Question What are the subsets of the set of complex numbers? Integers. Whole Numbers. Natural Numbers 3.4 Complex Numbers Essential Question What are the subsets of the set of complex numbers? In your study of mathematics, you have probably worked with only real numbers, which can be represented graphically

More information

P.6 Complex Numbers. -6, 5i, 25, -7i, 5 2 i + 2 3, i, 5-3i, i. DEFINITION Complex Number. Operations with Complex Numbers

P.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 information

Pre-Calculus Summer Packet

Pre-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 information

Math 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. 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 information

Section 6.6 Evaluating Polynomial Functions

Section 6.6 Evaluating Polynomial Functions Name: Period: Section 6.6 Evaluating Polynomial Functions Objective(s): Use synthetic substitution to evaluate polynomials. Essential Question: Homework: Assignment 6.6. #1 5 in the homework packet. Notes:

More information

To solve a radical equation, you must take both sides of an equation to a power.

To 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 information

SOLUTIONS FOR PROBLEMS 1-30

SOLUTIONS 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 information

5.1 Monomials. Algebra 2

5.1 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 information

B. Complex number have a Real part and an Imaginary part. 1. written as a + bi some Examples: 2+3i; 7+0i; 0+5i

B. 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 information

Chapter 8. Exploring Polynomial Functions. Jennifer Huss

Chapter 8. Exploring Polynomial Functions. Jennifer Huss Chapter 8 Exploring Polynomial Functions Jennifer Huss 8-1 Polynomial Functions The degree of a polynomial is determined by the greatest exponent when there is only one variable (x) in the polynomial Polynomial

More information

A2 HW Imaginary Numbers

A2 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 information

HONORS GEOMETRY Summer Skills Set

HONORS GEOMETRY Summer Skills Set HONORS GEOMETRY Summer Skills Set Algebra Concepts Adding and Subtracting Rational Numbers To add or subtract fractions with the same denominator, add or subtract the numerators and write the sum or difference

More information

SUMMER REVIEW PACKET. Name:

SUMMER 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 information

6-3 Solving Systems by Elimination

6-3 Solving Systems by Elimination Another method for solving systems of equations is elimination. Like substitution, the goal of elimination is to get one equation that has only one variable. To do this by elimination, you add the two

More information

Algebra I Unit Report Summary

Algebra 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 information

Complex Numbers. The Imaginary Unit i

Complex 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 information

Section 6.1/6.2* 2x2 Linear Systems and some other Systems/Applications

Section 6.1/6.2* 2x2 Linear Systems and some other Systems/Applications Section 6.1/6.2* 2x2 Linear Systems and some other Systems/Applications Solving 2x2 Linear Systems ax by c To solve a system of two linear equations means to find values for x and y dx ey f that satisfy

More information

Lesson 7.1 Polynomial Degree and Finite Differences

Lesson 7.1 Polynomial Degree and Finite Differences Lesson 7.1 Polynomial Degree and Finite Differences 1. Identify the degree of each polynomial. a. 3x 4 2x 3 3x 2 x 7 b. x 1 c. 0.2x 1.x 2 3.2x 3 d. 20 16x 2 20x e. x x 2 x 3 x 4 x f. x 2 6x 2x 6 3x 4 8

More information

) z r θ ( ) ( ) ( ) = then. Complete Solutions to Examination Questions Complete Solutions to Examination Questions 10.

) z r θ ( ) ( ) ( ) = then. Complete Solutions to Examination Questions Complete Solutions to Examination Questions 10. Complete Solutions to Examination Questions 0 Complete Solutions to Examination Questions 0. (i We need to determine + given + j, j: + + j + j (ii The product ( ( + j6 + 6 j 8 + j is given by ( + j( j

More information

1. Definition of a Polynomial

1. Definition of a Polynomial 1. Definition of a Polynomial What is a polynomial? A polynomial P(x) is an algebraic expression of the form Degree P(x) = a n x n + a n 1 x n 1 + a n 2 x n 2 + + a 3 x 3 + a 2 x 2 + a 1 x + a 0 Leading

More information

3 COMPLEX NUMBERS. 3.0 Introduction. Objectives

3 COMPLEX NUMBERS. 3.0 Introduction. Objectives 3 COMPLEX NUMBERS Objectives After studying this chapter you should understand how quadratic equations lead to complex numbers and how to plot complex numbers on an Argand diagram; be able to relate graphs

More information

Eby, MATH 0310 Spring 2017 Page 53. Parentheses are IMPORTANT!! Exponents only change what they! So if a is not inside parentheses, then it

Eby, MATH 0310 Spring 2017 Page 53. Parentheses are IMPORTANT!! Exponents only change what they! So if a is not inside parentheses, then it Eby, MATH 010 Spring 017 Page 5 5.1 Eponents Parentheses are IMPORTANT!! Eponents only change what they! So if a is not inside parentheses, then it get raised to the power! Eample 1 4 b) 4 c) 4 ( ) d)

More information

Quadratic Functions. College Algebra

Quadratic Functions. College Algebra Quadratic Functions College Algebra Imaginary Numbers We can find the square root of a negative number, but it is not a real number. If the value in the radicand is negative, the root is said to be an

More information

Order of Operations Practice: 1) =

Order of Operations Practice: 1) = Order of Operations Practice: 1) 24-12 3 + 6 = a) 6 b) 42 c) -6 d) 192 2) 36 + 3 3 (1/9) - 8 (12) = a) 130 b) 171 c) 183 d) 4,764 1 3) Evaluate: 12 2-4 2 ( - ½ ) + 2 (-3) 2 = 4) Evaluate 3y 2 + 8x =, when

More information

Chapter 2 Polynomial and Rational Functions

Chapter 2 Polynomial and Rational Functions Chapter 2 Polynomial and Rational Functions Overview: 2.2 Polynomial Functions of Higher Degree 2.3 Real Zeros of Polynomial Functions 2.4 Complex Numbers 2.5 The Fundamental Theorem of Algebra 2.6 Rational

More information

Math 1320, Section 10 Quiz IV Solutions 20 Points

Math 1320, Section 10 Quiz IV Solutions 20 Points Math 1320, Section 10 Quiz IV Solutions 20 Points Please answer each question. To receive full credit you must show all work and give answers in simplest form. Cell phones and graphing calculators are

More information

Section 4.3: Quadratic Formula

Section 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 information

CH 73 THE QUADRATIC FORMULA, PART II

CH 73 THE QUADRATIC FORMULA, PART II 1 CH THE QUADRATIC FORMULA, PART II INTRODUCTION W ay back in Chapter 55 we used the Quadratic Formula to solve quadratic equations like 6x + 1x + 0 0, whose solutions are 5 and 8. In fact, all of the

More information

CH 54 PREPARING FOR THE QUADRATIC FORMULA

CH 54 PREPARING FOR THE QUADRATIC FORMULA 1 CH 54 PREPARING FOR THE QUADRATIC FORMULA Introduction W e re pretty good by now at solving equations like (3x 4) + 8 10(x + 1), and we ve had a whole boatload of word problems which can be solved by

More information

P3.C8.COMPLEX NUMBERS

P3.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 information

A polynomial expression is the addition or subtraction of many algebraic terms with positive integer powers.

A 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 information

Chapter 2. Polynomial and Rational Functions. 2.5 Zeros of Polynomial Functions

Chapter 2. Polynomial and Rational Functions. 2.5 Zeros of Polynomial Functions Chapter 2 Polynomial and Rational Functions 2.5 Zeros of Polynomial Functions 1 / 33 23 Chapter 2 Homework 2.5 p335 6, 8, 10, 12, 16, 20, 24, 28, 32, 34, 38, 42, 46, 50, 52 2 / 33 23 3 / 33 23 Objectives:

More information

MathB65 Ch 4 IV, V, VI.notebook. October 31, 2017

MathB65 Ch 4 IV, V, VI.notebook. October 31, 2017 Part 4: Polynomials I. Exponents & Their Properties II. Negative Exponents III. Scientific Notation IV. Polynomials V. Addition & Subtraction of Polynomials VI. Multiplication of Polynomials VII. Greatest

More information

Factorizing Algebraic Expressions

Factorizing Algebraic Expressions 1 of 60 Factorizing Algebraic Expressions 2 of 60 Factorizing expressions Factorizing an expression is the opposite of expanding it. Expanding or multiplying out a(b + c) ab + ac Factorizing Often: When

More information

(x + 1)(x 2) = 4. x

(x + 1)(x 2) = 4. x dvanced 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 information

Day 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 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 information

Partial Fractions. June 27, In this section, we will learn to integrate another class of functions: the rational functions.

Partial Fractions. June 27, In this section, we will learn to integrate another class of functions: the rational functions. Partial Fractions June 7, 04 In this section, we will learn to integrate another class of functions: the rational functions. Definition. A rational function is a fraction of two polynomials. For example,

More information

Lesson 3 Algebraic expression: - the result obtained by applying operations (+, -,, ) to a collection of numbers and/or variables o

Lesson 3 Algebraic expression: - the result obtained by applying operations (+, -,, ) to a collection of numbers and/or variables o Lesson 3 Algebraic expression: - the result obtained by applying operations (+, -,, ) to a collection of numbers and/or variables o o ( 1)(9) 3 ( 1) 3 9 1 Evaluate the second expression at the left, if

More information

Equations in Quadratic Form

Equations 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 information

What you may need to do: 1. Formulate a quadratic expression or equation. Generate a quadratic expression from a description or diagram.

What you may need to do: 1. Formulate a quadratic expression or equation. Generate a quadratic expression from a description or diagram. Dealing with a quadratic What it is: A quadratic expression is an algebraic expression containing an x 2 term, as well as possibly an x term and/or a number, but nothing else - eg, no x 3 term. The general

More information

REAL WORLD SCENARIOS: PART IV {mostly for those wanting 114 or higher} 1. If 4x + y = 110 where 10 < x < 20, what is the least possible value of y?

REAL WORLD SCENARIOS: PART IV {mostly for those wanting 114 or higher} 1. If 4x + y = 110 where 10 < x < 20, what is the least possible value of y? REAL WORLD SCENARIOS: PART IV {mostly for those wanting 114 or higher} REAL WORLD SCENARIOS 1. If 4x + y = 110 where 10 < x < 0, what is the least possible value of y? WORK AND ANSWER SECTION. Evaluate

More information

Chapter 2 Formulas and Definitions:

Chapter 2 Formulas and Definitions: Chapter 2 Formulas and Definitions: (from 2.1) Definition of Polynomial Function: Let n be a nonnegative integer and let a n,a n 1,...,a 2,a 1,a 0 be real numbers with a n 0. The function given by f (x)

More information

P4 Polynomials and P5 Factoring Polynomials

P4 Polynomials and P5 Factoring Polynomials P4 Polynomials and P5 Factoring Polynomials Professor Tim Busken Graduate T.A. Dynamical Systems Program Department of Mathematics San Diego State University June 22, 2011 Professor Tim Busken (Graduate

More information

Midterm 3 Review. Terms. Formulas and Rules to Use. Math 1010, Fall 2011 Instructor: Marina Gresham. Odd Root ( n x where n is odd) Exponent

Midterm 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 information

Alg. 1 Radical Notes

Alg. 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 information

Dear Future Pre-Calculus Students,

Dear Future Pre-Calculus Students, Dear Future Pre-Calculus Students, Congratulations on your academic achievements thus far. You have proven your academic worth in Algebra II (CC), but the challenges are not over yet! Not to worry; this

More information

Algebra II (Common Core) Summer Assignment Due: September 11, 2017 (First full day of classes) Ms. Vella

Algebra II (Common Core) Summer Assignment Due: September 11, 2017 (First full day of classes) Ms. Vella 1 Algebra II (Common Core) Summer Assignment Due: September 11, 2017 (First full day of classes) Ms. Vella In this summer assignment, you will be reviewing important topics from Algebra I that are crucial

More information

Evaluate the expression if x = 2 and y = 5 6x 2y Original problem Substitute the values given into the expression and multiply

Evaluate the expression if x = 2 and y = 5 6x 2y Original problem Substitute the values given into the expression and multiply Name EVALUATING ALGEBRAIC EXPRESSIONS Objective: To evaluate an algebraic expression Example Evaluate the expression if and y = 5 6x y Original problem 6() ( 5) Substitute the values given into the expression

More information

LT1: Adding and Subtracting Polynomials. *When subtracting polynomials, distribute the negative to the second parentheses. Then combine like terms.

LT1: Adding and Subtracting Polynomials. *When subtracting polynomials, distribute the negative to the second parentheses. Then combine like terms. LT1: Adding and Subtracting Polynomials *When adding polynomials, simply combine like terms. *When subtracting polynomials, distribute the negative to the second parentheses. Then combine like terms. 1.

More information

Partial Fraction Decomposition Honors Precalculus Mr. Velazquez Rm. 254

Partial 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 information

Algebra II Chapter 5: Polynomials and Polynomial Functions Part 1

Algebra II Chapter 5: Polynomials and Polynomial Functions Part 1 Algebra II Chapter 5: Polynomials and Polynomial Functions Part 1 Chapter 5 Lesson 1 Use Properties of Exponents Vocabulary Learn these! Love these! Know these! 1 Example 1: Evaluate Numerical Expressions

More information

Polynomials. Exponents. End Behavior. Writing. Solving Factoring. Graphing. End Behavior. Polynomial Notes. Synthetic Division.

Polynomials. Exponents. End Behavior. Writing. Solving Factoring. Graphing. End Behavior. Polynomial Notes. Synthetic Division. Polynomials Polynomials 1. P 1: Exponents 2. P 2: Factoring Polynomials 3. P 3: End Behavior 4. P 4: Fundamental Theorem of Algebra Writing real root x= 10 or (x+10) local maximum Exponents real root x=10

More information

Roots 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

Roots 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 information

Topic 7: Polynomials. Introduction to Polynomials. Table of Contents. Vocab. Degree of a Polynomial. Vocab. A. 11x 7 + 3x 3

Topic 7: Polynomials. Introduction to Polynomials. Table of Contents. Vocab. Degree of a Polynomial. Vocab. A. 11x 7 + 3x 3 Topic 7: Polynomials Table of Contents 1. Introduction to Polynomials. Adding & Subtracting Polynomials 3. Multiplying Polynomials 4. Special Products of Binomials 5. Factoring Polynomials 6. Factoring

More information

Quadratic Functions. Key Terms. Slide 1 / 200. Slide 2 / 200. Slide 3 / 200. Table of Contents

Quadratic 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 information

Quadratic Functions. Key Terms. Slide 2 / 200. Slide 1 / 200. Slide 3 / 200. Slide 4 / 200. Slide 6 / 200. Slide 5 / 200.

Quadratic 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 information

Slide 1 / 200. Quadratic Functions

Slide 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 information

Never leave a NEGATIVE EXPONENT or a ZERO EXPONENT in an answer in simplest form!!!!!

Never leave a NEGATIVE EXPONENT or a ZERO EXPONENT in an answer in simplest form!!!!! 1 ICM Unit 0 Algebra Rules Lesson 1 Rules of Exponents RULE EXAMPLE EXPLANANTION a m a n = a m+n A) x x 6 = B) x 4 y 8 x 3 yz = When multiplying with like bases, keep the base and add the exponents. a

More information

Beginning Algebra. 1. Review of Pre-Algebra 1.1 Review of Integers 1.2 Review of Fractions

Beginning 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 information

UNIT 2B QUADRATICS II

UNIT 2B QUADRATICS II UNIT 2B QUADRATICS II M2 12.1-8, M2 12.10, M1 4.4 2B.1 Quadratic Graphs Objective I will be able to identify quadratic functions and their vertices, graph them and adjust the height and width of the parabolas.

More information

October 28, Complex Numbers.notebook. Discriminant

October 28, Complex Numbers.notebook. Discriminant 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

More information

Summer Mathematics Packet Say Hello to Algebra 2. For Students Entering Algebra 2

Summer Mathematics Packet Say Hello to Algebra 2. For Students Entering Algebra 2 Summer Math Packet Student Name: Say Hello to Algebra 2 For Students Entering Algebra 2 This summer math booklet was developed to provide students in middle school an opportunity to review grade level

More information

ECS 178 Course Notes QUATERNIONS

ECS 178 Course Notes QUATERNIONS ECS 178 Course Notes QUATERNIONS Kenneth I. Joy Institute for Data Analysis and Visualization Department of Computer Science University of California, Davis Overview The quaternion number system was discovered

More information

All work must be shown or no credit will be awarded. Box all answers!! Order of Operations

All work must be shown or no credit will be awarded. Box all answers!! Order of Operations Steps: All work must be shown or no credit will be awarded. Box all answers!! Order of Operations 1. Do operations that occur within grouping symbols. If there is more than one set of symbols, work from

More information

Solving Quadratic Equations Review

Solving 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 information

Skills Practice Skills Practice for Lesson 4.1

Skills 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 information

ALGEBRAIC LONG DIVISION

ALGEBRAIC LONG DIVISION QUESTIONS: 2014; 2c 2013; 1c ALGEBRAIC LONG DIVISION x + n ax 3 + bx 2 + cx +d Used to find factors and remainders of functions for instance 2x 3 + 9x 2 + 8x + p This process is useful for finding factors

More information

IES Parque Lineal - 2º ESO

IES Parque Lineal - 2º ESO UNIT5. ALGEBRA Contenido 1. Algebraic expressions.... 1 Worksheet: algebraic expressions.... 2 2. Monomials.... 3 Worksheet: monomials.... 5 3. Polynomials... 6 Worksheet: polynomials... 9 4. Factorising....

More information

2.1 Quadratic Functions

2.1 Quadratic Functions Date:.1 Quadratic Functions Precalculus Notes: Unit Polynomial Functions Objective: The student will sketch the graph of a quadratic equation. The student will write the equation of a quadratic function.

More information