Unit 4 Richardson

Bellringer 9/23/14 Simplify 64 Simplify 18 𝑥 3 = ±8 = 3𝑥 2𝑥

Simplifying Radicals Review and Radicals as Exponents

A radical expression contains a root, which can be shown using the radical symbol, . The root of a number x is a number that, when multiplied by itself a given number of times, equals x. For Example: 2 4 , 3 8 , 𝑛 𝑥 Simplifying Radicals Basic Review

Simplifying Radicals Steps Use a factor tree to put the number in terms of its prime factors. Group the same factor in groups of the number on the outside. Merge those numbers into 1 and place on the outside. Multiply the numbers outside together and the ones left on the inside together. 3 1080 3 2∗2∗2∗3∗3∗3∗5 2∗3 3 5 6 3 5

To add and/or subtract radicals you must first Simplify them, then combine like radicals. Ex: 2 18 + 2 12 − 2 50 2 2∗3∗3 + 2 2∗2∗3 − 2 5∗5∗2 3 2 2 +2 2 3 −5 2 2 2 2 3 −2 2 2 Simplifying Radicals Adding and Subtracting

Square Roots as Exponents 2 81 2 3∗3∗3∗3 3*3 9 81 1 2 Please put this in your calculator. What did you get? = 9

Bellringer 9/24/14 Please get the calculator that has your seat number on it, if there isn’t one please see me! Simplify: 4 32 Rewrite as an exponent and solve on your calculator: 5 1024 =2 4 2 =4

Exponent Rules and Imaginary Numbers - with multiplying and dividing square roots if we have time

Imaginary Numbers Can you take the square root of a negative number? Ex: 2 −4 → what number times itself ( 𝑥 2 ) gives you a negative 4? Can u take the cubed root of a negative number? Ex: 3 −8 → what number times itself, and times ( 𝑥 3 ) itself again gives you a negative 8? The imaginary unit i is used to represent the non-real value, 2 −1 . An imaginary number is any number of the form bi, where b is a real number, i = 2 −1 , and b ≠ 0.

a0 = 1 𝑎 ( −𝑚 𝑛 ) 1 = 1 𝑎 ( 𝑚 𝑛 ) Exponent Rules Zero Exponent Property Negative Exponent Property A base raised to the power of 0 is equal to 1. a0 = 1 A negative exponent of a number is equal to the reciprocal of the positive exponent of the number. 𝑎 ( −𝑚 𝑛 ) 1 = 1 𝑎 ( 𝑚 𝑛 ) Examples:

𝑎 𝑚 𝑎 𝑛 = 𝑎 𝑚−𝑛 𝑎 𝑚 ∗ 𝑎 𝑛 = 𝑎 𝑚+𝑛 Exponent Rules Quotient of Powers Property Product of Powers Property To multiply powers with the same base, add the exponents. 𝑎 𝑚 ∗ 𝑎 𝑛 = 𝑎 𝑚+𝑛 To divide powers with the same base, subtract the exponents. 𝑎 𝑚 𝑎 𝑛 = 𝑎 𝑚−𝑛 Examples:

( 𝑎 𝑚 ) 𝑛 = 𝑎 𝑚∗𝑛 (𝑎𝑏) 𝑚 = 𝑎 𝑚 ∗ 𝑏 𝑚 Exponent Rules Power of a Power Property Power of a Product Property To raise one power to another power, multiply the exponents. ( 𝑎 𝑚 ) 𝑛 = 𝑎 𝑚∗𝑛 To find the power of a product, distribute the exponent. (𝑎𝑏) 𝑚 = 𝑎 𝑚 ∗ 𝑏 𝑚 Examples:

( 𝑎 𝑏 ) 𝑚 = 𝑎 𝑚 𝑏 𝑚 Exponent Rules Power of a Quotient Property To find the power of a quotient, distribute the exponent. ( 𝑎 𝑏 ) 𝑚 = 𝑎 𝑚 𝑏 𝑚 Examples:

Bellringer 9/25/14 Simplify: 3 81 =3 3 3 Simplify: (6∗𝑥) −3 = 1 216 𝑥 3

Imaginary Numbers and Exponents 𝑖= 2 −1 𝑖 2 = ( 2 −1 ) 2 =−1 𝑖 3 = ( 2 −1 ) 3 = 2 −1 ∗( 2 −1 ) 2 =−1 2 −1 𝑖 4 = ( 2 −1 ) 4 = ( 2 −1 ) 2 ∗( 2 −1 ) 2 =−1∗−1=1 Patterns!! 𝑖 5 = 2 −1 𝑖 6 = −1 𝑖 7 =−1 2 −1 𝑖 8 = 1 And so on…

Roots and Radicals Review The Rules (Properties) Multiplication Division b may not be equal to 0.

Roots and Radicals b may not be equal to 0. The Rules (Properties) Multiplication Division b may not be equal to 0.

Roots and Radicals Review Examples: Multiplication Division

Roots and Radicals Review Examples: Multiplication Division

Intermediate Algebra MTH04 Roots and Radicals To add or subtract square roots or cube roots... simplify each radical add or subtract LIKE radicals by adding their coefficients. Two radicals are LIKE if they have the same expression under the radical symbol.

Complex Numbers

Complex Numbers All complex numbers are of the form a + bi, where a and b are real numbers and i is the imaginary unit. The number a is the real part and bi is the imaginary part. Expressions containing imaginary numbers can also be simplified. It is customary to put I in front of a radical if it is part of the solution.

Simplifying with Complex Numbers Practice Problem 1 𝑖+ 𝑖 3 𝑖+ 𝑖∗ 𝑖 2 𝑖+ 𝑖∗ −1 𝑖−𝑖 =0 Problem 2 3 −8 + 2 −8 3 (−2)(−2)(−2) + 2 (2)(2)(2)(−1) −2 3 1 +2 2 2 (−1) −2+2 2 2 ∗ 2 −1 =−2+2𝑖 2 2

Bellringer 9/26/14 Sub Rules Apply

Practice With Sub – simplify, i, complex, exponent rules

Bellringer 9/29/14 Is this your classroom? Write all of these questions and your response Is this your classroom? Should you respect other people’s property and work space? Should you alter Mrs. Richardson’s Calendar? How should you treat the class set of calculators?

Review Practice Answers Discuss what to do when there is a substitute

Bellringer 9/30/14. EQ- What are complex numbers Bellringer 9/30/14 *EQ- What are complex numbers? How can I distinguish between the real and imaginary parts? 1. How often should we staple our papers together? When should we turn in homework and where? When and where should we turn in late work? 4. What are real numbers?

Let’s Review the real number system! Rational numbers Integers Whole Numbers Natural Numbers Irrational Numbers

More Examples of The Real Number System

Now we have a new number! Complex Numbers Defined. Complex numbers are usually written in the form a+bi, where a and b are real numbers and i is defined as . Because does not exist in the set of real numbers I is referred to as the imaginary unit. If the real part, a, is zero, then the complex number a +bi is just bi, so it is imaginary. 0 + bi = bi , so it is imaginary If the real part, b, is zero then the complex number a+bi is just a, so it is real. a+ 0i =a , so it is real

Examples Name the real part of the complex number 9 + 16i? What is the imaginary part of the complex numbers 23 - 6i?

Check for understanding Name the real part of the complex number 12+ 5i? What is the imaginary part of the complex numbers 51 - 2i? Name the real part of the complex number 16i? What is the imaginary part of the complex numbers 23?

Name the real part and the imaginary part of each. 1. 2. 3. 4. 5.

Bellringer 10/1/14 *EQ- How can I simplify the square root of a negative number? For Questions 1 & 2, Name the real part and the imaginary part of each. 1. 2. For Questions 3 & 4, Simplify each of the following square roots. 3. 4.

Simply the following Square Roots.. 1. 2. 3. 4. How would you take the square root of a negative number??

Simplifying the square roots with negative numbers The square root of a negative number is an imaginary number. You know that i = When n is some natural number (1,2,3,…), then

Simply the following Negative Square Roots.. 1. 2. 3.

Let’s review the properties of exponents….

How could we make a list of i values?

Practice 1. 2. 3. Find the following i values.. 4. 5. Simply the following Negative Square Roots.. 1. 2. 3. Find the following i values.. 4. 5.

Bellringer 10/2/14 Simply the following Negative Square Roots: 1. −25 2. −18 3. 3 −24

How could we make a list of i values?

A negative number raised to an even power will always be positive Note: A negative number raised to an even power will always be positive A negative number raised to an odd power will always be negative.

How could we make a list of i values? 1 −1 =𝑖 𝑖∗𝑖= −1 ∗ −1 =−1 𝑖 2 ∗𝑖= −1 ∗ −1 =−𝑖 (𝑖 2 ) 2 = −1 2 =−1∗−1=1 (𝑖 2 ) 2 ∗𝑖= −1 2 ∗ −1 =1∗ −1 = −1 =𝑖 (𝑖 2 ) 3 = −1 3 =−1∗−1∗−1=−1

Bellringer 10/3/14 Turn in your Bellringers

Bellringer 10/13/14 Simplify the following: 𝑖 0 = 𝑖 1 = 1 𝑖 2 = 𝑖 3 = 𝑖 4 = 1 −1 𝑜𝑟 𝑖 −1 −𝑖 1

Review

Review – Work on your own paper

Review – Work on your own paper

Bellringer 10/14/14 Simplify the following: 2 9 +6𝑖 𝑖 2 𝑖 5 2 9 +6𝑖 𝑖 2 𝑖 5 4 𝑖 2 −8 𝑖 3

Review/practice Complex Numbers

Bellringer 10/16/14 Simplify the following: 𝑊ℎ𝑎𝑡 𝑖𝑠 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑖? 𝑊ℎ𝑎𝑡 𝑖𝑠 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑖 2 ? Name the real and imaginary parts of the following: -2-I 5+3i 7i 12

Bellringer 10/17/14 Find the value of 𝑖 16 Find the value of 𝑖 27 Simplify −9 Simplify −29 What is 𝑥 exponentially?

Bellringer 10/20/14 Simplify: −4𝑖∗7𝑖 −8 ∗ −5 𝑖 19

Remember 28

Answer: -i

Conjugate of Complex Numbers

Conjugates In order to simplify a fractional complex number, use a conjugate. What is a conjugate?

are said to be conjugates of each other.

Lets do an example: Rationalize using the conjugate Next

Reduce the fraction

Lets do another example Next

Try these problems.

Bellringer 10/21/14 What is 𝑖 equivalent to? What is the Conjugate of 6+5𝑖?

𝑖+6𝑖 6+𝑖 3−2𝑖 4+6𝑖+3 4−6𝑖 −1+𝑖 5𝑖∗−𝑖 3+4𝑖 2𝑖 5𝑖∗𝑖∗−2𝑖 −6(4−6𝑖) Review: Simplify 𝑖+6𝑖 6+𝑖 3−2𝑖 4+6𝑖+3 4−6𝑖 −1+𝑖 5𝑖∗−𝑖 3+4𝑖 2𝑖 5𝑖∗𝑖∗−2𝑖 −6(4−6𝑖) −2−𝑖 4+𝑖

Bellringer 10/23/14 What is 𝑖 equivalent to? Simplify 𝑖 12 What is the conjugate of 3𝑖−2? Simplify 2𝑖 3𝑖−2

Review

Bellringer 10/24/14 Write “𝑖= −1 ” 20 times Write “ 𝑖 2 =−1” 20 times

Review

Bellringer 10/28/14 What is 𝑖 equivalent to? Simplify 𝑖 21 Simplify 16 𝑖 2 ? Simplify 𝑖 21 Simplify 16 Essential Question: How are polynomial operations related to operations in the complex number system?

Bellringer 10/29/14 Simplify 𝑖 13 Simplify 9 What is a Polynomial? What type of polynomial is 4𝑚 3 −6𝑚+2 Essential Question: How are polynomial operations related to operations in the complex number system?

Polynomial Operations

A polynomial is an algebraic expression with one or more terms A polynomial can have constants (like 4 or -6), variables (like x or y), and exponents (like x2 or x100) A polynomial can not have negative exponents (like x-3) or variables in the denominator (like 1/(x+2))

Adding and Subtracting Polynomials Can only add or subtract LIKE TERMS (terms having the same variables and exponents) Add or subtract coefficients (leave exponents the same)

EXAMPLES 1&2  

Bellringer 10/30/14 Simplify 𝑖 50 Simplify 25 What is a Coefficient? Add the polynomials: 5𝑥 2 + 7𝑥 3 − 4𝑥 4 +( 2𝑥 2 − 2𝑥 4 +8) Essential Question: How are polynomial operations related to operations in the complex number system?

EXAMPLES 3&4  

Multiplying Polynomials Multiply coefficients and add exponents Terms do NOT need to be alike

(5)(x + 6)

(x2)(x + 6)

(-2x)(x2 – 4x + 2)

When each polynomial has 2 terms, distribute each term in the first polynomial to each term in the second, then combine like terms (x + 5) (x + 3)

Bellringer 10/31/14 Simplify 3 𝑖 16 Simplify 12 Subtract the polynomials: 5𝑥 2 + 7𝑥 3 − 4𝑥 4 −( 2𝑥 2 − 2𝑥 4 +8) Essential Question: How are polynomial operations related to operations in the complex number system?

Objective The student will be able to: multiply two polynomials using the FOIL method, Box method and the distributive property. SOL: A.2b Designed by Skip Tyler, Varina High School

There are three techniques you can use for multiplying polynomials. The best part about it is that they are all the same! Huh? Whaddaya mean? It’s all about how you write it…Here they are! Distributive Property FOIL Box Method Sit back, relax (but make sure to write this down), and I’ll show ya!

Using the distributive property, multiply 2x(5x + 8) + 3(5x + 8). Combine like terms. 10x2 + 31x + 24 A shortcut of the distributive property is called the FOIL method.

The FOIL method is ONLY used when you multiply 2 binomials The FOIL method is ONLY used when you multiply 2 binomials. It is an acronym and tells you which terms to multiply. 2) Use the FOIL method to multiply the following binomials: (y + 3)(y + 7).

(y + 3)(y + 7). F tells you to multiply the FIRST terms of each binomial.

(y + 3)(y + 7). O tells you to multiply the OUTER terms of each binomial.

(y + 3)(y + 7). I tells you to multiply the INNER terms of each binomial. y2 + 7y + 3y

y2 + 7y + 3y + 21 Combine like terms. y2 + 10y + 21 (y + 3)(y + 7). L tells you to multiply the LAST terms of each binomial. y2 + 7y + 3y + 21 Combine like terms. y2 + 10y + 21

Remember, FOIL reminds you to multiply the: First terms Outer terms Inner terms Last terms

The third method is the Box Method. This method works for every problem! Here’s how you do it. Multiply (3x – 5)(5x + 2) Draw a box. Write a polynomial on the top and side of a box. It does not matter which goes where. This will be modeled in the next problem along with FOIL. 3x -5 5x +2

3) Multiply (3x - 5)(5x + 2) 15x2 First terms: Outer terms: Inner terms: Last terms: Combine like terms. 15x2 - 19x – 10 3x -5 5x +2 +6x -25x 15x2 -25x -10 +6x -10 You have 3 techniques. Pick the one you like the best!

4) Multiply (7p - 2)(3p - 4) 21p2 First terms: Outer terms: Inner terms: Last terms: Combine like terms. 21p2 – 34p + 8 7p -2 3p -4 -28p -6p 21p2 -6p +8 -28p +8

Group and combine like terms. 5) Multiply (2x - 5)(x2 - 5x + 4) You cannot use FOIL because they are not BOTH binomials. You must use the distributive property. 2x(x2 - 5x + 4) - 5(x2 - 5x + 4) 2x3 - 10x2 + 8x - 5x2 + 25x - 20 Group and combine like terms. 2x3 - 10x2 - 5x2 + 8x + 25x - 20 2x3 - 15x2 + 33x - 20

5) Multiply (2x - 5)(x2 - 5x + 4) You cannot use FOIL because they are not BOTH binomials. You must use the distributive property or box method. x2 -5x +4 2x -5 2x3 -10x2 +8x Almost done! Go to the next slide! -5x2 +25x -20

5) Multiply (2x - 5)(x2 - 5x + 4) Combine like terms! +4 2x -5 2x3 -10x2 +8x -5x2 +25x -20 2x3 – 15x2 + 33x - 20

Bellringer 11/3/14 Answer the Essential Question: How are polynomial operations related to operations in the complex number system?

Bellringer 11/4/14 Answer the Essential Question: How are rational exponents and roots of expressions similar?

Bellringer 11/5/14 No Bellringer : Clear your desk except for a pencil

Bellringer 11/6/14 No Bellringer : Turn in your Review Packet Clear your desk except for a pencil

Bellringer 11/7/14 What is probability? What is a subset? What is the difference between a union and an intersection?