M3U3D4 Warm Up Divide using Synthetic division: (2x ³ - 5x² + 3x + 7) /(x - 2) 2x² - x + 1 + 9/(x-2)

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Presentation transcript:

M3U3D4 Warm Up Divide using Synthetic division: (2x ³ - 5x² + 3x + 7) /(x - 2) 2x² - x /(x-2)

Homework Check: Document Camera

QUIZ

Complex Numbers

Definition of pure imaginary numbers: Any positive real number b, where i is the imaginary unit and bi is called the pure imaginary number.

Definition of pure imaginary numbers: i is not a variable it is a symbol for a specific number

Simplify each expression.

Remember Simplify each expression. Remember

Distribute Imaginary Numbers Handout

Simplify. To figure out where we are in the cycle divide the exponent by 4 and look at the remainder. 0

Simplify. Divide the exponent by 4 and look at the remainder. 1

Simplify. Divide the exponent by 4 and look at the remainder. 2

Simplify. Divide the exponent by 4 and look at the remainder. 3

Definition of Complex Numbers Any number in form a+bi, where a and b are real numbers and i is imaginary unit.

Definition of Equal Complex Numbers Two complex numbers are equal if their real parts are equal and their imaginary parts are equal. If a + bi = c + di, then a = c and b = d

When adding or subtracting complex numbers, combine like terms.

Simplify.

Multiplying complex numbers. To multiply complex numbers, you use the same procedure as multiplying polynomials.

Simplify. F O I L

F O I L

How to… Divide complex numbers Remember when we needed to divide radical expressions in Math 1?? How did we do that?!?! As a refresher, how would you divide the following: √ √3 = 8 - 4√ = 8 - 4√3 = 1 The conjugate of 2 + √3!

How to… Divide complex numbers Multiply the numerator & denominator by the conjugate! Then complete your steps for multiplying complex numbers! (a + bi) has a conjugate of (a – bi) and (a – bi) has a conjugate of (a + bi) Goal NO IMAGINARY NUMBERS IN THE DENOMINATOR!!

How to… Divide complex numbers i = √-1 i ² = -1 i ² CANNOT be in the simplified answer!

Example #1 Divide complex numbers Write the quotient in standard form i 1  4i 7 + 5i 1 – 4i 7 + 5i 1 – 4i = 1 + 4i Multiply numerator and denominator by 1 + 4i, the complex conjugate of 1 – 4i i + 5i + 20i i – 4i – 16i 2 = Multiply using FOIL i + 20(–1) 1 – 16(–1) = Simplify and use i 2 = -1. – i 17 = Simplify – = i Write in standard form.

GUIDED PRACTICE i i(9 – i) 2. (3 + i)(5 – i) i i 5 2 – 5 2 i i i 3 – 2i Write the expression as a complex number in standard form. ANSWER

Classwork U4D4 Complex Numbers

Homework U4D4 What do you call... odds And What do you get from… odds