Warm–up #61. Simplify 2. Simplify

Warm–up #6 Solutions1. Simplify = )+2+2+= 3 + 4+4x

Warm–up #6 Solutions2. Simplify = = =

∙ 3 √2−2√33 √2−2√3

Homework LogThurs

9/24

Lesson 1 – 9

Learning Objective: To simplify radical expressions

Hw: #114 Pg. 85 #1 – 71 odd

Homework LogFri

9/25

Lesson 1 – 9

Learning Objective: To find zeroes of polynomials

Hw: #115 Pg. 86#73 – 77 odd, Pg. 56 #97, 101, Pg. 93 #85 – 153 eoo

9/24/15 Lesson 1 – 9 Complex Numbers Day 1 & 2

Advanced Math/Trig

Learning Objective To simplify complex numbers

To perform operations on complex numbers

Complex Numbers21 1

,

iff and

3 2 iff 2 and 3

i i

a bi a b

a bi c di a c b d

x i yi x y

RComplex Number:

Def of Equality:

real part imaginary part

3 4i

2

2

4 ( 1)(4) 1 4 2

25 5

7 7

4 9 ( 4)( 9) 36 6

2 3 6 6( 1) 6

36 6

36 4 (6 )(2 ) 12 12( 1) 12

i

i

i

i i i

i

i i i

Simplify

Careful!Right: Wrong!

1i

Power of i

Divide by 4, and the remainder is the new exponent!

1. = == – i2. = = = i

R1R2R3R0

Power of i3. = = = = = i

4. = = =– 15. = = 1

∙ 𝑖𝑖=  𝑖− 𝑖2=  𝑖

−(−1)

Adding & Subtracting6. (2 – 3i) + (7 + i)= 9 – 2i

7. (3i) + (8 – 2i)= 8 + i

8. (9 – 5i) – (6 + 4i)= 9 – 5i – 6 – 4i = 3 – 9i 9. (–2 + i) – (–5i)= –2 + i + 5i = –2 + 6i

Multiplying10. (3 + 7i) (2 – 5i)= (3)(2) – (3)(5i) + (7i)(2) – (7i)(5i) = 6 – 15i + 14i – 35= 6 – i – 35(–1)= 6 – i + 35= 41 – i

Multiplying11. (5 + 2i) (5 – 2i)= (5)(5) – (5)(2i) + (2i)(5) – (2i)(2i) = 25 – 10i + 10i – 4= 25 – 4(–1)= 25 + 4 = 29

Or difference of squares!= = 25 – 4= 25 – 4(–1) = 29

Conjugates a + bi & a – bi

Dividing12. i = , so there’s a in denominator!= = =

∙ (3−𝑖)(3−𝑖)

¿18−6 𝑖−6 𝑖+2 𝑖2

9−(−1)

¿18−12 𝑖−2

10 ¿16−12 𝑖

10¿8−6 𝑖

5

Put in standard form a + bi

9/25/15 Lesson 1 – 9 Complex Numbers Day 2

Advanced Math/Trig

Learning Objective To find zeroes of polynomials

Zero of Polynomial

2

2 2 2

Show 2 is a zero of 4

4 ( 2 ) 4 4 4 4( 1) 4 0

z i P z

P z i i

A ‘zero’ makes the poly = 0

Plug in –2i for z & see if makes P = 0

Zero of Polynomial1. Show that is a zero of

¿𝑦−𝑥𝑥𝑦 ∙ 𝑥𝑦𝑥+ 𝑦

Negative Exponents in Rational Expression2.

= = =

( 𝑥𝑥 )¿

𝑦𝑥𝑦 −

𝑥𝑥𝑦

𝑦𝑥𝑦 + 𝑥

𝑥𝑦

¿

𝑦− 𝑥𝑥𝑦𝑦+𝑥𝑥𝑦( 𝑥𝑥 )

( 𝑦𝑦 )( 𝑦𝑦 )

=  𝑦+𝑥𝑥𝑦 ∙ 𝑥2 𝑦2

(𝑦+𝑥)(𝑦−𝑥)

Negative Exponents in Rational Expression3.

= = =

( 𝑥𝑥 )¿𝑦𝑥𝑦 +

𝑥𝑥𝑦

𝑦 2

𝑥2 𝑦2−𝑥2

𝑥2 𝑦 2

¿

𝑦+𝑥𝑥𝑦

𝑦2− 𝑥2

𝑥2 𝑦2( 𝑥2

𝑥2 )( 𝑦𝑦 )( 𝑦 2

𝑦 2 )

Ticket Out the Door Show that z =1 – 2i is a zero of P = z2 – 2z + 5.

Explain what you did.

Homework#114 Pg. 85 1 – 71 odd

Homework#114 Pg. 86 73 – 77 odd

Pg. 56 97, 101

Pg. 92 85 – 153 eoo

Pg. 86 #77 mistake in book: 3Show 1 3 is a zero of 8z i P z