m3u3d4 warm up divide using synthetic division: (2x ³ - 5x² + 3x + 7) /(x - 2) 2x² - x + 1 +...
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M3U3D4 Warm UpM3U3D4 Warm Up
Divide using Synthetic Divide using Synthetic division:division:
(2x (2x³ - 5x² + 3x + 7) /(x - ³ - 5x² + 3x + 7) /(x - 2)2)
2x² - x + 1 + 9/(x-2)
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Complex NumbersComplex 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.
b2 b2 1 bi
Definition of pure imaginary numbers:
i 1
i2 1
i is not a variable it is a symbol for a specific
number
Simplify each expression.
1. 81 81 1 9i
2. 121x5 121x4 1 x 11x2i x
3. 200x 100 1 2x 10i 2x
4. 8i 3i 24i2 241
Remember i2 1
Simplify each expression.
24
5. 5 20 i 5i 20Remember that 1 i
i2 100 110Remember i
2 1
10
i0 1
Cycle of "i"
i1 i
i2 1
i3 i
i4 1
i5 i
i6 1
i7 i
Distribute Imaginary Numbers Handout
So i12 i0
i12
Simplify.
To figure out where we are in the cycle divide the exponent by 4 and look at
the remainder.
124 =3 with remainder 0
10
So i17 i1
Simplify.
Divide the exponent by 4 and look at the
remainder.
174 =4 with remainder 1
i
i17
1
So i26 i2
Simplify.
Divide the exponent by 4 and look at the
remainder.
264 =6 with remainder 2
1
i26
2
So i11 i3
Simplify.
Divide the exponent by 4 and look at the
remainder.
114 =2 with remainder 3
i
i11
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.
Ex: 8 3i 2 5i 8 2 3i 5i
10 2i
8 7i 1211i
8 12 7i 11i
418i
Simplify.
Simplify.
9 12 6i 2i
3 8i
9 6i 122i
Multiplying complex numbers.
To multiply complex numbers, you use the
same procedure as multiplying polynomials.
Simplify.
8 5i 2 3i
16 24i 10i 15i2F O I L
16 14i 15 31 14i
Simplify.
3018i 10i 6i2F O I L
3028i 6 2428i
62i 5 3i
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:
4
2 + √3
. 2 - √3
2 - √3=
8 - 4√3
4 - 3=
8 - 4√3= 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.
7 + 5i 1 4i
7 + 5i 1 – 4i
7 + 5i 1 – 4i= 1 + 4i
1 + 4i Multiply numerator and denominator by 1 + 4i, the complex conjugate of 1 – 4i.
7 + 28i + 5i + 20i2
1 + 4i – 4i – 16i2= Multiply using FOIL.
7 + 33i + 20(–1)1 – 16(–1)= Simplify and use i2 = -1.
–13 + 33i 17= Simplify.
1317 –= + 33
17 i Write in standard form.
GUIDED PRACTICE
1.
1 + 9i
i(9 – i)
2. (3 + i)(5 – i)
16 + 2i
3. 5 1 + i
52 – 5
2 i
1113 + 16
13 i
4. 5 + 2i 3 – 2i
Write the expression as a complex number in standard form.
ANSWER
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ClassworkClasswork U4D4 Complex Numbers
HomeworkHomeworkU4D4
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odds