Download Warm up

Unit 4
Operations &
Rules
Warm up
Combine Like Terms
1) 3x – 6 + 2x – 8
2) 3x – 7 + 12x + 10
5x – 14
15x + 3
Exponent Rules
3) What is 2x  3x?
6x2
Degree
The exponent for a
variable
Degree of the Polynomial
Highest (largest)
exponent of the
polynomial
Standard Form
Terms are placed in
descending order
by the DEGREE
Write all answers in Standard Form!
Leading Coefficient
Once in standard
st
form, it’s the 1
NUMBER in front of the
variable (line leader)
# of
Terms
Name by # of
Terms
1
Monomial
2
Binomial
3
Trinomial
4+
Polynomial
Degree
(largest exponent)
Name by
degree
0
Constant
1
Linear
2
Quadratic
3
Cubic
Special Names:
 2y  9
Linear
# of Terms Name:
Binomial
Leading Coefficient: -2
Degree Name:
Special Names:
34x
Degree Name:
3
Cubic
# of Terms Name:
Monomial
Special Names:
2
4 x  6x
Quadratic
# of Terms Name:
Binomial
Leading Coefficient: 4
Degree Name:
Special Names:
3
7y  y  2y
2
Cubic
# of Terms Name:Trinomial
Leading Coefficient: 1
Degree Name:
Adding
Polynomials
1.
2x
2
 4 x  3   x  5 x  1
2
2
3x
+x+2
2.
 6  x    2 x  8
2
2
x
+ 2x – 2
Subtracting
Polynomials
When SUBTRACTING polynomials
Distribute the NEGATIVE
3.
 3a
2
2
3a
 10a    8a  a 
2
+ 10a –
–
2
5a
2
8a
+ 11a
+a
4.
 3x
2
2
3x
 2 x  4    2 x  x  1
2
+ 2x – 4 –
2
x
2
2x
–x+1
+x–3
Multiplying
Polynomials
5.
2
-2x(x –
4x + 2)
2 x  8 x  4 x
3
2
6.
(x + 3) (x – 3)
x 9
2
7.
(3x – 1)(2x – 4)
6 x  14 x  4
2
8. Find the area of the rectangle.
7 x 10
4x  8
28 x  96 x  80
2
9. Find the volume.
x6
x3
x
x  9 x  18 x
3
2
Warm up
Find an expression for the area of
the following figure:
Challenge
Find an expression for the volume of
cylinder:
 ( x  x  8x  12)
3
2
Multiply:
( x  6)
Polynomial Ops
Q8 of 20
2
Find the perimeter
Find the Perimeter:
Find an expression for the volume of cylinder:
Polynomial Ops
Q11 of 20
Write an expression for the volume
of the box:
Polynomial Ops
Q14 of 20
Find the area of the label.
Polynomial Ops
Q16 of 20
Skills
Check
Square Roots
and
Simplifying
Radicals
Parts of a radical
index
radicand
No number where the index is
means it’s a square root (2)
1. No perfect square factors other
than 1 are under the radical.
2. No fractions are under the radical.
3. No radicals are in the denominator.
You try!
1.
20
2 5
You try!
2.
3 50
15 2
You try!
3.
 120
2 30
Variables Under Square Roots
Even Exponent – Take HALF out
(nothing left under
the radical)
ODD Exponent – Leave ONE under the
radical and take
HALF of the rest out
x
6
x
3
y
15
y
7
y
1026
a
a
513
b
783
b
391
b
13 24
a b
a b
6 12
a
5
18c d
 3c d
2
2
4
2c
th
N
Roots &
Rational
Exponents
Parts of a radical
root
radicand
No number where the root is
means it’s a square root (2)
Simplifying Radicals
Break down the radicand in to prime
factors.
Bring out groups by the number of
the root.
root
radicand
1.
Simplify
3
128
3
 2222222
3
4 2
Simplify
2.
3
27x
3
3
 333 x x x
 3x
Simplify
3.
4
7
32x
4
 22222 x x x x x x x
4
 2x 2x
3
Simplify
4.
9
3
x
27
3
x

3
Rewriting a
Radical to have a
Rational
Exponent
Rewriting Radicals to Rational Exponents

root
radicand

power
 radicand
Power is on top
Roots are in the ground
power
root
Rewriting Radicals to Rational Exponents

root
radicand

power
 radicand
Power is on top
Roots are in the ground
power
root
Rewrite with a Rational
Exponent
5.
10w  10w 
1
2
Rewrite with a Rational
Exponent
6.
3
7 p   7 p
1
3
Rewrite with a Rational
Exponent
7.

17

5
 17
5
2
Rewrite with a Rational
Exponent
8.
  y
8
y
2
y
2
8
1
4
Rewrite with a Rational
Exponent
9.
  z
3
z
6
6
3
z
2
Rewriting Rational
Exponents to Radicals
radicand
power
root


root
radicand

power
Rewrite with a Rational
Exponent
(don’t evaluate)
10. 12
3
5


5
12

3
Rewrite with a Rational
Exponent
(don’t evaluate)
11. 13
2
3


3
13

2
Rewrite with a Rational
Exponent
(don’t evaluate)
12. x
3
2

 
x
3
Warm up
1.
2.
3.
4.
1  i
 6i
96  4 i 6
325  5i 13
8575  35i 7
36
Powers of i
and Complex
Operations
“I one, I one!!”
Negatives in the middle.
1
i
i
2
i
3
i
4
i
1
i
1
Try these!
 i
29
i
i
251
 i
i
9536
1
i
5. i
6.
7.
8.
75
Add and Subtract
Complex
Numbers
Add/Subt Complex Numbers
1. Treat the i’s like variables
2. Combine the real parts then
combine the imaginary parts
3. Simplify (no powers of i higher
than 1 are allowed)
4. Write your answer in standard
form a + bi
Simplify
9. (3  2i)  (7  6i)
 10  8i
Simplify
10. (6  5i)  (1 2 i)
 6  5i  1 2i
 5  7i
Simplify
11. (9  4i)  (2  3i)
 9  4i  2  3i
 7  7i
Simplify
12. 9  (10  2i)  5i
 9  10  2i  5i
 1 7i
Simplify
4
3
4
3
4
3
13. (11i  4i )  (2i  6i )
4
 11i  4i  2i  6i
3
 111  4  i   2 1  6  i 
 9  10i
Multiplying
Complex
Numbers
Multiplying Complex Numbers
1. Treat the i’s like variables
2. Simplify all Powers of i
higher than 1
3. Combine like terms
4. Write your answer in
standard form a + bi
Multiplying Complex Numbers
14.  i(3  i)
 3i  i
2
 3i  (1)
 1 3i
15.
Multiplying Complex Numbers
(2  3i)(6  2i)
 12  4i  18i  6i
2
 12  22i  6(1)
 12  22i  6
 6  22i
Multiplying Complex Numbers
16.
 3  i 8  5i 
 24  15i  8i  5i
2
 24  15i  8i  5  1
 24  7i  5
 29  7i
Dividing
Complex
Numbers
What is a
Conjugate?
17. Dividing – Multiply top & bottom by
the Conjugate
3  4i
2  4i
 2  4i 
2
6  12i  8i  16i


2
2

4
i

 4  8i  8i  16i
6  12i  8i  16  1
10  20i


4  8i  8i  16  1
20
10 20i


20
20
1
  i
2