Definitions
What are integers?
Integers are natural numbers, whole numbers and negative numbers.
...-3,-2,-1,0,1,2,3...
Solve for i24
...1
What categories does pi belong to?
Irrational and Real Numbers.
What categories does 0 belong to?
Whole, Integers, Rational and Real Numbers.
Determine if the following claim is true or false and provide one example as evidence.
Natural numbers are closed under division.
False
The quotient of 1 and 2 is
1/2. Which is a fraction.
What is the difference between cubic and linear?
The highest exponent of a cubic is 3, while the highest exponent of a linear is 1.
What is the sum of -17+3i and 2+6i?
-17+3i+2+6i
-15+3i+6i
-15 +9i
What's the difference between 35+i2 and -52+6i2?
35+i2 - (-52+6i2)=
35+i2 + 52-6i2 =
35+i2+52-6i2=
35+(-1)+52-6(-1)=
35-1+52+6=
92
What is the product of (5+i) and (7+3i)?
35+15i+7i+3i2
35+15i+7i+3(-1)
35+15i+7i-3
32 + 22i
Solve i-12* i27 * i-13?
i-12* i27 * i-13 =
i-12+27+(-13) =
i2 = -1
What categories make up Non-Real Numbers?
Pure Imaginary and Complex Numbers.
Determine if the following claim is true or false and provide one example as evidence.
Integers are closed under multiplication.
True
-2 * 5 = -10
0 * 1,000 = 0
-1 * -13 = 13
Factor and Solve: 12x2+60x-168
12 ( x2+5-14)
12 (x+7) (x-2)
x+7= 0 and x-2=0
x=-7 and x= 2
How many roots are there in (x-3)7 and what category are the roots under?
There are 7 roots and all roots are real, rational, integers, whole, and natural numbers.
Also, all 7 roots are 3...
Determine if the following claim is true or false and provide one example as evidence.
Imaginary numbers closed under division.
False
i8/i4 = 1/-1 = -1
What categories of numbers make up Real Numbers?
Natural, Whole, Integers, Fractions, Rational, and Irrational.
Given the roots x= -3 and x = 5, what are your factored and standard forms?
Factored Form: (x+3)(x-5)
Standard Form: x2-2x-15
Determine if the following claim is true or false and provide one example as evidence.
The quotient of a quadratic and a cubic is a polynomial.
False
x2 x*x 1
___ = _____ = ___
x3 x*x*x x
Which is not a polynomial.
What's the quotient (10x-30)(14x+98) and
(2x+14)(10x-30)?
(10x-30)(14x+98) = 10(x-3)14(x+7)
(2x+14)(10x+70)= 2(x+7)10(x+7)
10(x-3)14(x+7)
_____________
2(x+7)10(x-3)
10*14 10*7*2 7
_____ = ______ = __ = 7
2*10 2*10 1
Determine if the following claim is true or false and provide one example as evidence.
The product of a quadratic and a quadratic is quadratic.
False
x2 * 2x2 = 2x4, which is not a quadratic.
In order for a function to be considered a polynomial it must __________________________________.
1. Not have negative, fractions or variables as exponents.
2. Not have variables in the denominator of any fractions.
3. Not contain the square root of a variable.
How many roots are there in x2+9x+12, and what category are the roots under?
The highest exponent is 2, so there are 2 roots. Using the quadratic formula we can see that both roots are irrational.
What's the difference between -x4-3x3+2x-13 and
2x5-x4-5x3+2x2-2x-15?
-x4-3x3+2x-13 - (2x5-x4-5x3+2x2-2x-15) =
-x4-3x3+2x-13 - 2x5+x4+5x3-2x2+2x+15 =
-2x5+2x3-2x2+4x+2
Determine if the following claim is true or false and provide one example as evidence.
Complex Numbers are closed under multiplication.
True
(2+i)(4+i) =
8+2i+4i+i2=
8+6i+i2=
8+6i+(-1)=
7+6i
What's the quotient (3x2-12x-63) and
(4x-28)(2x+10)?
(3x2-12x-63)= 3(x2-4x-21) = 3(x+3)(x-7) <- by BC
(4x-28)(2x+10)= 4(x-7)2(x+5)
3(x+3)(x-7) 3(x+3) 3(x+3)
___________ = ___________ = ________
4(x-7)2(x+5) 4*2(x+5) 8(x+5)