Download Your guid to Sections 1.2, 3.3, 4.2, 6.8, and 7.4

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Section 1.2
Field Axioms
Closure ­ a set of numbers is closed under a set through
addition and multiplication is by using addition and
multiplication, the result is a unique number within the set
{set}
X+Y= {Unique Number in Set}
XY= {Unique Number in Set}
Commutativity ­ when two numbers are added or
multiplied together, the order in which they appear does
not change the result
X+Y=Y+X XY=YX
Associativity ­ the order in which two numbers are added
or multiplied together does not change the result
(X+Y)+Z = X+(Y+Z)
Distributivity ­ X(Y+Z) = XY + XZ
Identity Elements ­ the element that in addition and
multiplication does not change the the number added or
multiplied to the element
X * 1 = X X + 0 = X
Inverses ­ the number in addition that causes the result to
be 0 and in multiplication cause the number to be 1
X + (­X) = 0 X * (1/X) = 1
Add a little bit of body text
Section 3.3:
Other Forms of Linear
Function Equation
y = mx + b
Slope-intercept form
y - y1 = m(x - x1)
Point-slope form, (x1 , y1) is a
Ax + By = C
“Ax + By = C” form. A, B, and C stand for
constants
Goals of this section:
Memorize every variant of a linear function
equation
Understand each equation in terms of the
different variables
Be able to graph each different form of a linear
function
Understand the concepts of converting from one
form of a linear function equation to another
form
Section 4.2:
Solutions of Systems of Linear
Equations
To solve a system of equations, you need to
determine which variable you can eliminate, by
multiplying both sides of one equation in the system
by a constant that can give the x OR y values to
become equal to subtract or add.
Ex. Given the system
3x + 4y =6
5x ­ 7y =14
You can multiply by the top equation by 5 and the bottom
equation by 3 to get a LCM of 15.
15x + 20y = 30
15x ­ 21y = 42
Alternately, you can multiply to get y equal to its LCM of 28
21x + 28y = 42
20x ­ 28y = 54
If you run into a system in which you get decimal answers,
round to the nearest significant figure.
If you run into an
equation like
Ex.
5x + 2y = 10
5x + 2y = 20
You can subtract, but
you will end up with the
answer: 0=­10. This is
when the equations in a
system are
inconsistent. This
means that these two
equations have not a
single common solution.
When a system has
identical solutions, they
are called dependent
equations. Those
graphs who normally
just intersect at one
point are called
independent equations.
Examples of the Types of
Graphs
Inconsistent: x­y=0, x­y = 10
Dependent: x+y=0, x+y=0
Independent: 4x­2y=8, 7x+3y = 0
Section 7.4
More Factoring and
Graphing
Sum and Difference of Two Cubes
Sum of Two Cubes X3+Y3 = (X+Y)(X2­XY+Y2)
Difference of Two Cubes X3­ Y3= (X­Y)(X2+XY+Y2) Splitting the Middle Term
Given the equation ax2+bx+c determine the factors by
splitting b into two different integers. First, you multiply
a to c. Next, look for two factors of ac that equal b or
vice versa. Lastly, you split the middle term based on
those results.
Discriminant Test For Prime Quadratics
When an expression is in the ax2+bx+c form, find the
discriminant of the expression. If the discriminant is a
perfect square, then the expression has factors that
have integer coefficients, if not, then the expression is
prime. Car Depreciation
Logarithms, or exponential functions, as described in
chapter 6, have the ability to model different things, such
as your fuel measurement to your distance, or the
population observed in a particular town. However, one
concept that is expressed as a word problem in 6.14 that
shows how exponential functions really work is question 7.
“A rule­of­thumb used by car dealers is that
the trade­in value of a car decreases by 30%
each year. That is, the value at the end of
any year is 70% of its value at the beginning
of the year (‘70% of…’ means ‘0.7 times…)”
This problem describes logarithms quite well because it shares
that as time goes on, your car value will depreciate. Meaning,
as soon as you drive your car out of the driveway of a
dealership, your car’s value drops immediately to 30% of its
MSRP, because the beginning of that first year of ownership is
when you drive your car for the first time. This even applies in
real life to a lot of things, like say an iPhone. You pay quite a bit
upfront to have the latest tech, however, if you try to trade it in a
year later, you will get back less than half of what you paid for
the device originally. This is a good example of logarithms in the
real world because it describes when time can increase, how
much a certain value will increase or decrease depending on
how much time has passed.
6.8: Exponential Equations
Solved By Logarithms
The word “logarithm” combines the words logic
and arithmetic. The log of a number is the
exponent it is with base 10.
Ex. log10 = 1, log3 = .4771212547…
Base 10 logarithms are as follows:
y = log(x) if and only if 10^y = x
Ex. 10^x = 5, log5 = x
log10^x = x, xlog10 = x
Ex. log10^18 = 18
1.2
Practice Problems
1) Are the sets closed under the field axiom?
{positive numbers} {rational numbers}
2) Show how distributivity does not work over
multiplication.
3.3
Find the slope of the following equation:
6x + 2y = -3
Convert the following equation into “Ax + Bx =
C” form:
y - 5 = ⅔(x - 7)
Graph the linear function:
y - 7 = ¾(x - 3)
4.2
1. x - y = -11
7x + 4y = -22
2. 11x - 5y = -38
9x + 2y = -25
6.8
1. 10^x = 0
2. Change to base 10, then solve:
7*5X = 4
7.4
Factor the polynomials completely
X9+Y9
8X6- 27Y6
X3- 729
2) Use the discriminant to determine whether
the polynomial is prime.
X9- 2X3Y3+Y9
5X2 - 3X + 15
X16 +14X4 - 81
3) Split the middle term or show that the
Polynomial is prime
20X2+39X+18
17X2+15X - 5
Solutions
1.2
1) Closed
2) X(YZ) does not equal XYXZ
3.3
Find the slope of the following equation:
6x + 2y = -3
Convert the following equation into “Ax + Bx = C”
form:
y - 5 = ⅔(x - 7)
Graph the linear function:
y - 7 = ¾(x - 3)
4.2
1.
2.
6.8
7.4
1) (X+Y)(X2-XY+Y2)(X6-X3Y3+Y6)
(2X-3Y)(4X2+6XY+9Y2)
(X-9)(X2+9X+81)
2.
a) Factorable
3.
b) Prime
a) (5X+6)(4X+3)
c) Prime
b) Prime