- The zeros of a function are the input values which result in an output value of zero.

One way of solving quadratic equations is using factoring
Examples are the following:
1) x2 + 5x + 6 = 0
Set this equal to zero:
(x + 2)(x + 3) = 0
Solve each factor: -2008 All Rights Reserved
x + 2 = 0 or x + 3 = 0
x = –2 or x = – 3
Then, clearly, the zeros are -2 and -3, since those are the input values which will result in an output value of zero.

2) x(x + 5) = 03) x2 – 4 = 0
x = 0 or x + 5 = 0 (x – 2) (x + 2) = 0
x = 0 or x = –5x – 2 = 0 or x + 2 = 0
x = 2 or x = –2

Another way of solving quadratic equations is using quadratic formula

Given, f ( x ) = 2 x2 + 8 x + 1, the zeros are the values of x which result in an output value of y = 0. Thus the zeros are the solutions of the equation
2 x2 + 8 x + 1 = 0

Using the quadratic formula, we get

[pic]

Another way of solving quadratic equations is using completing the square

...Review of Algebra
2
s
REVIEW OF ALGEBRA
Review of Algebra
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Here we review the basic rules and procedures of algebra that you need to know in order to be successful in calculus.
Arithmetic Operations
The real numbers have the following properties: a b b a ab a b c a b ab c ab ac In particular, putting a b and so b c b c ba c (Commutative Law) (Associative Law) (Distributive law)
ab c
a bc
1 in the Distributive Law, we get c 1 b c 1b 1c
EXAMPLE 1
(a) 3xy 4x 3 4 x 2y 12x 2y (b) 2t 7x 2tx 11 14tx 4t 2x 22t (c) 4 3 x 2 4 3x 6 10 3x If we use the Distributive Law three times, we get a b c d a bc a bd ac bc ad bd
This says that we multiply two factors by multiplying each term in one factor by each term in the other factor and adding the products. Schematically, we have a In the case where c or
1
b c
d
a and d a b
b, we have
2
a2
ba
ab
b2
a
b
2
a2
2ab
b2
Similarly, we obtain
2
a
b
2
a2
2ab
b2
REVIEW OF ALGEBRA
x
3
EXAMPLE 2
6x 2 3x (a) 2x 1 3x 5 (b) x 6 2 x 2 12x 36 2x 6 (c) 3 x 1 4x 3
10x 3 4x 2 12x 2 12x 2
5 x 3x 5x
6x 2
7x
5 12 12
3 2x 9 2x 21
Fractions
To add two fractions with the same denominator, we use the Distributive Law: a b Thus, it is true that a b c a b c b c b 1 b a 1 b c 1 a b c a b...

...Algebra is a way of working with numbers and signs to answer a mathematical problem (a question using numbers)
As a single word, "algebra" can mean[1]:
* Use of letters and symbols to represent values and their relations, especially for solving equations. This is also called "Elementary algebra". Historically, this was the meaning in pure mathematics too, like seen in "fundamental theorem of algebra", but not now.
* In modern pure mathematics,
* a major branch of mathematics which studies relations and operations. It's sometimes called abstract algebra, or "modern algebra" to distinguish it from elementary algebra.
* a mathematical structure as a "linear" ring, is also called "algebra," or sometimes "algebra over a field", to distinguish it from its generalizations.
A variable is a letter or symbol that takes place of a number in Algebra. Common symbols used are a, x, y, θ, and λ. The letters x and y are commonly used, but remember that any other symbols would work just as well.
Variables are used in algebra as placeholders for unknown numbers. If you see "3 + x", don't panic! All this means is that we are adding a number who's value we don't yet know.
Term: A term is a number or a variable or the product of a number and a variable(s).
An expression is two or more terms, with operations...

...
ANALYSIS
Physics has a lot of topics to cover. In the previous experiments, we discussed Forces, Kinematics, and Motions. In this experiment, the focus is all about Friction. Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction like fluid friction which describes the friction between layers of a viscous fluid that are moving relative to each other; dry friction which resists relative lateral motion of two solid surfaces in contact and is subdivided into static friction between non-moving surfaces, and kinetic friction between moving surfaces; lubricated friction which is a case of fluid friction where a fluid separates two solid surfaces; skin friction which is a component of drag, the force resisting the motion of a fluid across the surface of a body; internal friction is the force resisting motion between the elements making up a solid material while it undergoes deformation and sliding friction.
When surfaces in contact move relative to each other, the friction between the two surfaces converts kinetic energy into heat. This property can have dramatic consequences, as illustrated by the use of friction created by rubbing pieces of wood together to start a fire. Kinetic energy is converted to heat whenever motion with friction occurs, for example when a viscous fluid is stirred. Another important consequence of many types of friction can be wear,...

...
The case between Beauty and Stylish involves concept of a valid contract, pre-contractual statements, express term and misrepresentation.
A valid contract is established between Beauty and Stylish when an offer is accepted and there is intention for both parties to create legal relations. An offer refers to the expression of willingness of the offerer to be contractually bound by an agreement if his or her offer is properly accepted. It has to be clear and certain in terms. It must also be communicated to the offeree before it is being accepted. In addition, the acceptance has to be unqualified, unconditional and made by a positive act. In the case of Beauty and Stylish, a positive act refers to the signing of the contract. All terms of the offer must be accepted without any changes and cannot be subjected to any condition, taking effect only upon fulfillment of that condition. When Beauty and Stylish enter into the agreement, they must intend to bind and bound legally to each other by their agreement. This is the intention to create legal relations between two parties. In the meanwhile, this contract must possess consideration. A contract must therefore be a two-sided affair, with each side providing or promising to provide something of value in exchange for what the other is to provide.
Every contract, whether oral or written, contain terms. The terms of a contract set out the rights and duties of the parties. Terms are the promises and undertakings given by each...

...MATH 122 SYLLABUS
Faculty Information: Name: E-mail: Office: Office Hours:
Section 08 B-1-122 MAK MW 6:00 – 7:15 PM Fall 2013
Corrina Campau campauc@gvsu.edu A-1-132 MAK Phone: (616)331-2052 Tuesday, Thursday 3:45 – 4:30 PM Monday, Wednesday 3:30 - 4:30 PM Monday, Wednesday 7:15 – 8:00 PM by appointment only
Prerequisite:
MTH 110 (a grade of C or better is recommended) or assignment through GVSU Math Placement. You may wish to take the MTH 122 proficiency test which would allow you to waiver 122 and is offered during the first week of class and other times during the semester. For more information visit gvsu.edu/testserv and click on Math placement. College Algebra MTH 122 special edition for Grand Valley State University by John Coburn Students will be required to possess and make use of a TI-83 or TI-84 graphics calculator during the course. You are expected to have and use your calculator every class period. Students will not be allowed to share calculators on tests. Symbolic manipulating calculators (such as the TI-89) and calculators on cell phones, PDA’s, etc. will not be allowed on tests. Math 122 is part of the Mathematical Sciences General Education Foundation Category. Courses in the Foundations Categories introduce students to the major areas of human thought and endeavor. These courses present the academic disciplines as different ways of looking at the world. They introduce...

...Cami Petrides
Mrs. Babich
Algebra Period 4
April 1, 2014
Extra Credit Project
12. When you flip a light switch, the light seems to come on almost immediately, giving the impression that the electrons in the wiring move very rapidly.
Part A: In reality, the individual electrons in a wire move very slowly through wires. A typical speed for an electron in a battery circuit is 5.0x10 to the -4th meters per second. How long does it take an electron moving at that speed to travel a wire 1.0 centimeter, or 1.0x10 to the -2nd?
Part B: Electrons move quickly through wires, but electric energy does. It moves at almost the speed of light, 3.0x10 to the 8th meters per second. How long would it take to travel 1.0 centimeters at the speed of light?
Part C: Electrons in an ordinary flashlight can travel a total distance of only several centimeters .suppose the distance an electron can travel in a flashlight circuit is 15 centimeters, or 1.5x10 to the -1st meter. The circumference of the earth is about 4.0x10 to the 7th meters. How many trips around the earth could a pulse of electric energy make at the speed of light in the same time an electron could travel through 15 centimeters of a battery circuit in 5.0x10 to the -4th meters per second?
For part A, the first step is to put (5.0) to the 10th to the -4th. The numerator would be (0.00050) if someone were trying to put 5.0x10 to the -4th in the form it’s supposed to be in. For the second scientific...

...On this page we hope to clear up problems you might have with polynomials and factoring. All the different methods of factoring and different things such as the difference of cubes are covered. Click any of the links below or scroll down to start gaining a better understanding of polynomials and factoring.
Combining like terms
Multiplication of polynomials
Factoring
Factoring by grouping
Sums and differences of cubes
Quiz on Polynomials and Factoring
When terms of a polynomial have the same variables raised to the same powers, the terms are called similar, or like terms. Like terms can be combined to make the polynomial easier to deal with. Example:
1. Problem: Combine like terms in the following
equation: 3x2 - 4y + 2x2.
Solution: Rearrange the terms so it is easier
to deal with.
3x2 + 2x2 - 4y
Combine the like terms.
Probably the most important kind of polynomial multiplication that you can learn is the multiplication of binomials (polynomials with two terms). An easy way to remember how to multiply binomials is the FOIL method, which stands for first, outside, inside, last. Example:
1. Problem: Multiply (3xy + 2x)(x^2 + 2xy^2).
Simplify the answer.
Solution: Multiply the first terms of each bi-
nomial. (F)
3xy * x2 = 3x3y
Multiply the outside terms of each binomial. (O)
3xy * 2xy2 = 6x2y3
Multiply the inside terms of each...

...A2 Mathematics Coursework C3
Year 12
Numerical solutions of equations
Solving 0 = x5+x-5 using the “Change Of Sign” Method
The method I will use to solve 0 = x5+x-5 is the Change of Sign Method involving the Decimal Search method. I have drawn this graph using the Autograph Software, and the print screen of this is below:
From my graph above, I can see that the root of this equation is between x =1 and x = 1.5. The table of x values and f(x) values is shown below. I can work out the f(x) values by substituting the x-values into the equation.
x
1
1.1
1.2
1.3
1.4
1.5
f(x)
-3
-2.28949
-1.31168
0.01293
1.77824
4.09375
From my table of values above, it is clear that the change of sign from negative to positive occurs between x = 1.2 and x = 1.3. So, I can narrow these values down further to find another change of sign.
x
f(x)
1.21
-1.19626
1.22
-1.07729
1.23
-0.95469
1.24
-0.82837
1.25
-0.69824
1.26
-0.56420
1.27
-0.42616
1.28
-0.28403
1.29
-0.13769
1.30
0.01293
I can see that the change of sign is between x = 1.29 and x = 1.30.
x
f(x)
1.291
-0.12283
1.292
-0.10792
1.293
-0.09296
1.294
-0.07797
1.295
-0.06293
1.296
-0.04784
1.297
-0.03271
1.298
-0.01754
1.299
-0.00233
1.300
0.01293
The change of sign is in the interval [1.299, 1.300]
x
f(x)
1.2991
-0.000805
1.2992
0.000720
1.2993
0.002244
The root of this equation lies in the interval [1.2991, 1.2992]. This...