1) The document provides an overview of basic algebra concepts for a mathematics major, including defining algebraic expressions, polynomials, and operations on algebraic expressions.
2) It covers laws of exponents, factoring expressions using greatest common factor and difference of squares, and special products like the square and cube of a binomial.
3) The goal is to prepare students for the Licensure Examination for Teachers (LET) by focusing on competencies involving algebraic expressions, polynomials, rational expressions, and functions.
Q1 week 1 (common monomial,sum & diffrence of two cubes,difference of tw...Walden Macabuhay
It consists of ten units in which the first unit focuses on the special products and factors. Its deals with the study of rational algebraic expressions. It aims to empower students with life – long learning and helps them to attain functional literacy. The call of the K to 12 curriculum allow the students to have an active involvement in learning through demonstration of skills, manifestations of communication skills, development of analytical and creative thinking and understanding of mathematical applications and connections.
OBJECTIVES
Revision On:
Simplify of Algebraic Fraction
Perform Operations on Algebraic Fraction
Solve Equations Involving Algebraic Fraction
Make Substitution in Algebraic Fraction
Solve Simultaneous Equations Involving Algebraic Fraction
Undefined value of an Algebraic Fraction
Represent Algebraic Fractions Graphically.
Q1 week 1 (common monomial,sum & diffrence of two cubes,difference of tw...Walden Macabuhay
It consists of ten units in which the first unit focuses on the special products and factors. Its deals with the study of rational algebraic expressions. It aims to empower students with life – long learning and helps them to attain functional literacy. The call of the K to 12 curriculum allow the students to have an active involvement in learning through demonstration of skills, manifestations of communication skills, development of analytical and creative thinking and understanding of mathematical applications and connections.
OBJECTIVES
Revision On:
Simplify of Algebraic Fraction
Perform Operations on Algebraic Fraction
Solve Equations Involving Algebraic Fraction
Make Substitution in Algebraic Fraction
Solve Simultaneous Equations Involving Algebraic Fraction
Undefined value of an Algebraic Fraction
Represent Algebraic Fractions Graphically.
These slides about functions between two sets. Topics included are
1- The definition of a function as relations between two sets
2- Examples of infinite functions
3- Relations which are not functions
4- Domain and range of real functions
5- Identical functions
Videos explaining these slides are available in the following links
1- The definition of a function as relations between two sets
https://youtu.be/Vi3N2vLySd0
2- Examples of infinite functions
https://youtu.be/Fex82-Ml55c
3- Relations which are not functions
https://youtu.be/abhbALKcHn8
4- Domain and range of real functions
https://youtu.be/82LJ5MXAKRQ
5- Identical functions
https://youtu.be/ZOIt5JxoBxo
The reference book for these slides is
A Transition to Advanced Mathematics 8th Edition,
by Douglas Smith, Maurice Eggen, Richard St. Andre. ISBN-13: 978-1285463261, published by Cengage Learning (August 6, 2014).
https://www.cengagebrain.co.uk/shop/i...
This an animated slides for students. Introduce basis concept of proofs to students. Direct proofs. Please search for slides Proof methods-teachers. If you want to teach using these slides.
How to find the minimum and maximum value of a quadratic equation
How to find the Y-intercept of a quadratic graph and equation
How to calculate the equation of the line of symmetry of a quadratic curve
How to find the turning point (vertex) of a quadratic curve, equation or graph.
These slides about functions between two sets. Topics included are
1- The definition of a function as relations between two sets
2- Examples of infinite functions
3- Relations which are not functions
4- Domain and range of real functions
5- Identical functions
Videos explaining these slides are available in the following links
1- The definition of a function as relations between two sets
https://youtu.be/Vi3N2vLySd0
2- Examples of infinite functions
https://youtu.be/Fex82-Ml55c
3- Relations which are not functions
https://youtu.be/abhbALKcHn8
4- Domain and range of real functions
https://youtu.be/82LJ5MXAKRQ
5- Identical functions
https://youtu.be/ZOIt5JxoBxo
The reference book for these slides is
A Transition to Advanced Mathematics 8th Edition,
by Douglas Smith, Maurice Eggen, Richard St. Andre. ISBN-13: 978-1285463261, published by Cengage Learning (August 6, 2014).
https://www.cengagebrain.co.uk/shop/i...
This an animated slides for students. Introduce basis concept of proofs to students. Direct proofs. Please search for slides Proof methods-teachers. If you want to teach using these slides.
How to find the minimum and maximum value of a quadratic equation
How to find the Y-intercept of a quadratic graph and equation
How to calculate the equation of the line of symmetry of a quadratic curve
How to find the turning point (vertex) of a quadratic curve, equation or graph.
Suma, Resta y Valor numérico de Expresiones algebraicas.
Multiplicación y División de Expresiones algebraicas.
Productos Notables de Expresiones algebraicas.
Factorización por Productos Notables.
This booklet is a brief introduction to Algebra, one of the most main field of mathematics. Further it introduces learners to the worlds of quadratics and concepts of factorization and expansion.
GR 8 Math Powerpoint about Polynomial Techniquesreginaatin
-This is a powerpoint inspired by one of Canva displayed presentation.
- This is about Math Polynomials and good for highschoolers presentation for school.
- It consists of 39 pages explaining each of the Polynomial Techniques.
- Good for review or inspired powerpoint.
Harnessing WebAssembly for Real-time Stateless Streaming PipelinesChristina Lin
Traditionally, dealing with real-time data pipelines has involved significant overhead, even for straightforward tasks like data transformation or masking. However, in this talk, we’ll venture into the dynamic realm of WebAssembly (WASM) and discover how it can revolutionize the creation of stateless streaming pipelines within a Kafka (Redpanda) broker. These pipelines are adept at managing low-latency, high-data-volume scenarios.
Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
Cosmetic shop management system project report.pdfKamal Acharya
Buying new cosmetic products is difficult. It can even be scary for those who have sensitive skin and are prone to skin trouble. The information needed to alleviate this problem is on the back of each product, but it's thought to interpret those ingredient lists unless you have a background in chemistry.
Instead of buying and hoping for the best, we can use data science to help us predict which products may be good fits for us. It includes various function programs to do the above mentioned tasks.
Data file handling has been effectively used in the program.
The automated cosmetic shop management system should deal with the automation of general workflow and administration process of the shop. The main processes of the system focus on customer's request where the system is able to search the most appropriate products and deliver it to the customers. It should help the employees to quickly identify the list of cosmetic product that have reached the minimum quantity and also keep a track of expired date for each cosmetic product. It should help the employees to find the rack number in which the product is placed.It is also Faster and more efficient way.
Understanding Inductive Bias in Machine LearningSUTEJAS
This presentation explores the concept of inductive bias in machine learning. It explains how algorithms come with built-in assumptions and preferences that guide the learning process. You'll learn about the different types of inductive bias and how they can impact the performance and generalizability of machine learning models.
The presentation also covers the positive and negative aspects of inductive bias, along with strategies for mitigating potential drawbacks. We'll explore examples of how bias manifests in algorithms like neural networks and decision trees.
By understanding inductive bias, you can gain valuable insights into how machine learning models work and make informed decisions when building and deploying them.
An Approach to Detecting Writing Styles Based on Clustering Techniquesambekarshweta25
An Approach to Detecting Writing Styles Based on Clustering Techniques
Authors:
-Devkinandan Jagtap
-Shweta Ambekar
-Harshit Singh
-Nakul Sharma (Assistant Professor)
Institution:
VIIT Pune, India
Abstract:
This paper proposes a system to differentiate between human-generated and AI-generated texts using stylometric analysis. The system analyzes text files and classifies writing styles by employing various clustering algorithms, such as k-means, k-means++, hierarchical, and DBSCAN. The effectiveness of these algorithms is measured using silhouette scores. The system successfully identifies distinct writing styles within documents, demonstrating its potential for plagiarism detection.
Introduction:
Stylometry, the study of linguistic and structural features in texts, is used for tasks like plagiarism detection, genre separation, and author verification. This paper leverages stylometric analysis to identify different writing styles and improve plagiarism detection methods.
Methodology:
The system includes data collection, preprocessing, feature extraction, dimensional reduction, machine learning models for clustering, and performance comparison using silhouette scores. Feature extraction focuses on lexical features, vocabulary richness, and readability scores. The study uses a small dataset of texts from various authors and employs algorithms like k-means, k-means++, hierarchical clustering, and DBSCAN for clustering.
Results:
Experiments show that the system effectively identifies writing styles, with silhouette scores indicating reasonable to strong clustering when k=2. As the number of clusters increases, the silhouette scores decrease, indicating a drop in accuracy. K-means and k-means++ perform similarly, while hierarchical clustering is less optimized.
Conclusion and Future Work:
The system works well for distinguishing writing styles with two clusters but becomes less accurate as the number of clusters increases. Future research could focus on adding more parameters and optimizing the methodology to improve accuracy with higher cluster values. This system can enhance existing plagiarism detection tools, especially in academic settings.
Online aptitude test management system project report.pdfKamal Acharya
The purpose of on-line aptitude test system is to take online test in an efficient manner and no time wasting for checking the paper. The main objective of on-line aptitude test system is to efficiently evaluate the candidate thoroughly through a fully automated system that not only saves lot of time but also gives fast results. For students they give papers according to their convenience and time and there is no need of using extra thing like paper, pen etc. This can be used in educational institutions as well as in corporate world. Can be used anywhere any time as it is a web based application (user Location doesn’t matter). No restriction that examiner has to be present when the candidate takes the test.
Every time when lecturers/professors need to conduct examinations they have to sit down think about the questions and then create a whole new set of questions for each and every exam. In some cases the professor may want to give an open book online exam that is the student can take the exam any time anywhere, but the student might have to answer the questions in a limited time period. The professor may want to change the sequence of questions for every student. The problem that a student has is whenever a date for the exam is declared the student has to take it and there is no way he can take it at some other time. This project will create an interface for the examiner to create and store questions in a repository. It will also create an interface for the student to take examinations at his convenience and the questions and/or exams may be timed. Thereby creating an application which can be used by examiners and examinee’s simultaneously.
Examination System is very useful for Teachers/Professors. As in the teaching profession, you are responsible for writing question papers. In the conventional method, you write the question paper on paper, keep question papers separate from answers and all this information you have to keep in a locker to avoid unauthorized access. Using the Examination System you can create a question paper and everything will be written to a single exam file in encrypted format. You can set the General and Administrator password to avoid unauthorized access to your question paper. Every time you start the examination, the program shuffles all the questions and selects them randomly from the database, which reduces the chances of memorizing the questions.
HEAP SORT ILLUSTRATED WITH HEAPIFY, BUILD HEAP FOR DYNAMIC ARRAYS.
Heap sort is a comparison-based sorting technique based on Binary Heap data structure. It is similar to the selection sort where we first find the minimum element and place the minimum element at the beginning. Repeat the same process for the remaining elements.
Water billing management system project report.pdfKamal Acharya
Our project entitled “Water Billing Management System” aims is to generate Water bill with all the charges and penalty. Manual system that is employed is extremely laborious and quite inadequate. It only makes the process more difficult and hard.
The aim of our project is to develop a system that is meant to partially computerize the work performed in the Water Board like generating monthly Water bill, record of consuming unit of water, store record of the customer and previous unpaid record.
We used HTML/PHP as front end and MYSQL as back end for developing our project. HTML is primarily a visual design environment. We can create a android application by designing the form and that make up the user interface. Adding android application code to the form and the objects such as buttons and text boxes on them and adding any required support code in additional modular.
MySQL is free open source database that facilitates the effective management of the databases by connecting them to the software. It is a stable ,reliable and the powerful solution with the advanced features and advantages which are as follows: Data Security.MySQL is free open source database that facilitates the effective management of the databases by connecting them to the software.
Water billing management system project report.pdf
Basic algebra
1. | Mathematics Major [5]
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Philippine Normal University
LICENSURE EXAMINATION FOR TEACHERS (LET)
Refresher Course
I. Algebraic Expressions
A mathematical phrase that contains a variable is an open phrase. A number phrase is an expression
that does not contain a variable. It is also referred to as a numerical expression. The English phrase ‘a certain
number added to 5’ may be translated to the open phrase ‘n + 5’ where n stands for a certain number. The
English phrase ‘seven added to 5’ may be translated to the number phrase ‘7 + 5’.
Expressions like 8 + 2, 12 – 2, 5 x 2, and 20 ÷ 2 are some number phrases for the number 10. Expressions
like
5
x
, - 2a, 2n + 8, 2(l + w) are examples of open phrases. Another name for open phrase is algebraic
expression. An expression composed of constants, variables, grouping symbols, and operation symbols, is
called an algebraic expression. It is the result of adding, subtracting, multiplying, dividing (except by 0), or
taking roots on any combination of constants and variables.
Example a) xx 32 2
b. 1
2
xx c) 524
3
2 2
3
xx
x
d) 2
11
xx
A. Polynomials
A term of a polynomial is a constant, a variable or the product of a constant and one or more variables
raised to whole number exponents. The constant preceding the variable in each term is called coefficient of
the variable. In 2153 23
xx , the coefficient of x3
is 3, and the coefficient of x2
is –15. In algebra, a number is
frequently referred to as a constant, and so the last term –2 in 2153 23
xx is called the constant term.
A polynomial is a term or a finite sum of terms, with only non-negative integer exponents permitted
on the variables. If the terms of a polynomial contain only the variable x, then the polynomial is called
polynomial in x.
WHAT TO EXPECT
MAJORSHIP: MATHEMATICS
FOCUS: Basic Algebra
LET COMPETENCIES:
1. Perform operations on Algebraic Expressions
2. Simplify a given algebraic expression with series of operations
3. Apply the Laws of Exponents in Multiplying and Dividing Algebraic Expressions
4. Factor polynomials
5. Use factoring in simplifying rational expressions
6. Perform operations on Rational Expressions
7. Perform operations on Radical Numbers
8. Identify the domain and/or the range of a given function
9. Identify/ describe the graph of a function
10. Solve problems on
a) Linear equations
b) Systems of linear equations
11. Compute the value of a function f(n), where n is a counting number
PART I – CONTENT UPDATE
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Examples:
a) 32
8 sr b) x2
– 2xy – 3y2
c) 524
3
2 2
3
xx
x
d) 3
38 yx e) 132
24
sr f) 5118 24
xxx
B. Degree of a Polynomial
The degree of a polynomial in one variable is the highest exponent of the variable in the polynomial. A
term containing more than one variable has degree equal to the sum of all the exponents appearing on the
variables in the term. The degree of a polynomial in more than one variable is the highest degree of all the
terms appearing in the polynomial. If a monomial consists of a constant term then the expression is called
constant polynomial. The degree of a constant polynomial is zero.
Examples:
1) The degree of the polynomial xyyx 36 23
is 5.
2) The degree of the polynomial 354 2
xx is 2.
3) The degree of the polynomial yyxx 752 22
is 3.
C. Like Terms
If two terms contain the same powers of the same variables, they are called like terms or similar
terms. For example, 2
3x and 2
5x are like terms, whereas 2
3x and 3
2x are not like terms. Other examples
are yxandxy 54 , 22
53 yxandyx , and yxxandyxx 74 .
D. Kinds of Polynomials
Certain polynomials have special names depending on the number of terms they have.
1. Monomial is a polynomial that has only one term.
Examples: a) 32
8 sr b)
4
5 32
yx
2. Binomial is a polynomial that has two terms.
Examples: a) 3
38 yx b) xy
yx
3
5
6 2
3. Trinomial is a polynomial that has three terms.
Examples: a) 354 2
xx b)
5
3
5
3
2 2
2
yx
x
4. Multinomial is a polynomial that has four or more terms
Examples: a) 754 23
xxx b) aba
baba
52
45
2 2
2334
E. Evaluating Algebraic Expressions
In Algebra, we replace a variable with a number. This is called substituting the variable. To evaluate an
algebraic expression, substitute the variable by a number and then calculate. Evaluating an algebraic
expression means obtaining or computing the value of the expression where value/s of the variable/s is/are
assigned.
Examples: Evaluate.
1)
44
yxyx
for x = 12 & y = 8 2)
x
yxy 3
2
for x = 2 & y = 4
=
4
812
4
812
=
2
)4(3
2
24
=
4
4
4
20
= 5 + 1 =
2
12
2
6
= 3 + 6
=
4
24
= 6 =
2
18
= 9
F. Operations on Algebraic Expressions
H. Simplifying Algebraic Expressions Involving Grouping Symbols
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II. Laws of Exponents
A. Product Law
If m and n are integers and a 0, then
nmnm
aaa
Examples:
1) 532
xxx 2) 642
333
B. Power of a Power Law
If m and n are integers and a 0, then
mnnm
aa
Examples:
1) 2446
423
xxx 2) 632
33
C. Power of a Product Law
If m is an integer and a 0 and b 0, then
mmm
baab
Examples:
1) 66223
422 xxx 2) 1836636122
33242
333 yxyxyx
D. Quotient Law
If m and n are integers and m > n, and a 0, then
nm
n
m
a
a
a
If m and n are integers and m < n, and a 0, then
mnn
m
aa
a
1
If m and n are integers and m = n, and a 0, then
10
aa
a
a nm
n
m
Examples:
1) 3222
2
2 549
4
9
2) 242)4(22
4
111
xxxx
x
E. Power of a Quotient Law
If n, a, and b are integers, and b 0, then
n
nn
b
a
b
a
Examples:
1) 3
33
y
x
y
x
2)
81
16
3
2
3
2
4
44
III. Special Products and Factoring
A. Special Products
A. Product of the Sum and Difference of Two Terms
Examples:
a) 22
bababa b) 22
2595353 bababa
The product of the sum and difference of two terms is obtained by subtracting the product of the
last terms from the product of the first terms, and is called the difference of two squares (DTS).
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Square of a Binomial
Product of Two Binomials of the Form dycxandbyax
Product of a Binomial and Trinomial of the Form 22
babaandba
The product of a binomial ba and a trinomial 22
baba where the first term of the trinomial is
the square of the first term of the binomial, the middle term of the trinomial is the negative of the product of
the two terms of the binomial, and the third term of the trinomial is the square of the second term of the
binomial, is equal to the product of the first terms plus or minus the product of the last terms of the
binomial and trinomial factors.
Cube of a Binomial
The cube of a binomial ba is equal to ba ba ba or 3
ba .
3
ba = 2
ba ba = bababa 22
= 3223
33 babbaa
B. Factoring
Factoring is the reverse of multiplying. To factor an expression means to write an equivalent expression
that is a product of two or more expressions.
1) Common Monomial Factoring
1 Get the GCF of the terms in the polynomial.
2 Divide the polynomial by the GCF of the terms in the polynomial.
3 Write as factors the GCF and the quotient.
Examples:
Factor
a) 34
205 xx = 45 3
xx
b) 422224
482416 yxyxyx = 2222
6328 yxyx
2) Factoring the Difference of Two Squares (DTS)
For a binomial to be a difference of two squares, two conditions must hold.
1. The given binomial is a difference of two terms.
2. The two terms must be perfect squares.
Examples:
Factor
a) ,254 2
x b) 68
2536 yx
Procedure:
a) 254 2
x =
2
2
)2(
a
x -
2
2
)5(
b
=
a
x2( +
b
)5
a
x2( -
b
)5
b) 68
2536 yx =
2
24
)6(
a
x -
2
23
)5(
b
y =
a
x4
6( +
b
y )5 3
a
x4
6( -
b
y )5 3
The product of the binomials dycxandbyax where a, b, c, & d are real numbers, is equal
to dycxbydycxax .
The square of a binomial is the sum of the square of the first term, twice the product of the two terms,
and the square of the last term.
22222
22 bababbaaba
22222
22 bababbaaba
1) 3322
babababa (Sum of Two Cubes)
2) 3322
babababa (Difference of Two Cubes)
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3) Factoring the Perfect Square Trinomial (PST)
The square of a binomial is often called the perfect square trinomial.
Use the following to help recognize a perfect square trinomial:
a) Two of the terms (1st
& 3rd
) must be squares, 22
banda , and are both positive.
b) If we multiply a and b and double the result, we get the middle term, 2ab, or its additive
inverse, - 2ab.
To factor perfect square trinomial, use the following relationships:
a) babababababababa
2222
2
b) babababababababa
2222
2
Remember to factor out a common factor first, if there is any.
4) Factoring the Quadratic Trinomial
In the polynomial cbxx 2
, recall that c is the constant term. If that c is not a perfect square, the
trinomial cannot be factored using perfect square trinomial type. It may, however, be possible to factor it as
the product of two different binomials.
Some points to consider in factoring cbxx 2
, where candb are constants.
a) If the sign of the constant term is positive, look for its factors whose sum is the numerical
coefficient of the middle. The signs of these factors must be the same. The sign of the middle
term becomes the signs of the factors of the constant term.
Some points to consider in factoring cbxax 2
, where candba, are constants.
To factor cbxax 2
, we look for binomials in the form
________ xx
where the products of constants in the blanks are as follows.
a) The constants in the first blanks of the binomials have product a.
b) The constants in the last blanks of the binomials have product c.
c) The product of the constants in the extremes and the product of the
constants in the means have a sum of b.
5) Factoring the Sum or Difference of Two Cubes
If we divide 33
ba by ba , we get the quotient 22
baba and no remainder. So
ba and 22
baba are factors of 33
ba . Similarly, if we divide 33
ba by ba , we get the quotient
22
baba and no remainder. So ba and 22
baba are factors of 33
ba .
6) Factoring by Grouping
When the given expression is a multinomial, it may be factored by grouping.
Examples:
a) 6496 23
xxx b) 222
2 byxyx
Procedure:
a) 6496 23
xxx
There is no factor common to all the terms other than 1. We can, however, group the terms as
)64()96( 23
xandxx and factor these separately.
= )64()96( 23
xxx Grouping the terms
= )32(2)32(3 2
xxx Factoring each binomial
= 2332 2
xx Factoring out the common factor (2x–3)
b) 222
2 byxyx
The terms do not have any common factor other than 1 but the terms can be grouped as
222
)2( bandyxyx . Both groups can be expressed as squares. Thus, 22
byx
= 22
byx Rewriting the expression as DTS.
= byxbyx Factoring the DTS
= byxbyx Simplifying each factor
Factoring a Difference or a Sum of Two Cubes
33
ba = ( ba ) ( 22
baba )
33
ba = ( ba ) ( 22
baba )
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7) Factoring by Completing the Square
There are polynomials that cannot be factored by using any of the preceding methods. Completing the
square can factor some of these polynomials.
Examples: Factor
a) 644
x b) 4224
49374 yyxx
Procedure:
a) 644
x
If the given expression is a binomial whose terms are perfect squares and positive, these two terms can
be considered as the first and last terms of the perfect square trinomial. It means that in the binomial 644
x ,
the middle term is missing. The middle term (mt) can be found by using the formula:
mt = 2 ltft
where ft is the first term and lt is the last term.
Using the formula,
mt = 2 644
x = 22
1682 xx
So,
= 0644
x Additive Identity Property
= 224
166416 xxx 01616 22
xx
= )16()6416( 224
xxx Grouping the terms
= 222
48 xx Rewriting the expression as DTS
= xxxx 4848 22
Factoring the DTS
848464 224
xxxxx Simplifying and arranging the terms
b) 4224
49374 yyxx
In the expression 4224
49374 yyxx , the first and last terms are squares . For the given to be a PST,
the middle term must be
mt = 2 44
494 yx = 2222
28722 yxyx
Thus, we rename 22
37 yx as 2222
928 yxyx , and we have
4224
49374 yyxx = 224224
949284 yxyyxx
= )9()49284( 224224
yxyyxx Grouping the terms
= 3222
372 xyyx Rewriting the expression as DTS
= xyyxxyyx 372372 2222
Factoring the DTS
4224
49374 yyxx = 2222
732732 yxyxyxyx Simplifying & arranging the terms
IV. Rational Expressions
Definition:
A rational expression, or a fraction, is a quotient of algebraic expressions (remember that division by zero
is not defined).
A. Signs of Rational Expressions
1)
)(
)(
)(
)(
b
a
b
a
b
a
b
a
b
a
b
a
2)
)(
)(
)(
)(
b
a
b
a
b
a
b
a
b
a
b
a
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Examples:
a.
yx
x
xy
x
xy
x
xy
x
)(
)(
)(
b.
yx
yx
yx
xy
yx
xy
)()( 333333
B. Relations of Rational Expressions
1) Equivalent rational expressions
These are rational expressions with equal values. Note that
d
c
b
a
if and only if bcad .
Examples:
a)
8
4
2
1
b) 24
23
2
21
ba
ba
a
c) 33
22
1
yx
yxyx
yx
2) Similar rational expressions
These are rational expressions with the same denominator.
Examples:
a)
4
9
,
4
7
,
4
3
b)
ab
y
ab
x
ab
2
,
1
,
2
c)
yx
yx
yx
x
yx
x
2
,
2
1
,
2
2
3) Dissimilar rational expressions
These are rational expressions having different denominators.
Examples:
a)
6
1
,
8
3
,
9
5
b) 22
3
,,
3
ab
x
ab
x
ba
c) 22
32
,
1
,
2
yx
yx
yxyx
C. Reduction of Rational Expression to Simplest Form
A rational expression is said to be in lowest or simplest form if the numerator and denominator are
relatively prime. The process of reducing fractions to their simplest form is dividing both numerator and
denominator by their greatest common factor (GCF).
Examples:
1) Reduce 33
24
42
35
ba
ba
to lowest terms.
Solution:
b
a
bbbaaa
bbaaaa
ba
ba
6
5
732
75
42
35
33
24
Or
b
a
baba
baba
6
5
742
735
2333
2324
where the greatest common factor (GCF)between the numerator and the denominator is 23
7 ba .
2) Reduce
xxx
xxx
23
6
23
23
to lowest term.
Solution:
We will first factor the members of the given fraction and then proceed as above.
)23(
)6(
23
6
2
2
23
23
xxx
xxx
xxx
xxx
Common factoring by x
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)2)(1(
)2)(3(
xxx
xxx
Factoring quadratic trinomials
=
1
3
x
x
Canceling x and (x – 2)
3) Reduce
)1(2)1(3
23
xxx
x
to simplest form.
Solution:
=
)1(2)1(3
23
xxx
x
Given
=
)23)(1(
23
xx
x
Common factoring or by grouping
=
1
1
x
Canceling (3x – 2)
D. Operations on Rational Expressions
1) Multiplication and Division of Rational Expressions
a) Product of fractions:
bd
ac
d
c
b
a
The product of two or more given rational expressions is a fraction whose numerator is the product of
the numerators of the given fractions and whose denominator is the product of the given denominators. The
product should be reduced to lowest terms.
Examples:
1)
addc
cb
da
dc
cb
ba 2
5
4
18
20
4
9
37
53
23
46
46
32
2) 22
22
22
22
2
34
42
48
372
4
baba
baba
ba
ba
baba
ba
=
))(2(
)3)((
)2(2
)2(4
)3)(2(
)2)(2(
baba
baba
ba
ba
baba
baba
Factoring
=
ba
ba
)(2
Canceling common factors and get the product
b) Quotient of fractions:
bc
ad
c
d
b
a
d
c
b
a
To obtain the quotient of two rational expressions, we multiply the dividend by the reciprocal of the divisor.
Examples:
1) 4
33
3
42
42
60
30
100
ac
ba
bc
ba
Solution:
= 33
4
3
42
60
42
30
100
ba
ac
bc
ba
Getting the reciprocal of the divisor
then proceed to multiplication
=
)20)((3
)6)((7
)6)(5(
)20)(5(
32
3
3
32
baa
cac
cb
bab
Factoring the numerator and
the denominator
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=
3
7c
Canceling the common factors of the numerators and the
denominators and simplify the result.
2) Divide
132
23
2
2
xx
xx
by
232
2
2
2
xx
xx
Solution:
=
)2)(1(
)12)(2(
)1)(12(
)1)(2(
xx
xx
xx
xx
Factoring and multiplying the dividend by the
reciprocal of the divisor
=
1
2
x
x
Canceling common factors
c) Addition and Subtraction of Rational expressions
Sum or difference of rational expressions:
Similar:
d
ca
d
c
d
a
Dissimilar:
bd
bcad
d
c
b
a
Examples:
a) Find
16
39
16
57
16
12
x
x
x
x
x
x
.
Solution:
=
16
395712
x
xxx
Writing the rational expressions as a single
fraction with a common denominator
=
16
)351()972(
x
xxx
Collecting like terms in the numerator
=
16
1
x
Performing the indicated operations in the
numerator
b) Express 222
49
8
23
3
23
2
yx
y
xyx
x
yx
in simplest form.
Solution:
=
)23)(23(
8
)23(
3
)23(
2
yxyx
y
yxx
x
yx
Factoring the
denominators
Make the rational expressions similar by getting the LCD, divide the LCD by the
given denominator and multiply the quotient by the given numerator
=
)23)(23(
)(8
)23)(23(
)23(3
)23)(23(
)23(2
yxyxx
xy
yxyxx
yxx
yxyxx
yxx
=
)23)(23(
86946 22
yxyxx
xyxyxxyx
Getting the products in the
numerators
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=
)23)(23(
23 2
yxyxx
xyx
Combining like terms in the
numerator
=
)23)(23)((
)23)((
yxyxx
yxx
Factoring the numerator
=
)23(
1
yx
Reducing the result to
simplest form
E. Simplification of Complex Rational Expressions
Complex rational expression is a rational expression in which the numerator or denominator is a
rational expression.
Examples:
1) Simplify the complex rational expression
x
x
y
y
x
y
2
2
.
Solution:
=
x
xy
y
xy
x
xy
y
xy
2222
22
22
=
y
x
xy
x
y
xy
22
22
Canceling common factors
and then multiplying
the dividend by the reciprocal
of the divisor.
V. Radical Expressions and Negative Exponents
Any expression involving an nth
root can be written in radical form. The symbol is called the radical
symbol.
The number a is called the radicand. The number n is called the index of the radical. Remember that
n
a = n
a
1
. So n
a is the positive nth
root of a when n is even and a is positive, whereas n
a is the real nth root
of a when n is odd and a is any real number. The expressions 4 , 4
16 , and 6
2 are not real numbers
because there are no even roots of negative numbers in the real number system.
Whenever the exponent of a base is in rational form
n
m
where n 0, the expression can always be
expressed in radical form.
Radicals
If n is a positive integer and a is a real number for which n
a
1
is defined, then the expression n
a is
called a radical, and n
a = n
a
1
.
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Example 1: Write each exponential expression using radical notation.
a) 4
3
a b) 2
5
a c) 3
2
8
Solution:
a) 4
3
a = 4 3
a b) 2
5
a = 52
)(
a c) 3
2
8 = 3 2
8
Example 2: Write each radical expression using exponential notation.
a) 36 b) 3
8 c) 3 6
a
Solution:
a) 36 = 2
1
36 b) 3
8 = 3
1
)8( c) 3 6
a = 3
6
a
A. Simplifying Radical Expressions
Examples: Simplify each of the following using the Product Rule for Radicals.
a) 4
32 b. 6
12x c) 3 95
54 yx
Solution:
a) 4
32 = 4
16 4
2 = 4 4
2 4
2 = 4
22
b) 6
12x = 6
4x 3 = 232
)(2 x 3 = 32 3
x
c) 3 95
54 yx =3 93
27 yx 3 2
2x = 3 933
3 yx 3 2
2x = 3 23
23 xxy
Examples: Simplify each of the following using the Quotient Rule for Radicals.
a) 3
53
27
8 yx
b) 3
5
125
24y
Solution:
a)
3
2
3
)2(
3
8
27
8 3 23 2333
3 3
3 233
3
53
yxyyyxyyxyx
b)
5
2
5
2
5
38
125
24 3 23 233
3 3
3 23
3
5
yyyyyyy
Product Rule for Radicals
The th
n root of a product is equal to the product of the th
n roots of the factors. In symbols,
n
ab = n
a n
b ,
provided that all of the expressions represent real numbers.
Quotient Rule for Radicals
The th
n root of a quotient is equal to the quotient of the th
n roots of the numerator and denominator .
In symbols n
n
n
b
a
b
a
provided that all of the expressions are real numbers and 0b .
Rationalizing the Denominator
Multiplying both the numerator and denominator by another radical that makes the denominator rational.
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Examples: Rationalize the denominator of each of the following:
a)
6
10
b)
3
3 42
4
24
xy
yx
c)
31
2
Solution:
a)
3
15
6
152
6
152
6
60
6
6
6
10
6
10 2
2
b)
3
3 42
4
24
xy
yx
= 3
32
3 333
3 633
3 33
3 64
3 22
3 22
3
3 42
6
2
62
2
62
8
48
2
2
4
24
xy
xy
xxy
yx
xyx
yx
yx
yx
yx
xy
yx
c) To rationalize letter c, use the conjugate of the two-term denominator as the multiplier of both numerator
and denominator of the given expression.
31)31(
2
)31(2
31
)31(2
31
)31(2
31
31
31
2
2
B. Operations on Radical Expressions
1) Addition and Subtraction
Like radicals are radicals that have the same index and the same radicand. To simplify the sum
of 2523 , we can use the fact that 3x + 5x = 8x is true for any value of x. So, 282523 . The
expression 3223 cannot be simplified because they are unlike radicals. There are radicals that need to
be simplified before adding or subtracting them.
Examples:
Find:
1) 3
2x - 2
4x + 5 3
18x 2) 3 43
16 yx - 3 43
54 yx
Procedure:
1) 3
2x - 2
4x + 5 3
18x Given
= xx 2
2 - 22
2 x + 5 xx 22
23 Product rule for radicals
= xx 2 - 2x + 15x x2 Simplifying each radical
= 15x x2 - 2x Adding like radicals
2) 3 43
16 yx - 3 43
54 yx Given
= 3 333
22 yyx - 3 333
23 yyx Product rule for radicals
= 3
22 yxy - 3
23 yxy Simplifying each radical
= 3
2yxy Adding like radicals
2) Multiplication of Radicals
The product rule for radicals, n
ab = n
a n
b , allows multiplication of radicals with the same index.
Examples: Find the product of the radicals in simplest form.
1) 62 34 2) )2( 3 233
aaa 3) 32( + )5 32( - )5
Procedure:
1) 62 34 = 2242)3)(8(328188 2
2) )2( 3 233
aaa =
3 33 2
2 aa = aa 23 2
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3) 32( + )5 32( - )5 = 2
34 - 2
5 = 4(3) – 5 = 12 – 5 = 7
C. Simplifying Expressions with Negative Exponents
To simplify algebraic expressions with negative exponents, mean to express the given expression into
an equivalent quantity where the exponents become positive.
Examples: Simplify the following expressions:
a) 3234
342
3
9
cba
bca
b) 21
42
23
49
yx
yx
Procedure:
a)
a
b
cba
cba
cba
bca
cba
bca
cba
bca 3
3244
3244
3234
344
3234
3422
3234
342
3
3
3
3
3
)3(
3
9
b) 2
2
22
22
224
24
42
42
21
42
23
)23(
)23)(23(
23
49
23
49
xy
xy
xyxy
xyxy
yxxy
xy
yx
yx
yx
yx
VI. Relations, Functions, and their Zeros
A. Definition of Relation, Function, Domain and Range
Example 1:
The table of values below shows the relation between the distance of the movie projector from the
screen and the size of a motion picture on the screen.
Distance 1 2 3 4
Screen Size 1 4 9 16
The numbers in the table above could be written as ordered pairs (x, y) where x is the
First member or first coordinate and y is the second member or second coordinate. We can express these
numbers as the set of ordered pairs.
16,4,9,3,4.2,1,1
The set 16,4,9,3,4.2,1,1 is a relation. The domain of the relation is 4,3,2,1 , and the range
is 16,9,4,1 .
Example 2:
The set of ordered pairs below shows a similar relation. Each person is paired with a
number representing his or her height.
Person Carl Dan Em Frank
Height (cm) 202 142 138 142
The relation is 142,,138,,142,,202, FrankEmDanCarl . The domain is
FrankEmDanCarl ,,, and the range is 202,142,138 . Notice that for each person there is
exactly one height. This is a special kind of relation called a function.
A relation is a set of ordered pairs. The domain of a relation is the set of first coordinates. The
range is the set of second coordinates. Relations are often defined by equations with no domain stated. If
the domain is not stated, we agree that the domain consists of all real numbers that, when substituted for the
independent variable, produce real numbers for the dependent variable.
For any rational number a except 0, and for all whole numbers m,
m
m
a
a
1
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Example 3:
Let f be the function defined by 3, 2
xyyxf . Find the domain and range of the function.
Solution:
The domain is the set of all real numbers. Since the expression 2
x is nonnegative, the smallest value
that y can take is 3 (when 0x ). Hence the range is is the set of all real numbers greater than or equal to 3 or
3yy .
Example 4:
Find the domain and range of the function defined by
1
,
x
x
yyxf .
Solution:
The domain is the set of all real numbers except 1 , which is 1 xRx . To determine the range,
express x in terms of y, that is
1
y
y
x . From this result, y cannot be equal to 1 . Therefore the range of
function is the set of all real numbers except 1 , which is 1 yRy
B. Values of Functions
The symbol xf (read """" xatforxoff ) denotes the particular value of the function that
corresponds to the given value of x . The variable x is called the independent variable while the variable y is
called the dependent variable because y is usually expressed in terms of x when their relationship is given in
the form of an equation. If there is an equation that is a function, we may replace y in the equation
with xf , since xf represents y . For example, if the given function is 13 xy , we may write it
as 13 xxf .
To evaluate a function for a specific value of x , replace each x in the function with the given value, and
then perform the indicated operation/s.
Example 1: Find the value of 253 2
xxxf when 2x , denoted by 2f .
Solution:
253 2
xxxf
225232
2
f
210432 f 8122 f 42 f
In the example, the ordered pair 4,2 belongs to the function f , in symbols f4,2 . We also say that 4 is
the image of 2 under f , and 2 is the pre-image of 4 under f .
Example 2: Given that f is the function defined by 342
xxxf , find the following:
a) 0f b) 2f c) hf 2 d) 2
3hf e) hxf
VII. Equations
Examples:
a) 523 b) 5243 xx c) 092
x
A function is a relation that assigns to each member of the domain exactly one member of the
range. It is a set of ordered pairs of real numbers yx, in which no two distinct ordered pairs have the
same first coordinate. The set of all permissible values of x is called the domain of the function, and the set
of all resulting values of y is called the range of the function.
Definition
An equation is a mathematical sentence that uses an equal sign to state that two expressions represent
the same number or are equivalent.
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An equation that contains at least one variable is called an open sentence. Equations cb & above are
examples of open sentences. In equation b, only -1 makes the sentence true or satisfies the equation.
However, more than one number might satisfy an equation. For example, +3 and -3 satisfy the equation
092
x . Any number that satisfies an equation is called a solution or root to the equation. The set of
numbers from which you can select replacements for the variable is called the replacement set.
1) Solving Equations
To solve an equation means to find all of its solutions. The most basic method for solving equations
involves the properties of equality.
Equations that have the same solution set are called equivalent equations. Using the properties of
equality, we can derive equations equivalent to the original equation. The equations 1042 x and
xx 421 are equivalent equations because 7 satisfies both equations.
2) Types of Equations
Examples: a) 26132 xx b) 11 xx c) 1
x
x
Both equations 26132 xx and 11 xx are considered identities because both sides of each
equation are identical. They can be satisfied by any real numbers. The equation 1
x
x
can be satisfied by all
real numbers except 0 because
0
0
is undefined.
Examples: a) 5243 xx b) 092
x
The only value that makes equation 5243 xx true is – 1. While 092
x has the solution set 3,3 .
Solution set
The set of all solutions to an equation is called the solution set to the equation.
Properties of Equality
For any rational numbers a, b and c,
a) aa is always true Reflexive Property
b) If ba , then ab Symmetry Property
c) If ba , cb , then ca . Transitive Property
d) Adding the same number to both
sides of the equation does not
change the solution set of the
equation. In symbols, if ba ,
then cbca . Addition Property
e) Multiplying both sides of the
equation by the same nonzero
number does not change the
solution set of the equation.
In symbols, if ba then
bcac . Multiplication Property
Identity is an equation that is satisfied by every number by which both sides of the
equation are defined. The number of solutions is infinite.
Conditional equation is an equation that is satisfied by at least one number but is not an
identity. The number of solutions is finite.
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Examples: a) 2 xx b) xxx 794635
Simplifying either of the two equations using only the properties of equality will yield an equation that is false.
A. Linear Equations
Examples:
a) 24 xy c) 4523 xx e. 1
2
5
3
2
xx
b) 952 x d) 024 yx
Techniques in Solving Equations
1) Simplifying the equation first
Example: Solve the equation .345)4(2 xx
Solution:
Before using the properties of equality, we simplify the expression on the left of the equation:
.345)4(2 xx Given
34582 xx Distributive Property
3487 x Combining like terms
834887 x Addition property by 8
427 x Simplify
427
7
1
x Multiplication property
x = 6 Solution set is {6}
2) Using the properties of equality
Example: Solve xx 421 .
Solution:
We want to obtain an equivalent equation with only an x on the left side and a constant on the other.
xx 421 Given
214421421 xxxx Addition Property
213 x Combining like terms
213
3
1
x Multiplication Property by
3
1
x = 7
Checking: Replacing x by 7 in the original equation gives us
- 7 – 21 = - 4(7)
- 28 = - 28
which is correct. So the solution set to the original equation is {7}.
Inconsistent equation is an equation whose solution set is the empty set.
Definition
An equation is linear if the variables occur as first powers only, there are no products of
variables, and no variable is in a denominator. The graph of the linear equation is a straight line. A linear
equation is also called a first-degree equation.
Linear Equations in One Variable
A linear equation in one variable x is one in the form 0 bax where a and b are real
numbers, with 0a .
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We may remove fractions by multiplying by the lowest common denominator (LCD).
Example: Solve the equation .3
2
5
4
3
xx
Solution:
Find the LCD first. The LCD is 2x.
.3
2
5
4
3
xx
Given
3
2
5
4
3
2
xx
x Multiply the equation by 2x.
xx 6586 Distributive property
)6()6(65)6()6(86 xxxx Addition property
12 x Combining like terms
12
2
1
x Multiplication property
x =
2
1
Solution set is {
2
1
}.
Decimals may be removed from an equation before solving. Multiply by a power of 10 large enough to
make all decimal numbers whole numbers. If you multiply by 10, you move the decimal point in all terms one
place to the right. If you multiply by 100, you move the decimal point in all terms two places to the right.
Example: Solve the equation 5.475.01.0 xxx
Solution: Because the highest number of decimal places is 2 in the term 0.75x, we multiply the equation by
100, thus
)5.475.01.0(100 xxx Multiply the equation by 100
4507510100 xxx Distributive Property
)75(45075)75(10100 xxxxx Addition property
45015 x Combining like terms
45015
15
1
x Multiplication property
x = 30 Solution set is {30}
Applications of Linear Equations in One Variable
1) Number-Related problems
Example:
There are two numbers whose sum is 50. Three times the first is 5 more than twice the second. What
are the numbers?
2) Time, Rate, and Distance or Motion-Related Problems
Example:
A passenger bus starts from Tuguegarao City and heads for Santiago City at 40 kph. Two hours later, a car
leaves the same station for Santiago City at 60 kph. How long will it be as the car overtakes the passenger bus?
3) Age- Related Problems
Example:
Mother is four times as old as Mary. Five years ago, she was seven times as old. How old will each be in
5 years?
4) Work-Related Problems
Example:
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Jerry can build a garage in 3 days, and Sam can build a similar garage in 5 days. How long would it take
them to build a garage working together?
5) Investment-Related Problems
Example:
Gary invested P50,000, part of it at 6% and the other part at 8%. The annual interest on the 6%
investment was P480 more than that from the 8% investment. How much was invested at each rate?
6) Digit-Related Problems
Example:
The tens digit of a certain number is 3 less than the units digit. The sum of the digits is 11. What is the
number?
7) Geometry-Related Problems
Example:
The length of a rectangular piece of property is one meter longer than twice the width. If the perimeter
of the property is 302 meters, find the length and width.
8) Mixture-Related Problems
Example:
What amounts (in ounce) of 50% and 75% pure silver must be mixed to produce a solution of 15 ounces
with 70% pure silver?
B. Systems of Equations
Types of Systems of Linear Equations
1) A system of equations that has one or more solutions is called consistent. The graphs of the
equations either intersect at a point or coincide. The set of coordinates of the intersection is
the solution set of the system.
a) Consistent Independent is a system of linear equations with only one point as its solution. The graphs of the
equations in the system intersect at one point only.
Example:
a) Find the solution set 6 yx and 2 yx by using the substitution method.
Procedure:
The second equation states that x and 2y are equal, thus in the first equation, we can replace x
with 2y .
6 yx Equation 1
62 yy Replace x in equation 1 with 2y
Since this equation now has only one variable, we can solve for y .
42 y
2y
Next, replace y with 2 in either equation to solve for x .
Equation 1 6 yx Equation 2 2 yx
62 x 22 x
4x 4x
Definition:
Any collection of two or more equations taken as one is called a system of equations. If the system
involves two variables, then the set of ordered pairs that satisfy all of the equations is the solution set of the
system.
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Checking: Replace x with 4 and y with 2 in either equation.
Equation 1 6 yx Equation 2 2 yx
624 224 4 = 2 + 2
66 44
The solution set of the system is {4, 2}. The system is consistent because there is only one solution.
b) The sum of two numbers is 115. Their difference is 21. Find the numbers.
Procedure: (By Elimination Method)
Let x = 1st
number (greater number) & y = 2nd
number (smaller number)
Formulate the equations:
Equation 1: 115 yx
Equation 2: 21 yx
Here, the elimination method can be done by adding the equations or by subtracting one equation
from the other. Thus,
+
221
1115
Equationyx
Equationyx
1362 x
From the sum, compute for the value of the retained variable x .
68x MPE
To compute for the value of y , replace x with 68 in either equation.
Equation 1 115 yx
11568 y
68115y
47y
The numbers are 68 and 47. The system is consistent because there is only one solution.
b) A system of equations that has infinitely many solutions is called consistent dependent. The graphs of the
equations coincide. The equations in the system are equivalent.
Examples:
a) xy 22 Equation 1 b) )3(213 xy
42 yx Equation 2 323 xy
Expressing y as a function of x in each of the equations in each system gives equal expressions.
Graphing can also show dependence between the two equations in each system. The graphs of the equations
will coincide.
2) A system of equations that has no solution is called inconsistent. The graphs of the equations do not
intersect or are parallel. The solution set is an empty set.
Examples:
a) 632 yx Equation 1 b) 754 xy
323 xy Equation 2 1254 xy
Solving the system by elimination or by substitution will result into the inequality of two constants
where both variables are dropped. The elimination of both variables implies that no solution can be obtained.
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Directions: Choose the best answer from the choices given. Write the letter that corresponds to your
answer.
1. Which of the following equations is inconsistent equation?
a. xx 22 c. 1223 xx
b. xx 1232 d.
xx 2
7
3
3
2
2. What is the simplest form of the expression )]3(7)2(3[]3)2(5[ yyxx ?
a. 1742 yx c. 1742 yx
b. 1742 yx d. yx 42
3. The product of zyx 734 and zyx 734 can easily be obtained if terms were regrouped as
a. zyx 734 and zyx 734 c. zyx 734 and zyx 734
b. zyx 734 and zyx 734 d. zyx 734 and zyx 734
4. When xx 647
is factored completely, it is equal to
a. )164)(4( 242
xxxx c. )42)(42)(2)(2( 22
xxxxxxx
b. )164)(2)(2( 24
xxxxx d. )42)(42)(2)(2( 22
xxxxxxx
5. If the cost of a basket of apples is xxxx 4843 245
and there are xxxx 241473 234
apples,
how much is a piece of apple?
a. xx 22
b. xx 22
c. 2x d. 2x
6. What is the simplest form of the rational expression
212
125
9
3
ca
ca
?
a. 7
4
3a
c
b.
a
c
3
c. 8
4
9a
c
d.
ac3
1
7. Which of the following is equivalent to
x
x
2
1
?
a.
2
1
b.
2
1
x
x
c.
2
1
x
x
d.
2
1
x
x
8. Which of the following sets of ordered pairs defines relation as a function?
a. 5,4,4,3,4,2 c. 10,3,5,2,5,2
b. 6,3,6,2,4,2 d. 1,1,1,,1,
9. The following statements are true EXCEPT
a. In baf , a is called the pre-image of b under f .
b. The range of the function is the image of its domain.
c. In baf , b is called the image of a under f .
d. Any relation is also a function.
10. Which of the following is a polynomial in x?
a. 42
xx b. 53 xx c. 13
32
xx d. 13
4
x
x
11. The domain of the function
52
14
x
x
xf is
a. 0xx b. 5xx c.
2
5
xx d.
2
5
xx
12. The system of equations 2 yx and 822 yx is an example of a/an
a. Independent system. c. Inconsistent system
b. Dependent system. d. Consistent system.
13. Which of the following is true?
a. baba 22
c. aaaaa 2234
2
b. (a + b)2
= a2
+ b2
d.
c
a
b
a
cb
a
PART II – ANALYZING TEST ITEMS
21. | Mathematics Major [5]
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14. Which of the following are similar radicals when written in simplest form?
a. 32 and 12 b. 3
16 and 3
54 c. 4
32 and 4
48 d. 5
64 and 5
96
15. The product of aa
xx 53
34 and aa
xx 42
2 is
a. aaa
xxx 975
328 c. aaa
xxx 975
328
b. aaa
xxx 2075
328 d. aaa
xxx 2075
328
16. If n is a perfect square, what is the next larger perfect square?
a. 122
nn b. 12
nn c. 12
n d. 12 nn
17. If the binomial factor of a difference of two cubes is yx 52 , the other factor is
a. 22
25204 yxyx c. 22
25204 yxyx
b. 22
25104 yxyx d. 22
25104 yxyx
18. Which of the following statements is/are true?
I. 36 is a real number.
II. 144 is a rational number.
III. yxyx 3 33
IV.
a. I only b. II only c. I and II only d. I, II and III
19. )8(
2
1 y
x
is equal to
a. 23xy
b. 43y - x
c. 2x - 3y
d. 4xy
20. The sum of
xx
xx
xx
x
810
56
23
45
2
2
2
and
13
23
143
49
2
2
x
x
xx
x
is
a. 2 b.
x
x
2
123
c.
xx
xx
22
366
2
2
d. 344 2
xx
21. What is the simplest form of
2
1
2
1
2
1
xxx ?
a. 8
7
x b. 6
5
x c. 3
2
x d. 8
1
x
22. What is
2
3
2)1(
3)3(
2)2(
8)2(
x
x
xx
xxx
x
xx
in simplest form?
a. 2x b. x2 c. x2 d. 2x
23. The simplest form of the complex rational expression
x
x
x
x
1
1
1
1
1 is
a.
1x
x
b.
1
1
2
xx
xx
c.
1
1
2
xx
xx
d.
1
1
2
2
xx
x
24. In his motorboat, a man can go downstream in 1 hour less time than he can go the same
distance upstream. If the rate of the current is 5 kph, how fast can he travel in still water if it
takes him 2 hours to travel the given distance upstream?
a. 5 kph b. 8 kph c. 12 kph d. 15 kph
25. A man, 32 years old, has a son 8 years of age. In how many years will the man be twice as
old as his son?
a. 16 years b. 24 years c. 32 years d. 48 years
26. How many gallons of milk containing 5% butterfat must be mixed with 90 gallons of milk
with 1% butterfat to obtain a mixture of milk with 2% butterfat?
a. 10 gal b. 20 gal c. 30 gal d. 40 gal
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27. The length of a rectangle is two times its width. If the length is decreased by 5 cm and the
width is increased by 5 cm, the area is increased by 75 square cm. Find the dimensions of the
original rectangle.
a. 28 cm & 56 cm b. 25 cm & 50 cm c. 22 cm & 44 cm d. 20 cm & 40 cm
28. There are two numbers, x and y . The sum of the first and twice the second is 34, while the
difference of the first and four times the second is 4. Which of the following describes the
relationship between x and y ?
a.
44
342
yx
yx
b.
44
342
yx
yx
c.
44
342
yx
yx
d.
44
342
yx
yx
29. Which system of linear equations has the solution 3,2 ?
a.
4
952
yx
yx
b.
5
54
yx
yx
c.
1810
1989
yx
yx
d.
18
2010
yx
x
30. If 7 yx and 3 yx , what is yx 2 ?
a. 18 b. 8 c. 10 d. 12p