Mathematics Major Focus: Basic Algebra

| 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 0b .
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 0x ). Hence the range is is the set of all real numbers greater than or equal to 3 or
 3yy .
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 2x , 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  f4,2 . 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)  2f 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 0a .

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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 2y are equal, thus in the first equation, we can replace x
with 2y .
6 yx Equation 1
62  yy Replace x in equation 1 with 2y
Since this equation now has only one variable, we can solve for y .
42 y
2y
Next, replace y with 2 in either equation to solve for x .
Equation 1 6 yx Equation 2 2 yx
62 x 22 x
4x 4x
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 .
68x MPE
To compute for the value of y , replace x with 68 in either equation.
Equation 1 115 yx
11568  y
68115y
47y
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.  2x d.  2x
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.  0xx b.  5xx 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

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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. 2x b. x2 c. x2 d. 2x
23. The simplest form of the complex rational expression
x
x
x
x





1
1
1
1
1 is
a.
1x
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

Mathematics Major Focus: Basic Algebra

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