ALGEBRA

Engr. Earl Jayson

Alvaran
 “We are like a Projectile.
Our dream is the trajectory.
In order to hit the mark,
we must aim above that mark”
Which of the following is not a
number?
 Zero
 First
 pi
 None of the Above
NUMBER SYSTEM
REAL NUMBERS:
 __________
Natural Numbers: 1, 2, 3,4, 5, 6, 7 …
 __________
Whole Numbers: 0, 1, 2, 3,4, 5, 6, 7 …
 __________
Integers Numbers: 0, ±1, ± 2, ±3, ±4, ± 5 …
 __________Numbers:
Rational 0, ± ½, ±1, ±4/3, ±2…
 __________Numbers:
Irrational e, π, √2 , 3√5
Transcendental Surd
The number 0.06906907 is a..
 rational
 transcendental
 surd
 irrational
Which of the following is a surd?
 3.1416159
 2.6457513
 2.8743729
 2.7940316
Solve i^41 + i^33 + i
0
1
 2i
 3i
Number Systems
Imaginary Numbers: √-1 , I, j

i^0 = ________
1
i^1= ________
i
i^2= ________
-1
i^3= ________
-i
i^4= ________
1
Which of the following is a prime
number?
 15
 17
 39
 93
Which of the following is not a
prime number?
 3331
 3053
 3571
 2803
GREATEST COMMON FACTOR
 Largest
natural number that divides two or
more numbers evenly
Greatest common divisor of
180 and 280?
2
 10
 20
 80
LEAST COMMON MULTIPLE
 Smallestinteger exactly divisible by all of
given sets of numbers
Least common multiple of 180
and 280?
 20
 80
 2520
 705600
LAW OF NATURAL NUMBERS
 Closure: a+b and axb are Natural numbers
 Order doesn’t matter
Commutative Law: ________
A+B = B + A
 Grouping
Associative Law: _________ ____doesn’t matter
(AB)C = A(BC)
 Distributive Law:
A(B+C) = AB + AC
 Identity
0
A+______=A
1
Ax______=A
 Inverse
-A
A + ________ =0
1/A =1
A x ________
The multiplicative inverse of
(x/3) is
 -x/3
1
 3/x
0
The terms of a sum may be
grouped in any manner
without affecting the result?

 Commutative
 Associative
 Distributive
 Reflexive
EXPONENTS
Given: an
 a is called the BASE and n is called the
exponent.
 0n = _______
0 ; 1n = ________
1 a
; a1 = ________
 a0 = _______
1 if a ≠0
 a-n = 1/an if a ≠0
 am+n
aman = __________
 (am)n= __________
amn
 am / an = ____________
am-n
RADICALS
 Represent roots as powers with rational
exponents and apply the properties and
operations with powers
 NOTE: nth root of a = a^(1/n)
LOGARITHM
 The logarithm of a positive number b
to given base a is the _____________
exponent of
power c to which the base a must be
raised to produce b.
LOGARITHM
 loga b=c -> ac = b
 a^ (loga b) = b
 loga 1 = 0
 loga a = 1
 loga (bk) = k loga b
 loga (bc) = loga b + loga c
 loga (b/c) = loga b - loga c
 loga b = log b / log a
If 3^(X+1) – 3^(x-1) = 72, then
x=?
2
4
3
5
Solve for the value of x that will
satisfy equation (x+2)^(1/2) = -
1
 -1
2
3
 No Solution
If 8^x = 3, find the value of
2^(6x)
9
2
8
4
Given log6 + xlog4 = log4 + log
(32+4^x), find the value of x
2
4
3
5
QUADRATIC EQUATION
A quadratic equation in x is an equation
that can be written in the general form:
Ax2 + Bx + C = 0

 Quadratic Formula:
-B ± (B2 – 4AC )
X=
2A
QUADRATIC EQUATION
 Thequantity under the radical sign is called
___________________.
discriminant

 NATURE OF ROOTS:
 B2 – 4AC > 0 REAL and UNEQUAL ROOTS
 B2 – 4AC =0 REAL AND EQUAL ROOTS
 B2 – 4AC <0 COMPLEX CONJUGATE
Find the value of k if 4x^2 + 6x
+ k will have equal roots
 8/3
2
4
 9/4
CUBIC EQUATION
General form: x3 +ax2+bx+c=0

 Relation of Roots:
 x1 + x2 + x3 = -a
 (x1)(x2)(x3)=-c
 (x1)(x2)+(x2)(x3)+(x3)(x1)=b
If the roots of the equation are
-1,2 and 4, what is the
equation
 X3 – 5x2 + 2x + 8 =0
 X3 – 4x2 + 3x + 8 = 0
 X3 – 5x2 – 3x + 6 = 0
 X3 – 4x2 + 2x + 6 = 0
Pascal’s Triangle
 An
array of numbers having a 1 at the top
and at the ends of each line. All the other
numbers are made by adding the pair of
numbers closes to them in the line above.
Find the 6th term in the
expansion of (a+2b)^8
 1792 a3 b5
 1792 a5 b3
 56 a3 b5
 448 a3 b5
BINOMIAL EXPANSION
 Rthterm in (x+y)n:
 (nCr-1)xn-r+1 yr-1
POLYNOMIAL EXPANSION
 SUM OF THE COEFFICIENTS IN THE
EXPANSION:
(ax + by +…)n :
Substitute 1 to every variable and simplify.

 SUM OF THE EXPONENTS IN THE

EXPANSION:
(ax + by +…)n :
n(n+1)
What is the sum of the
coefficients in the expansion
(x+y+z)^8?
0
1
2
3
What is the sum of the
coefficients in the expansion
(2x-1)^20?
0
1
2
4
THEORY OF EQUATIONS
 -Number of Roots in an Equation:
“every rational integral solution of f(x)=0 of the
nth degree has exactly n roots”

 -Remainder Theorem:
“If f(x) is divided by (x-r), the remainder is f(r)”

 -Factor Theorem:
“If (x-r) is a factor of (x-r), then the remainder
f(r)=0
If 4x^3 + 18x^2 + 8x – 4 is
divided by 2x+3, the
remainder is
 10
 11
 12
 13
X = sqrt (1-sqrt(1-sqrt(1-…)))
Solve x
 0.4
 0.5
 0.6
 0.7
SEQUENCE AND PROJECTIONS
 Sequence – ordered set of objects
 Progression – sequence of terms having a
constant relation

 Ex. 3,5,7,9,.. ,3+2(n-1)

2,10,50,.. ,2(5)n-1
ARITHMETIC PROGRESSION
 a sequence of numbers each differing by a
constant amount called Common Difference
an = a1 + (n-1) d

 SUM (arithmetic series)

Sn = (n/2)(2a1 + (n-1) d) = (n/2)(a1 + an)

 Arithmetic Mean
Xmean = (x1 + x2 + x3 + …. + xn) / n
Find the 30th term of the AP
4,7,10…
 88
 91
 94
 97
GEOMETRIC PROGRESSION
 a sequence of numbers each differing by a
constant factor called Common Ratio
gn = g1 rn-1
ginfinite = g1 / (1-r)

 SUM (geometric series)

Sn = g1[(1-rn)/(1-r)]

 Geometric Mean
gmean = nth root of (g1 * g2 * g3 * … * gn)
The 4th term of a Geometric
progression is 189 and 6th term
is 1701, find 8th term
 5103
 10305
 15309
 20107
HARMONIC PROGRESSION
a sequence of terms in which each term is
the RECIPROCAL of the corresponding
term in the arithmetic progression.

 For
HARMONIC Progression / Series, just
get the reciprocal of each term and use
the formula for arithmetic Progression
What is the 11th term of the
harmonic progresion if the 1st
term is ½ and 3rd term is 1/6?
 1/20
 1/21
 1/22
 1/24
RATIO
 The ratio of two numbers a and b:
a to b a:b a/b

 where a is called the ___________

Antecedent
 and b is called the ____________
Consequent
PROPORTION
A proportion is a statement of equality
between two ratios:
 a:b = c:d (a/b) = (c/d)

 where a and d are called ____________

Extremes

 b and c are called _________

Means

 d is called the FOURTH PROPORTIONAL to

a,b,c
Find the 3rd proportional to -2
and 4
 -2
 -4
4
 -8
VARIATION PROBLEM
 Direct Variation: y=kx
 Inverse Variation: y=k(1/x)
Joint Variation: y=kxz
 Combined Variation y=k(x/z)
Given that “w” varies directly as the
product of x and y and inversely as
square of z, and that w=4 when x=2 y=6
and z=3. Find the value of w when x=3,
y=6 and z=3

2
3
6
9
WORD PROBLEMS

 Tosolve a worded problem, one must

transform the sentences into accurate
mathematical expression, and then apply
algebraic operations to solve the solution.
I. AGE PROBLEM
 “The DIFFERENCE of the ages between two persons is
CONSTANT”

 “Time elapsed for persons concerned are equal”

 Present Age: “ages now” “at present” “is” “are”

 Past Age: “k years ago” “was at that time” “in the
last k years”
 Future Age: “k years from now” “years hence” “years
after”

 Technique:
a) create table of variables
b) transform the sentences into equations using the
variable table
A father is 4 times as old as his son
now. 6 years ago he was 7 times as old
as his son during that time. Find their
present ages
8 41
 10 40
 12 45
 12 48
Mary is 24 yrs old. Mary is twice
as old as Ann was when Mary
was as old as Ann is now. How
old is Ann now?
 16
 18
 12
 15
II. MIXTURE PROBLEMS
 Technique:
a) create table of mixtures wherein first column
is the Volume/Mass of the solution, Second
column as the rate expressed in percent
form, and Third as the mass/volume of the
quantity for the given rate.

b) Complete the table using multiplication

across and addition downwards.

c) Solve for the unknown

10 liters of 25% salt solution and 15 liters
if 35% salt solution are poured into a
drum originally containing 30 liters of
10% salt solution. What is the percent
concentration of the mixture?
 10%
 20%
 30%
 40%
How much water must be evaporated
from 10 kg solution which has 4% salt to
make a solution of 10% salt?
4 kg
 5 kg
 6 kg
 7 kg
III. WORK RATE
 Work Rate = 1
Total Time to Finish the Job Done

 Work-Rate Analysis
Sum of individual Rate = Combined Rate

1+1+1…+1=1
t1 t2 t3 tn T
A certain pipe can fill the tank in 7 hrs
and another pipe can fill the same
tank in 5 hrs. A drainpipe can empty
the full content of it in 25 hrs. With all 3
pipes open, how long will it take to fill
the tank?
 2.5 hrs
 3.3 hrs
 1.92 hrs
 2.8 hrs
IV. MOTION PROBLEMS
 V = D/T

 Meeting Situation = sum of individual distance

travelled
Dt = D1 + D2

 Departing Situation = sum of individual distance

travelled
Dt = D1 + D2

 Overtaking Situation = total distance at the

overtaking point is equal
A jogger starts a course at a steady
rate of 8kph. Five minutes later, a
second jogger starts the same course
at 10kph. How long will it take the
second jogger to catch the first?

 20 min
 21 min
 22 min
 18 min
IV. MOTION PROBLEMS
 Against the Wind (current)
V = VA – VR

 With the Wind (current)

V = VA + VR
An airplane flying with the
wind took 4 hrs to travel 1000
km and 8 hrs in travelling
back. What was the wind
velocity in kph?
 50.25kph
 62.5kph
 70.75kph
 40.5kph
IV. MOTION PROBLEMS
 Motionin Circular Path (same direction):
Circumference = D2 – D1

 Motion in Circular Path (opposite

direction):
Circumference = D2 + D1
Coyote and Bipbip run at constant
speeds along a circular track 1350m in
circumference. If they run in opposite
directions, they meet after 3 minutes. If
they run in the same direction, they are
together after 27 minutes. Determine
the speed of Coyote.
 250 m/min
 180 m/min
 150 m/min
 230 m/min
V. CLOCK PROBLEM
 Angle of Minute Hand = 6m

 Angle of Hour Hand = 30h + 0.5m

 Difference of Angle of Hands = Angle of

Separation
What time after 2:00 will the
hands of the clock extend in
the opposite direction for the
1st time?
 2:43.64
 2:43.46
 2:34.64
 2:34.46
VI. COIN PROBLEMS

 Penny = 1 cent
 Nickel = 5 cents
 Dime = 10 Cents
 Quarter = 25 cents

 ∑(Quantity*Value) = Total Amount

Cabby opened her coin purse and found
pennies, nickels and dimes. If there are
twice as many pennies as there are nickels
and dimes combined, how many pennies,
dimes and nickels are there if she has a
collection of 90 coins?
a) P=50, N=20, D=20
b) P=60, N=15, D=15
c) P=60, N=30, D=10
d) P=40, N=30, D=20
DIOPHANTINE EQUATIONS
A system of linear equations in which the
number of equations is less than the value
of unknowns

 Technique: Create an equation solving for

one of the unknowns. Use Calc or trial and
error to determine which value produces
a whole number result.
Jom bought 20 chickens for P200.00.
The cocks cost P30.00 each, hens
P15.00 each and chicks P5.00 each.
How many cocks did he buy?

3
4
5
6
VIII. NUMBER PROBLEM
 Basis of Analysis

 For 2-digit number:

10t + u -> Original Number
10u + t -> Reversed Number

 Technique:
Check the choices which
among them satisfies the question
Thrice the middle digit of a 3-digit
number is the sum of other 2. If the
number is divided by the sum of its
digits, the answer is 69 and the
remainder is 3. If the digits are reversed,
the number becomes smalley by 693.
Find the number
 224
 563
 831
 944
The sum of two numbers is 21.
One number is twice the
other. Find the numbers
 6, 15
 7, 14
 8, 13
 9, 12