Algebra Aero Engr Board Review Lecture

7/28/2019 Algebra Aero Engr Board Review Lecture

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Integers and Exponents

Polynomials, Operations, Special Products

Binomial Expansion (Binomial Theorem)

Factoring Polynomials

Rules of Exponents; Simplification of RationalExpressions; Operations on Rational Expressions

Properties Radicals; Simplification of Radicals Operations on Radicals

Complex Numbers

MATHEMATICS: ALGEBRA AERONAUTICAL ENGINEERING BOARD EXAM REVIEW


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Linear Equations; Literal Equations

Quadratic Equations in One Variable

Word Problems

Other Equations in One Variable: Radical, Fractional,Quadratic in Form

Polynomial Equation of Degree n

Inverse Functions

Exponential and Logarithmic Functions

Exponential and Logarithmic Equations

As per CHED CMO 28 s2007 forAero Eng.

Reference: College Algebra and

Trigonometry by Louis Leithold



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Im thinking what in

the earth will myapple benefit from

this bullshit algebra



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Principles, examples, techniquesMATHEMATICS: ALGEBRA AERONAUTICAL ENGINEERING BOARD EXAM REVIEW


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The key technique isPRACTICE...PRACTICE



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So Come On....LetsGet It On!!!MATHEMATICS: ALGEBRA AERONAUTICAL ENGINEERING BOARD EXAM REVIEW


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So well

then..What is

Algebra???



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Well...well thats funny, full of

terminologies, but please give me

the formal definition of algebra. Idont have all the time in the

world, Ill still have to work out my

theory of relativity



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A branch of mathematics that substitutes letters for numbers. An algebraic

equation represents a scale, what is done on one side of the scale with anumber is also done to the other side of the scale. The numbers are the

constants. Algebra can include real numbers, complex numbers, matrices,

vectors etc.

- about.com

A branch of mathematics in which arithmetical operations and

relationships are generalized by using alphabetic symbols to

represent unknown numbers or members of specified sets ofnumbers

- thefreedictionary.com



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And most of all it isincluded in the boardexams so you better studyit!!!MATHEMATICS: ALGEBRA AERONAUTICAL ENGINEERING BOARD EXAM REVIEW


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Hmmm..so it seems Algebra is

not bad at all, its interesting

and maybe useful to my

apple..Ok, Id like to find outmore of it..Please do proceed.



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A is an item that describes amagnitude or a position.



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a. numbers which allow usto count the objects or ideas in a given collection.

Examples:

b. state the position ofindividual objects in a sequence.

Examples:



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are symbols or combination of

symbols which describe a number

written in Latin

Alphabet



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o - to multiply it by 100 times

V = 500

- (bar above the number) to multiply thenumber by 1000 times

V = 5,000

- to multiply the number by 1000000V = 5,000,000



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Natural Numbers

Whole Numbers

Integers

Rational Numbers

Irrational Numbers



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numbers which are considered as thecounting numbers



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may variously refer to:

natural numbers beginning 1, 2, 3, ...; the positive

integers

natural numbers beginning 0, 1, 2, 3, ...; the non-negative integers

are simply the numbers 0, 1, 2, 3, 4, 5, (and

so on)



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all the natural number, the negative ofthe natural numbers and the number

zero.

Examples: -4, -1, 0, 3, 8...



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are real numbers which can be expressed as a ratio of two

integers.a. Integers (positive and negative integers)

b. Fractions (positive and negative fractions)

c. Zero

Examples: 0.5, 2/3, -3, 0.333...0.5 can be expressed as and -3 can be expressed as -

6/2, hence the two examples are rational numbers. Thenumber 0.333 can also be expressed as 1/3 and therefore arational number.

The number 0.333...is a repeating and non terminatingdecimal. As a rule, a non terminating but repeating (orperiodic) decimal is always a rational number.



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are real numbers which cannot be

expressed as a ratio or quotient of twointegers.

Examples: 2, , e,...The numbers above can never be

expressed exactly as a quotient of two

integers. They are in fact, a nonterminating number with non terminatingdecimal.



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Its not your favorite snack....

Nor the famous Piscine Molitor Patel (Pi) whois the main protagonist in the movie Life of

Pi.....



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mathematical constant that is the ratio of a circle's circumference to

its diameter, and is approximately equal to 3.14159. It has been represented by

the Greek letter "" since the mid-18th century, though it is also sometimeswritten as pi. is an irrational number, which means that it cannot be expressed

exactly as a ratio of two integers (such as 22/7 or other fractions that are

commonly used to approximate ); consequently, its decimal representation never

ends and never settles into a permanent repeating pattern.



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An is denoted asi which is equal to square root of

negative one.

Imaginary numbers and its equivalent:


1

1

1

1

4

3

2

i

ii

i

i


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A is an expression of

both real and imaginary number combined.It takes the form ofa + bi, where a and bare real numbers

If a = 0, then pure imaginary number isproduced while real number is obtainedwhen b= 0.

Example: The number -3 + 6i is a complexnumber whose real part is -3 and whoseimaginary part is 6



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Imaginary

Numbers



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The of a number is the

numerical value of the number neglecting the

sign.

The absolute value of -5 is 5 whilez is z. The

absolute value [b] is either positive or zero

but can never be negative.



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The absolute value of a number is

its distance from zero on the number line.

Find the absolute value of4:



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are numbers which are in the

form a/b, where a is called the numerator

which may be any integer while b is called

the denominator which may be any

integer greater than zero. Fraction is alsodefined as a part of a whole.



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numerator and denominatorare both integers. Also called common fraction.(2/3, -4/9).

numerator is smaller thanthe denominator. (5/8, 2/3)

numerator is greater thanthe denominator. (5/2, 8/3).

a fraction that results ininterchanging the numerator and the

denominator. ( 4 is the reciprocal of ) numerator anddenominator are both fractions. (3/4 7/8 )



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- two or more

simple fractions that have the samedenominator. (1/5, 2/5)

- a fraction with

a denominator of zero. ( 5/0 )- a fraction

which has no quantitative meaning. (0/0)

- combination of aninteger and a proper fraction. (5 , 3 )



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Are positive integers that have more than twopositive whole number factors. (6 is a compositenumber because its factors are 1,2,3, and 6)

o

An integer greater than 1 that is divisible only by1 and itself. (2, 3,5,7)

Note: The number 1 is the only natural number that isneither composite nor prime. The number 2 is theonly prime number which is an even number.



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- are set of two consecutive odd

primes which differ by two.Examples: 3, 5 5, 7 11, 13

- pair of

prime numbers that are the same distance from agiven number in a number line.

Examples: Number: Symmetric Prime:

4 3, 5

5 3, 7

6 5, 7



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prime number that remains a prime

when its digits are reversed.

Examples: 11, 13, 17, 31, 71, 73, 79,

97, 101



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an integer that is equal to the sum of all

its possible divisors, except the numberitself.

Examples: 6, 28


Im

Perfect!!!


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In the case of 6, the factors or divisors are 1, 2

and 3. When the factors are added, the sum is

equal to the number itself.

1 + 2 + 3 = 6



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denoted by n!, represents the product ofall positive integers from 1 to n inclusive.

Example: 4! = 4 x 3 x 2 x 1 = 24

3! = 3 x 2 x 1 = 6



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Are digits that define the numerical value

of a number

A digit is considered significant unless it isused to place a decimal point.

The significant digit of a number beginswith the first non zero digit and ends withthe final digit, whether zero or non zero.

( 15.95 has 4 significant figures, 1.674 x103 has 4 significant figures, 0.0016 has 2significant figures)



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Are you

significant

enough??



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means replacing the number withanother number having fewer significant decimaldigits or for integer number, fewer value carrying(non zero) digits.

Examples:

3.14159 shall be rounded up to 3.1416

2.1334 shall be rounded up to 2.133 refers to dropping of the next

digits in order to obtain the degree of accuracybeyond the need of practical calculations.

Example:3.1415 is truncated to 3 decimalas 3.141



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Is a statement that the two ratios areequal

Example: a:x = y:c, a and c are calledextremes, while x and y are called means

a:x = a/x, a is called antecedent, whilex is called consequent



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1. If a/y = x/d , then a:x = y:d

2. If a/b = c/d , then a/c = b/d

3. If a/b = c/d , then b/a = d/c

4. If a/b = c/d , then a b/b = c d/d5. If a/b = c/d , then a + b/b = c + d/d

6. If a/b = c/d , then a + b/a b = c + d/c - d



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Refers to product of several prime numbersoccurring in the denominators each taken with itsgreat multiplicity.

Example: What is the least common

denominator of 8, 9, 12, and 15Soln: 8 = 23

9 = 32

12 = 3 . 22

15 = 3 . 5LCD = 23 (32)(5)

LCD = 360



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Is the lowest multiple of two numbers

Example: What is the least common

multiple of 15 and 18?Soln: 15 = 3 . 5

18 = 32 . 2

LCM = 32 (5)(2)

LCM = 90



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Is the largest number that divides into

two or more numbers evenly.



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The Field axioms of real numbers:

1. Closure property: a+b = integer, ab = integer

2.Commutative property: a+b = b+a, ab = ba

3. Associative property: (a+b)+c = a+(b+c), (ab)c = a(bc)

4. Identity property: a+0 = a, the number 0 is called the additive

identity, a x 1 = a, the number 1 is called the multiplicative

identity

5. Inverse property: a+(-a) = 0, the numbera is called the additive

inverse, a(1/a) =1, the number 1/a is called the multiplicativeinverse

6.Distributive property: a(b+c) = ab+ac



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1. Reflexive property: a = a

2. Symmetric property: if a = b, then b = a3. Transitive property: if a = b and b = c, then

a = c

4. Substitution property: if a = b, then a can bereplaced by b in any expression involving a

5. Addition/Subtraction property: if a = b, thena+c = b+c, if a = b, then a c = b c

6. Multiplication/Division property: if a = b, thenac = bc, if a = b then a/c = b/c with c not equal

to 0.7. Cancellation Property: if a + c = b + c, then

a = b. If ac = bc and c 0, then a = b



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1. a + 0 = a and a 0 = a

2. a(0) = 0

3. 0/a = 0, with a not zero

4. a/0 is undefined5. If ab = 0, then a= 0 or b = 0. This is called

as the zero factor property



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Any

Questions???..

Everything clearso far???


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Number that gives the power to which a

base is raised. 32 the base is 3 and the

exponent is 2. 32 = 9 the number 9 is called

the power.



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Find the following products:

a. (4x4y3)(-5x5y2)

= -20x9y5

b. (4nx5n)(5nx4n)

= 20n2

x9n

c. (2x3y2z)3(-x2y3z4)4

= 8x17y18z19

o Find the following quotients:a. 8a5b2 = 4a2 b. 6nx6n = 2x3n

2a3b8 b6 3nx3n



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Write as simple fractions with only

positive exponents x-3y4z-5 -2

x6y-2z-4


)2(4)2(2)2(6

)2(5)2(4)2(3

zyxzyx

8

10

4

8

12

6

.. z

z

y

y

x

x

21218

81048)12(6

zyxzyx

2

12

18

.

1

. zyx12

218

y

zx


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an algebraic expression involving

only non negative integer powers of one ormore variables and containing no variable in thedenominator.

Example: 2x, 5x2 + 5, 3x2 + 4x + 5 arepolynomials in the variable x...3x2y3, 6x2 +8y2 are polynomials in the variable x and y.

a symbol whose domain contains

only one elementExample: 6x2 + 2x + 5, 6, 2, and 5 are constants



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used to mean a

constant, a variable, or combinations of variablesand constants involving a finite number ofindicated operations

Example: 3x2, 5x + 2

- a constant or a constant multiplied bynon negative integer powers of variables.

Example: 5x2 + 3x + 5 has three (3) terms

any factor of a product is a said tobe the coefficient of the other factors.

Example: 5xyz, the coefficient of 5x is yz, thecoefficient of x is 5yz and so on



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a coefficient that is a

constantExample: 5xyz, 5 is the constant coefficient of xyz

terms that may differ only in theirconstant coefficients

Example: 6x2 and 3x2 are like terms a polynomial that has one term

a polynomial that has two terms

a polynomial that has three terms

exponent of thevariable in a monomial

Example: 5x3 has a degree 3



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If a polynomial has more than onevariable, its degree is the sum of theexponents of all the variables that appear. Thedegree of 3x2y5 is 7, the degree of 4 is 0, the

constant zero (0) has no degree. highest degree

in the polynomial

Example: 7x

2

4x + 2 is a second degreepolynomial



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Operations on


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7a2b - 3a2b + 8ab2 - 5ab2 - 6ab2

= 4a2b - 3ab2


8374 23 yyy

6y3 - 2y2 + 4

4351023

yyy


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(2x + 7)(3x 4)

= 2x(3x 4) + 7(3x 4)

= 6x2 - 8x + 21x - 28

= 6x2 + 13x - 28


We will discussFOIL Method of

multiplying

polynomials later

on our discussionFOIL Method can

also be used!!!


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Find the product (3x2 2x + 1)(3 + x 4x2)

3x2 - 2x + 1

-4x2 + x + 3-12x4 + 8x3 - 4x2

3x3 - 2x2 + x

9x2 - 6x + 312x4 + 11x3 + 3x2 - 5x + 3



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Divide 6x2 + 13x 28 by 3x 4

2x + 7

3x 4 6x2 + 13x 28

6x2 8x

21x 28

21x 280



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Divide 2x3 5x2 + 6x - 3 by x 2

2x2 x + 4

x 2 2x

3

5x

2

+ 6x 32x3 4x2

-x2 + 6x

-x2 + 2x

4x 34x 8

5

2

542: 2

x

xxAns



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2

345

2

61210

x

xxx 2

3

2

4

2

5

2

6

2

12

2

10

x

x

x

x

x

x

xxx 365 23


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Divide 2x3 5x2 + 6x - 3 by x 2

2 2 - 5 + 6 - 3

4 - 2 + 8

2 -1 + 4 + 5

ans: 2x2 x + 4 remainder 5


remainder


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SPECIAL PRODUCTS OF POLYNOMIALS


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(x + y)(x y) = x2 y2

(x + y)2 = x2 + 2xy + y2

(x y)2 = x2 2xy + y2

(x + y) 3 = x3 +3x2y +3xy2 +y3

(x y)3 = x3 3x2y + 3xy2 y3

x3 y3 = (x y)(x2 + xy + y2)

x3 + y3 = (x + y)(x2 xy + y2)

(x + y + z)2 =x2 + y2 + z2 + 2xy + 2xz +2yz



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This is thesensational FOIL

Method of

multiplying Binomials


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22 5102 yxyxyx



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Form: ac + bc = c(a + b)

Form: a2 b2 = (a +b) (ab)

Form: a2 + 2ab + b2 = (a + b)2

a2 2ab + b2 = (a b)2

Form: a2

+ b2

+ c2

+ 2ab + 2ac + 2bc =(a + b + c)2



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a. Grouping to produce a common factora2 b2 + a b = (a2 b2) + (a b)

= (a + b)(a b) + (a b)

= (a b)[(a + b) + 1]

= (a b )(a + b + 1)b. Grouping to produce a difference of two squares

a2 + b2 c2 2ab = (a2 2ab + b2) c2

= (a b)2 c2

= [(a b) + c][(a - b) - c]= (a b + c)(a b c)



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Form: a3 + b3 = (a + b)(a2 ab + b2)

a3 b3 = (a b)(a2 + ab + b2)



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2456 8246 yyyy

)423(22342

yyyy



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14 2 x

)12)(12( xx


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Factor 16t2 + 40t + 25 = (4t + 5)2

perfect square trinomial

Factor x2 + 3x 28 = (x 4)(x + 7)

Factor 9x4 37x2 + 4

We note that if the middle term were -12x2, the polynomial would be a perfect

square trinomial. Hence we add 25x2 to the middle term and then subtract 25x2 from

the resulting expression

9x4

37x2

+ 4 = 9x4

37x2

+ 25x2

+4 25x2

= (9x4 12x2 + 4) 25x2

= (3x2 2)2 - (5x)2

= (3x2 2 + 5x)(3x2 2 5x) = (x + 2)(3x 1)(3x + 1)(x 2)



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Factor the polynomials

a. 16a2 8ab + b2 c2 + 6c 9

b. a3 b3 a + b

c. 2xy + 8x + 3y + 12

a. 16a2 8ab + b2 c2 + 6c 9

= (16a2 8ab + b2) (c2 6c + 9)= (4a b)2 (c 3)2

= [(4a b) + (c 3)][(4a - b) (c 3)]

= (4a b + c 3)(4a b c + 3)

Grouping to produce

Difference of Two squares



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b. a

3

b

3

a + b = (a

3

b

3

) (a b)= (a b)(a2 + ab + b2) (a b)

= (a b)[(a2 + ab + b2) 1]

= (a b)(a

2

+ ab + b

2

1)

c. 2xy + 8x + 3y + 12 = (2xy + 8x) + (3y + 12)

= 2x(y + 4) + 3(y + 4)= (y + 4)(2x + 3)

Grouping to produce a common

factor



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Factor the binomial 125x3 + y9= (5x)3 + (y3)3

= (5x + y3)[(5x)2 (5x)(y3) + (y3)2]

= (5x + y3)(25x2 5xy3 + y6)

Factor the binomial 8 b3

= 23 b3

= (2 b)(22 + 2b + b2)= (2 b)(4 + 2b + b2)



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Factor the polynomials

a. x6

64 b. x6

16x3

+ 64

a. x6 64 = (x3)2 82

= (x3 + 8)(x3 8)

= (x + 2)(x2 2x + 4)(x 2)(x2 + 2x + 4)

b. x6 16x3 + 64 = (x3)2 + 2(-8)x3 + (-8)2

= (x3

8)2

= [(x 2)(x2 + 2x + 4)]2

= (x 2)2(x2 + 2x + 4)2



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Factor x3 + 6x2 + 11x + 6

Get its factors! (1, -1, 2, -2, 3, -3, 6, -6) and try

the different factors, the right factors

are with zero remainder

1 + 6 + 11 + 6

-1 - 5 - 6

1 + 5 + 6 + 0

- 2 - 61 + 3 + 0

- 3

1 + 0

Therefore the

factors are:(x + 1)(x + 2)(x + 3)


-2

-1

-3


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How are we doing so far?

Everything Clear?


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nnnnnnbba

nnnba

nnba

naba ......

3

2

2

1

12

1

11)(

33221

In elementary algebra, the Binomial Theorem describes the algebraic

expansion of powers of a binomial. According to the theorem, it is

possible to expand the power (x+ y)n into a sum involving terms of

the form axbyc, where the exponents b and c are nonnegative

integers with b + c = n, and the coefficient a of each term is a

specific positive integer depending on n and b.

OMG, What

a tedious


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Expand (2x y)5


nba )( xa 2 yb

555445

335225155

)()2(5

45

4

35

3

25

2

15

1

5)()2(

4

35

3

25

2

15

1

5

)()2(3

25

2

15

1

5)()2(

2

15

1

5)()2(

1

5)2(

yxyx

yxyxyxx

54322345 1040808032 yxyyxyxyxx

task!!!


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Example: Find the fifth term of (2x2 + 3y)9

Since the fifth term is desired, and our formula is for the (r + 1) th term, we

have r + 1 = 5 and r = 4. Therefore the exponent of a = 2x2 is n 4 = 5. The

exponent of b = 3y is 4. The fifth term is,

9 . 8 . 7 . 6 (2x

2

)

5

(3y)

4

= 126(2x

2

)

5

(3y)

4

1 . 2 . 3 . 4

= 326,529 x10y4

rrn bar

rnnnn ...............3.2.1

)1)......(2)(1(



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If the numerator and the denominatorof a fraction are polynomials, then the

fraction is called a rational expression

(3x + 2)/(x2 4), 2/5rs are examples of

rational expressions. A rational

expression is said to be in lowest termsif the numerator and denominator have

no common factor other than 1 and -1.

Dont you ever


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Don t you ever

dare to forget

that!!!



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52

45

21

7

ba

ba

22

22

49

2918

yx

yxyx

b

a

3

3

)7(3

)7(42

423

bab

baa

)23)(23(

)23)(6(

yxyx

yxyx

)23(

6

yx

yx


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xx 6

7

4

52

212

)2(7)3(5

x

x

212

1415

x

x


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444

45 22 xxx

2

)2(

4

)2)(2(

5

xxx

2)2(

4

)2)(2(

5

xxx

)2()2()2(4

)2()2()2(5

22

xxx

xxx

)2()2(

)2(4)2(52

xx

xx

)2()2(

841052

xx

xx

)2()2(

182

xx

x


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Find each of the following products in lowest terms.

a. b.


2

3

4.

8

5.

t

ta

)3)(2(

)2)(5(23

2

t

t

t6

5

23

4.

8

5

t

t

5112

9124.

94

52

2

2

xx

xx

x

x


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5112

9124

.94

5

. 2

2

2

xx

xx

x

x

b

)5)(12(

)32(.

)32)(32(

5 2

xx

x

xx

x

)]5)(32)[(12)(32(

)]5)(32)[(32(

xxxx

xxx

)12)(32(

32

xx

x


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Find the quotient of


2

22

2

22 12694

xxy

yxyx

yxy

yx

)43)(32(

)(.

)(

)32)(32(

yxyx

xyx

yxy

yxyx

)43)(32)((

))(32)(32(

yxyxyxy

xyyxyxx

)43(

)32(

yxy

yxx

xyy

xyx

34

322

2


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2


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Evaluate in its simplest term,


1

11

21

aa

a

1

1

1

1

1

)1(

2

aa

a

a

aaaa

a

1

11

2

2

a

aaa

a

a

1

2

2

2

a

aa

aa

)2(

1.

2

aa

a

a

a

)2(

)2)(1(2

aa

aa2

1

a

a

21


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21

11

1

2

2

1

xx

x

x

x

xEvaluate in its simplest terms

2

1)2)(1(

1

1)2)(1(

1

2)2)(1(

2

1)2)(1(

x

xx

x

xx

x

xxx

x

xxx

)1()2(

)2)(2()1)(1(

xx

xxxx

Continued on next slide.........


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Continued.........


12

4412 22

xx

xxxx

132

x

x23


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128/293


32/3 )9(9 33

27

233/2 )8(8 22

4

433/4 )27()27(

4)3(

81


129/293


3/2

3/2

8

1

8

23 )8(

1

22

1

4

1

23/13/2 )8(8 2

3/18

1

2

2

1

4

1


130/293


4/1

3/1

xx 4/13/1 1.x

x4/13/1

.

xx

4/1)3/1( x

12/1x


131/293


3/53/2

58 xx )58(3/2

xx 2/12/1 )1()1(2 xx

)]1(2[)1( 2/1 xx

)1()1(2/1

xx


132/293

,


133/293



134/293



135/293

Simplify each of the following radicals

a.

b.

3 3481 yx

3 3443 yx

3 33 xxy

4 7880 yx

4 784 5.2 yx4 32 52 yyx



136/293

322298


2.16222.49

242227

25


137/293

6 43 2 2535 xx


6/143/12 )25(3)5( xx

6/146/22

)25(3)5( xx )25(3)5( 6/46/16/46/2 xx

6/146/22

)25(3)5( xx 6/1426/22 )5(3)5( xx

6/226/22 )5(3)5( xx

)31()5( 6/22 x

3 254 x


138/293

Find the product and simplify the result:

3 22126 tsr 3 2236 trs

X


3 222 7.3.2 tsr 3 2222 3.2 trs

3 34343 7.3.2 tsr

3 33333 )7.3)(3.2( stsr

3 3)3.2( rst3 7.3 s

3 216 srst

X


139/293

And thats

how you

rationalize!!!



140/293

x23

32

2

2

3

y

x


x2

3 X

x

x

2

22)2(

6

x

x

x

x

2

6

3 22

3 22

)2)(2(

)2)(3(

yy

yx

3 3

3 2

)2(

12

y

yx

y

yx

2123

2


141/293

Express with a rationalized denominator

3 5

1

y


3 223

3 22

55

51

yy

y

3 3

3 22

)5(

5

y

y

y

y

5

253 2


142/293

Rationalize the fraction

5 - 2 = (5 - 2)(5 - 2 )

5 2 (5 2)(5 - 2)

= 52 - 210 22

52 - 22

= 5 - 210 2 = 7 - 210

5 2 3



143/293

Rationalize the denominator

2 = 2(x - 3y)

x 3y (x 3y)(x - 3y)

= 2x - 6y = 2x - 6y

x2 - 9y2 x - 9y



144/293


States that if a polynomial in an

unknown quantity x is divided by a

first degree expression in the same

variable, (x k), where k may be any

real number or complex number, the

remainder to be expected will be

equal to the sum obtained when thenumerical value of k is substituted for

x in the polynomial.


145/293

Divide 2x3 5x2 + 6x 3 by x 2

ans. 2x2 x + 4 + 5

x - 2

Using the remainder theorem, we can find the remainder by,

P(x) = 2x3 5x2 + 6x 3, subs. x = 2

= 2(2)3 - 5(2)2 + 6(2) 3= 5 the remainder


remainder

Previous problem


146/293

states that if a polynomial is divided by (x k)

will result to a remainder of zero, then thevalue (x k) is a factor of the polynomial.

Example:

Show that x 4 is a factor of 2x3 6x2 5x 12

If P(x) = 2x3 6x2 5x 12, then

P(4) = 2(4)3 6(4)2 5(4) 12

= 0 , therefore x 4 is a factor



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148/293


As a review when a real number and an

imaginary number unite together they

form a complex number


149/293

If p is a positive number, then the principal square root of

- p, denoted by , is defined by

= i

Example:

= i ,the square roots of -5 areand -i

= = 4i, the square roots of -16are 4i and -4i

Note: A complex number is said to be in standard formwhen it is written as a + bi, where a and b are realnumbers



150/293

Write the complex number in standard form:

1. = = 3i = 0 + 3i

2. 5 - = 5 6 (i ) = 5 6(2i) = 5 + (-12i)



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If a + bi and c + di are complex numbers, then

(a + bi) + (c + di) = (a + c) + (b + d)i

(a + bi)(c + di) = (ac bd) + (ad + bc)i

Example:

Find the sum and product of the complex numbers 5 4i and -2 + 6i

( 5 4i) + ( -2 + 6i) = 5 - 2 - 4i + 6i = 3 + 2i

(5 4i)(-2 + 6i) = -10 + 30i + 8i - 24i2

= -10 + 38i - 24(-1)

= -10 + 38i + 24 = 14 + 38i

Find the Difference of the complex numbers(5 4i) - (-2 +6i) = 5 4i + 2 6i = 7 10i



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a. The conjugate of 3 + 2i is 3 2i

b. The conjugate of -4 5i is -4 (-5i) or

equivalently -4 + 5i



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Find the quotient of the complex number

5 - 4i = (5 - 4i)(-2 - 6i)

-2 + 6i (-2 + 6i)(-2 - 6i)

= -10 - 22i + 24i2

4 - 36i2

= -10 - 22i + 24(-1)

4 - 36(-1)

= -34 - 22i40

= - 17 - 11 i

20 20



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(2 - -9) (2 -9)

(2 - -9) = (2 - 3i)(2 - 3i)

(2 -9) (2 + 3i)(2 - 3i)

= 4 - 6i - 6i + 9i2

4 - 9i2

= 4 - 12i + 9(-1)

4 - 9(-1)= -5 - 12i = -5 - 12 i

13 13 13



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157/293

Algebraic Equation in the variable x is a

statement that the two algebraic expressions

are equal



158/293


0

bax

Where a and b

are real numbers

and a 0


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8)6(4)42(31 xx

2

3

32

)134(164)134(136

164136

84241261

x

x

xxxx

xx

xx



160/293


23

2

4

3

xx

2

147

6829

8269

)4(2)23(3

x

x

xx

xx

xx


161/293


584

44

12

4

52

12

xx

x

xx

)12)(52(

44

12

4

52

1

xx

x

xx


162/293

2

5

251044216

4420812

44)52(412

x

xxx

xxx

xxx



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164/293


If a rectangle has a length that is 3 cm less


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If a rectangle has a length that is 3 cm less

than four times its width and its perimeter is19 cm, what are the dimensions?

Soln:4w - 3

w

Let w = width of the rectangle

4w 3 = length of the rectangle

P = 2L + 2W

19 = 2(4w - 3) + 2(w)

19 = 8w 6 + 2w

25 = 10w

w = 5/2

Thus, w = 5/2 cm and length

is 4w 3 = 4(5/2) 3 = 7 cm


One angle of the triangle is half the second,


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g g

and the third is 30 more than the second.

Find the angles it being known that the sum of

the interior angles of any triangle is 180

Soln: Let: x = size of the second angle

x = the first anglex + 30 = the third angle

Working Equation:

x = 60 degree the second angle

1/2x = 30 degree the first angle

x + 30 = 90 degree the third angle


180)30(2

1 xxx

A man invested part of $15 000 at 12 percent


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A man invested part of $15,000 at 12 percent

and the remainder at 8 percent. If his annualincome from the two investments is $1456,

how much does he have invested at each

rate?Let x = the number of dollars invested at 12 percent

15,000 x = the number of dollars invested at 8 percent

No. of Dollars Invested x rate = Number of

Dollars in

Interest12 % investment x 0.12 0.12x

8% investment 15,000 x 0.08 0.08(15,000 x)


1456)00015(080120


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Thus the man has $6400 invested at 12% and $8600 at 8%


6400

25604.0

145608.0120012.0

1456)000,15(08.012.0

x

x

xx

xx

86006400000,15000,15 x

A fruit vendor goes to market to buy fruits for resale at her store. She

spends half her money for mangoes and one third of what remains for


169/293

spends half her money for mangoes, and one third of what remains for

bananas. She spends P150 for other fruits and still has P200 left from the

amount she originally had. How much money did she have at the start?

Soln: Let: x = original amount she had

x = amount spent for mangoes

1/3(x x) = 1/6 x = amount spent for bananas

Total amount spent =

Amount remaining = P200

Working Equation:

x = P1050, the original amount


1503

2150

6

1

2

1 xxx

xx 2001503

2

A goldsmith has two alloys of gold, the first being 70 % pure gold, and the

d 60 % ld H f h t b d t k


170/293

second 60 % pure gold. How many grams of each must be used to make

100 grams of an alloy which will be 66 % pure gold?

Soln: Let: x = the weight in grams of 70% pure gold

100 x = the weight in grams of 60% pure gold

% of gold No. of grams Total Alloy

70 % pure gold 0.70 x 0.70x60 % pure gold 0.60 100 x 0.60(100 x)

66 % pure gold 0.66 100 0.66(100)

Working Equation:

Therefore 60 grams is needed for 70% alloy., and 40 grams for 60%

alloy.


gramsx

xx

60

)100(66.0)100(60.070.0

Determine how many liters of a 7% acid


171/293

Determine how many liters of a 7% acid

solution and how many liters of a 12% acidsolution should be mixed by a chemist to

obtain 6 liters of a 10 % acid solution.

Let x = the number of liters of the 7 % acid solution6 x = the number of liters of the 12 % acid solution

% of Acid x No. of Liters of Soln. = No. of Liters of Acid

7% acid 7% 7% x 0.07x

12 % acid 12% 6 x 0.12(6 x)Mixture 10% 10% 6 0.10(6)


)6(10.0)6(12.007.0 xx

600120720070 xx


172/293

Therefore the chemist should use 2.4 liters of the 7% acid

solution and 3.6 liters of the 12% acid solution.


60.012.072.007.0 xx

72.060.005.0 x

12.005.0 x

05.0

12.0

x

4.2x

4.266 x

6.3

One runner took 3 min 45s to complete a race


173/293

One runner took 3 min 45s to complete a race

and another runner required 4 min to run thesame race. The rate of the faster runner is 0.4

m/s more than the rate of the slower runner.

Find their rates.Let r = the number of meters per second in the rate of the slower runner

r + 0.4 = the number of meters per second in the rate of the faster runner

Rate in m/s x Time in seconds. = Distance in metersSlower runner r 240 240 r

Faster runner r + 0.4 225 225(r + 0.4)


Hence the equation is


174/293

Hence the equation is,

Therefore the rate of the slower runner is 6 m/s, and the fasterrunners rate is 6.4 m/s


)4.0(225240 rr

90225240 rr

9015 r

6r

4.64.064.0 r

Two cars A and B start at the same time from


175/293

Two cars A and B start at the same time from

two places 136 km apart toward each other. Atravels 10 km/hr and B 8 km/hr. If B rests 1 hr

on the way, in how many hours will they

meet?Soln: Let: t = time in hrs A travelst 1 = actual time of travelling for B

10t = distance in km travelled by A

8(t 1) = distance in km travelled by B

Working Equation:


136)18(10 tt

hrst 8

Two ninjas decided to compete at each other Using their


176/293

Two ninjas decided to compete at each other. Using their

supernatural powers, starting from the same point at thesame time, they run in the same direction at constant speed

of 34 and 45 kilometres per hour, respectively. In how many

hours will they be 33 kilometers apart?

Let t = the unknown number of hours

34t = distance travelled by a slower ninja

45t = distance travelled by a faster ninja (naruto)

45t

34t

33 kms.



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Ninjas Rate Time Distance

Faster 45 t 45t

Slower 34 t 34t

Working Equation:

Therefore the two ninjas will be 33 kilometers apart in 3 hours


333445

tthrst 3

A motorboat has an average speed of 20 km/hr in still


178/293

A motorboat has an average speed of 20 km/hr in still

water. It can travel upstream 5 km against a rivercurrent in the same time it can travel downstream 7

km with the river current. What is the speed of the

current?

Soln: Let v = speed of the river current in km/hr20 v = speed of the motorboat upstream

20 + v = speed of the motorboat downstream

Working Equation:

v = 3 and 1/3 km/hr


vv 20

7

20

5

One painter can paint a room in 12 hr and another


179/293

can paint the same room in 10 hr. How long will it

take to paint the room if they work together?

Let x = the number of hours in the time to paint the room when

they are working together

Fractional Part x No. of hours = Fractional Part of

of work done/hr worked work done

First Painter 1/12 x x/12Second Painter 1/10 x x/10


xx


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Hence it takes the painters 60/11 hr to paint the room together


11012

xx

)1(60

10

60

12

60

xx

6065 xx

6011 x11

60x

One pipe can fill a tank alone in 6 hours; another


181/293

pipe can fill it alone in 8 hours. A drain pipe can

empty the tank in 12 hours. If the tank is empty, and

all the three pipes are open, how long will it take to

fill the tank?

Soln: Let: x = the number of hours needed to fill the empty tankif all pipes are open

Working Equation:

Therefore the tank can be filled in 4.8 hours

12

1

8

1

6

11

xhoursx 8.4


A boy is one third as old as his brother and 8 years


182/293

y y

younger than his sister. The sum of their ages is 38years. How old is each of them?

Soln: Let: x = age of boy 3x = age of brother

x + 8 = age of sister

x = 6 years (age of boy)

3x = 18 years (age of brother)

x + 8 = 14 years (age of sister)

38)8(3 xxx


6x

8 years ago, Manny was three times as old as

h l ld d


183/293

Ronnie. Now he is only twice as old as Ronnie. Find

their ages now.

Soln: Let x = Ronnies age 8 years ago

3x = Mannys age 8 years ago

x 8 = Ronnies age now

3x 8 = Mannys age now

Working Equation:

x = 8 years (Ronnies age 8 years ago)3x = 24 years (Mannys age 8 years ago)

x 8 = 16 years (Ronnies age now)

3x 8 = 32 years (Mannys age now)

)8(283 xx8x


The difference between two numbers is 24


184/293

and their sum is 60. Find the numbersSoln: Let: x = one number

x + 24 = the other number

Working Equation:

Therefore the numbers are: 18 and 42

60)24( xx18x

4224 x


If the smaller of two consecutive odd integers is


185/293

doubled, the result is 7 more than the larger of the

two integers. Find the smaller integer.

Soln: Let: x = the smaller integer

2x = the smaller of two consecutive odd integers doubled

x + 2 = the larger integer

Working Equation:

Therefore the smaller integer is 9, the larger would be 11


7)2(2 xx

9x112 x182 x

Find the number which is 7/4 times the


186/293

excess of the number over 3Soln: Let: x = the required number

x 3 = the excess of the number over 3

Working Equation: )3(4

7 xx

7x



187/293

is an expression or an equation thatcontains the variable squared but not raised to a


188/293

contains the variable squared, but not raised to a

higher power.

, a is not zero

Can be Solved By:

1. Solution by Factoring

2. Solution by Completing the Square3. Solution by Quadratic Formula

02 cbxax

a

acbbx

2

42


The quantity in the above equation isacb 42


189/293

known as the The discriminantwill determine the nature of the roots of thequadratic equation.

: Nature of Roots:0 Only one root (real and equal)

> 0 Real and Unequal

< 0 Imaginary and Unequal

acb 42


Determine the character of the roots of the equations


190/293

a. 3x2

2x 6 = 0 b. 4x2

12x + 9 = 0

a. 3x2 2x 6 = 0 a = 3, b = -2, c = -6

b2 - 4ac = (-2)2 - 4(3)(-6)

D = 76, the discriminant is positive, hence theroots are real and unequal

b. 4x2 12x + 9 = 0 a = 4, b = -12, c = 9

b2 - 4ac = (-12)2 - 4(4)(9)D = 0, the discriminant is zero, therefore theroots are equal real numbers


Find the solution set of the equation


191/293

32

651

2

xx

6

3

)2(

6

516

2

xx

2)2()5(6 2xx

2

456 xx

0654 2 xx

0)2)(34( xx

24

3

2

1

x

x

The solution sets are -3/4 and 2


Find the solution set by completing the square

06232 112


192/293

06232 xx

3

0

3

6

3

2

3

32

xx

23

2

023

2

2

2

xx

xx

9

1

3

1

2

3

2

2

2

2

2

b

9

1

29

1

3

22

xx

9

19

3

12

x

3

19

3

1x

3

191x

The solution set is

3

191



193/293


194/293


The quadratic formula is derived by completing the square of

the general formula of the quadratic equation.


195/293

02 cbxax

aa

c

a

bx

a

ax 02

acx

abx

2

22

2

22

a

b

a

c

a

bx

a

bx

a

c

a

b

a

bx

2

22

42

2

22

4

4

2 a

acb

a

bx

a

acb

a

bx

2

4

2

2

a

acb

a

bx

2

4

2

2

aacbbx

24

2


Use the quadratic formula to find the solution


196/293

of set of the equation 6x2

= 10 + 11xSoln: 6x2 - 11x - 10 = 0 a = 6, b = -11, c = -10

The solution sets are 5/2 and -2/3


aacbbx

24

2

)6(2

)10)(6(4)11()11( 2 x

3

2

25

2

1

x

x

Sum of the Roots:

B


197/293

Product of the Roots:

Finding the quadratic equation givenits roots:


A

B

rr 21

A

Crr 21

0)(2121

2

rrxrrx

Find the sum and product of roots of 4x2 + 8x 12 = 0


198/293

r1 + r2 = -b/a = -8/4 = -2

r1r2 = c/a = -12/4 = -3

4x2

+ 8x 12 = 04(x2 + 2x 3) = 0

4(x + 3)(x 1) = 0

x1 = -3 x2 = 1

r1 + r2 = -3 + 1 = -2

r1r2 = (-3)(1) = -3


Find the quadratic equation given its roots

2/3 5/6


199/293

2/3, -5/6

x2 - (r1 + r2)x + r1r2 = 0

x2 - (2/3 5/6)x + (2/3)(-5/6) = 0

18x2 + 3x 10 = 0



200/293

A park contains a flower garden, 50 m long and 30 m wide, and a


201/293

p g , g ,

path of uniform width around it. If the area of the path is 600 m2

,what is its width?

30 meters

5

0meters

(30 + 2w) m

(50+2w)m

w meters

wm

eters

Let: w = width of the path

The area of the park

minus the area of the garden is

equal to the area of the path

Working Equation: 600)30)(50()230)(250( ww

600150041601500 2


202/293

Using quadratic formula:

Because w must be a positive number, we reject the negative

root.

w = -20 + 5(4.69) = 3.45 m, thus the width of the path is 3.45 m

015040

06001604

600150041601500

2

2

2

ww

ww

ww

aacbbw

242

)1(2

)150)(1(4)40(40 2 w

22520 w


203/293

- involves fraction


204/293

containing the variable in the denominator - an equation in which

constants are also represented by letters

- an equation involving avariable with a fractional exponent or an

equation in which a variable appears in a

radical


Solve for x:

3 + 4 2 + 5 0


205/293

3x + 4 - 2x + 5 = 0

6x 5 4x 1

(3x + 4)(4x 1) - (2x + 5)(6x 5) = 0

(6x 5)(4x 1)

(12x2 3x + 16x 4) - (12x2 10x + 30x 25) = 0

-7x + 21 = 0

x = 3


Multiplying both sides of the eqn by the LCD and simplifying, we have..

(k + x)2 + 5k = (3 + x)2 - 2k


206/293

( ) ( )

where: k is a constant


Solve the equation,

552 xx


207/293

Soln:

the solution sets are 2, 10, but by substitution, it can be found that only 2is the solution to the equation.10 is therefore called the

extraneous solution to the equation

552 xx

xx 55222 )5()52( xx 2102552 xxx

020122 xx0)2)(10( xx 2

10

2

1

x

x



32333 xx 02392 xx


208/293

By substituting the solution sets

to the equation, it can be found

that both solution sets does not

satisfy the given equation.Therefore the solution sets are

both extraneous. There is no

real solution set to the

equation

32333 xx

23333 xx

22 )233()33( xx

23236933 xxx

86236 xx

43233 xx

22 )43()233( xx

16249)23(9 2 xxx

162491827 2 xxx

0239 xx

0239 2 xx

0)13)(23( xx

32

1 x 312 x


)43(2)233(2 xx

An equation in a single variable x is said to be

quadratic in form if it can be written as


209/293

quadratic in form if it can be written as

where a 0 and u is an algebraic expression in x


02 cbuau


0152 24 xx


210/293

Let: u = x2, The equation then becomes,

u2 2u 15 = 0

Solving for u: (u 5)(u + 3) = 0

u = 5 u = -3

Replace u with x2 and solve the resulting equations

x2 = 5 x2 = -3

x = 5 x = i 3

The solution set of the original equation is therefore (5, i3)

0152xx



112


211/293

Let: The equation then becomes

3u2 4u 15 = 0

(3u + 5)(u 3) = 0

u = -5/3 u = 3

Replacing u with and solve for x

The solution sets will be -3/4, -1/4, 1/3, and 1

015

1

44

1

43

xxxx

xxu

14

xx1

4



212/293


213/293


214/293


Linear Inequalities


215/293

Linear Inequalities

Polynomial and Rational Inequalities

Inequalities involving Absolute Value


Find the solution set of the inequality


216/293

3x - 8 < 7

3x - 8 + 8 < 7 + 8

3x < 15

1/3 (3x) < 1/3 (15)

x < 5

5-5 0

The solution set is the interval (-, 5)



x - 7 x


217/293

x 7 x

4(4) x - 7 (4) x

4

x - 7 4x

x - 7 7 4x 7x 4x 7

-3x 7

-1/3 (-3x) (-1/3)7

x -7/3

The solution set is the interval [-7/3, )

5-7/3 0


Find the solution set of

3 < 4x 7 15


218/293

Solve each of the inequalities separately,

A value of x will be a solution of the given inequality if and only if -1 < xand x 2. Therefore the solution set is the interval (-1, 2]

-1 0 2

3 < 4x + 7

3 - 7 < 4x + 7 - 7

-4 < 4x

(-4) < (4x)

-1 < x

4x 7 15

4x + 7 - 7 15 - 7

4x 8

(4x) (8)

X 2


Find the Solution Set

x2 2x 15


219/293

The given inequality is equivalentto....

01522 xx

0)3)(5( xxThe critical numbers are -5 and 3 plotted in the

number line

3-5 0

The following intervals are determined: (-, -5), (-5, 3), (3, )


Interval Test Number

k

Sign of x + 5 at

k

Sign of x -3 at

k

Sign of (x + 5)

(x 3) on

interval


220/293


interval

(-, -5) -6 - - +

(-5, 3) 0 + - -

(3, ) 4 + + +

3-5 0

+ - +

Therefore (x + 5)(x - 3) > 0 if x is ineither ( - , -5 ) or ( 3, )

Thus the solution set of

the given inequality is:

( - , - 5 + U * 3, )

Find the Solution Set

(x + 1)(2x2 - 5x + 2) > 0


221/293

The given inequality is equivalent

to....

0)2)(12)(1( xxx

The critical numbers are -1, and 2 plotted in thenumber line

2-1


Interval Test

Number

Sign of

x + 1 at k

Sign of

2x -1 at k

Sign of

x 2 at k

Sign of (x + 1)

(2x 1) (x 2)


222/293

k on interval

(-, -1) -2 - - - -

(-1, ) 0 + - - +

(1/2 , 2) 1 + + - -

(2, ) 3 + + + +

2-1

+-+-

We conclude that (x + 1)(2x - 1)(x 2) > 0 if x is

in either ( - 1, ) or ( 2, )

Thus the solution set of

the given inequality is:

( - 1, ) U ( 2, )


Find the solution set of

13 x We now multiply on both


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162 xx

016

132

xx

x

06

)6(132

2

xx

xxx

06

542

2

xx

xx

sides of the inequality signby -1

0

6

542

2

xx

xx

0)2)(3(

)1)(5(

xx

xx

0)2)(3(

0)1)(5(

xx

xx


The critical numbers are 5, -1, 3, and -2


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3-2 -1 5

Interval Test

Number

k

Sign of

x + 2 at k

Sign of

x +1 at k

Sign of

x 3 at

k

Sign of

(x 5) at k

Sign

(x 5)(x + 1)

(x 3)(x + 2)

On interval

(-, -2) -3 - - - - +

(-2,-1 ) -3/2 + - - - -

(-1 , 3) 0 + + - - +

(3, 5 ) 4 + + + - -

(5, ) 6 + + + + +

3-2 -1 5

+ - + - +

As we can see in the table, the rational expression is (+) if

x is in one of the following intervals: (- , -2), (-1, 3), and


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g ( , ), ( , ),

(5, )

Therefore the solution set is (- , -2] U [-1, 3) U *5, )


Statement 1:


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bE is equivalent to bEb

bE is equivalent to bE

Statement 2:

Statement 1 is valid if the symbol < is replaced by

Statement 2 is valid if the symbol > is replaced by

and the symbol < is replaced by

bE or




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2x 7 < 9

-9 < 2x 7 < 9

-9 + 7 < 2x 7 + 7 < 9 + 7

-9 + 7 < 2x < 16

-2 < 2x < 16

-1 < x < 8

Therefore the solution set is the open interval (-1, 8)

From statement 1:

-1 0 8



352

x


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353

x

353

2

x

9152 x

242 x

12x

35

3

2x

9152 x

62 x

3x

From statement 2

The solution set is the interval ( - , 3+ U * 12, )


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0 3 12


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- the logarithm of a number or variable x to base b, logbx, isthe exponent of b needed to give x


231/293

Log216 = 4 may be written as 24 = 16

-logarithm to the base 10

logarithm to the base e which isapproximately 2.7182

y = lnx if and only if x = ey

If log x = a; then x is called the antilogarithm of a



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Find the value of each of the following logarithms:

1


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49log.) 7a 5log.) 5b 6log.)6c

2

77

497

2

y

y

y

2

1

55

55

2

1

y

y

y

1

66

6

16

1

y

y

y


xb log


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xb73

7log3 510 5log 1 0

ybyb log

52log5

2

310log3

10

01log b


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1log bbxy ln if and only if y

ex

1ln e

xex

ln

and xex

ln

1. yxxy loglog)log(


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2.

3.

4.

5.

6.


yxy

xlogloglog

xnxn loglog

bxx

blogloglog

a

x

x

b

ba

log

loglog

1log aa

Express each of the following in terms of logarithms of x,

y, and z, each of which represents a positive number.


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432log.) zyxab 2

log.)yz

xb

b5

3

2

log.)z

xyc

b

432432 loglogloglog.) zyxzyxabbbb


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zyx bbb log4log3log2 2

2logloglog.) yzx

yz

xb

bbb

)log(loglog 2zyxbbb

zyxbbb

log2loglog

2

5

2

log1

log.)xyxy

cbb


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33 5 zz

)loglog(log5

1 32zyx bbb

)log(log5

1 32 zxybb

)log3log2(log5

1zyx

bbb

zyxbbb

log5

3log

5

2log

5

1



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Solve 3x = 16


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x = log 3 16

x = log 16/ log 3

x = 2.5237



08.05 13 x


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08.0log5log 13 x

08.0log5log)13( x

08.0log5log5log3 x5log08.0log5log3 x

5log3

5log08.0log

x

1897.0x


3)32(loglog33

xx


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We reject the negative root since bothlogarithms are not defined. Therefore thesolution set is 9/2

3)32(log3

xx32 332 xx

027322

xx 0)92)(3( xx2

9

3

x

x



244/293

In a particular bacterial culture, if f (t) bacteria

are present at t minutes, then


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where B is a constant. If there are 1500bacteria present initially. Determine how

many minutes elapse until there are 30,000

bacteria present in the culture.

tBetf 04.0)(


Let T represent the number of minutes that

elapse until there are 30,000 bacteria present.


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Then in the given equation we substitute T fort and we have

TeTf 04.01500)( Te 04.01500000,30

Te 04.020

20ln04.0 T

9.74T

Therefore 1 hr, 14 min, and 54 secelapse until there are 30,000 bacteria

present


If $ 1000 is deposited into a savings account

that pays an annual interest rate of 6 %


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compounded continuously, and nowithdrawals or additional deposits made, how

long will it take until there is $1500 on

deposit?


itPeA

Te 06.010001500


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5.106.0 Te5.1ln06.0 T

06.0

5.1lnT

758.6T

Therefore it will take 6 years, 9

months, and 3 days until there is

$1500 on deposit


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Variation problems are problems in algebra

h h h h l h b h


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which show the relationship between thevariables in terms of expressions such as

directly proportional or inversely

proportional or simply proportional. k

is called constant of proportionality.


y varies directly as x, y is proportional to x xkykx

y ,


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z varies jointly with x and y,

z is directly proportional to x2 and inversely proportional to y,

,2

kx

y

xky

kxy

z k

x

zy

2


The time required for an elevator to lift a

i h i di l i h h i h d h


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weight varies directly with the weight and thedistance through which it is to be lifted and

inversely as the power of the motor. If it takes

30 seconds for a 10 hp motor to lift 100 lbs

through 50 ft., what size of the motor is

required to lift 800 lbs in 40 seconds through 40

ft.?

Mechanical Engineering Board October 1992


P

WSkt

Wh t 30 W 100 S 50 d P 10


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When t = 30, W = 100, S = 50 and P = 10

10

)50)(100(30 k

06.0k

When t = 40, W = 800, and S = 40

P

)40)(800()06.0(40 hpP 48


254/293


255/293

1. First Degree in Equation in Two variables and Linear

S t f E ti


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Systems of Equationsax + by + c = 0 Can be solved by:

Method of Elimination

Method of Substitution

2. Systems of First Degree Equations in Three Variables

ax + by +cz + d = 0 Can be solved by:

Method of Eliminationand Substitution

Determinants


1. The intersection of the two solution sets containstl d d i Th h i t t tl


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exactly one ordered pair. The graph intersects exactlyat one point. The equations are said to be consistentand independent.

2. The intersection of the two solution sets is the emptyset. The graphs are distinct parallel lines. Theequations are said to be inconsistent.

3. The solution sets of the two equations are equal. Thegraphs are the same line. The equations are said tobe dependent.


Find the solution set of the system of

equations

32


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103532 yx

yx

)32(3 yx

10)35( yx

936 yx1035

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Algebra Aero Engr Board Review Lecture

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