338839 329000 20-1Topic 1 Fundamentals of Algebra PDF

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Lecture notes for Topic 1 Algebra. The topic includes Real Numbers, Algebra Essentials Polynomials, Factoring Polynomials, Synthetic Division, Rational Expressions, and nth Roots; Rational Exponents....

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PAM0135

Algebra

Topic 1

Faculty of Information Science & Technology (FIST)

PAM 0135 Algebra

Foundation in Life Science Foundation in Information Technology

ONLINE NOTES

Topic 1 Fundamentals of Algebra

TOPIC 1: FUNDAMENTALS OF ALGEBRA

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Algebra

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Reference: Michael Sullivan, et.al (2011). Algebra and Trigonometry. Pearson, Prentice Hall. Objectives: 1. Know the classification of numbers. 2. Graph inequalities, find distance on the real number line, use the Laws of Exponents and evaluate square roots. 3. Add and subtract polynomials, multiply polynomials, know formulas for special products and divide polynomials using long division. 4. Factor the difference of two squares and the sum and the difference of two cubes, factor perfect squares, factor a second-degree polynomial and factor by grouping. 5. Divide polynomials using synthetic division. 6. Reduce a rational expression to lowest terms, multiply and divide rational expressions, add and subtract rational expressions and simplify complex rational expressions. 7. Work with nth roots, simplify radicals, rationalize denominators and simplify expressions with rational exponents Contents: R.1 Real Numbers R.2 Algebra Essentials R.3 Polynomials R.4 Factoring Polynomials R.5 Synthetic Division R.6 Rational Expressions R.7 nth Roots; Rational Exponents

R.1

REAL NUMBERS

Classification of Numbers 1.

Natural Numbers (N)

{1, 2, 3, . . .}

2.

Whole Numbers (W)

{0, 1, 2, . . .}

3.

Integers (Z) Consist of:

{. . ., -3, -2, -1, 0, 1, 2, 3, . . .} - positive integers {1, 2, 3, . . . } - negative integers { . . , -3, -2, -1} - zero {0}

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4.

Algebra

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Rational Numbers (Q) - numbers that can be expressed as a quotient of two integers. a Q  b

a and b are integers,

 b 0  

- may be represented as decimals: terminating, or non-terminating with repeating digits. Example:

2,

3 0.375 , 8

3 0.272727... 0.27 11

5.

Irrational Numbers - represented by decimals that neither repeats nor terminates. 2 1.414213... ,  3.14159... Example:

6.

Real Numbers (R ) - all rational numbers together with all irrational numbers

Equality a a 1. The reflexive Property: a b , then b a . 2. The symmetric property: If 3. The principle of substitution: If a b , we may substitute b for a in any expressions with a. Properties of Real Numbers For all the following properties, a , b , c  R :

a  b is a real number; ab is a real number.

1.

Closure

2.

Commutative Properties:

a  b b  a ;

3.

Associative Properties:

a   b  c   a  b   c a  b  c a  b c   a b  c a b c

4.

Distributive Properties:

a b  c  a b  a c  a  b c a c  b c

5.

Identity Properties:

a b b a

0  a a  0 a ;

a 1 1 a a

0 - additive identity; 1 - multiplicative identity. a  ( a )  a  a 0 ; (-a) - negative of a.

6.

Additive Inverse Property:

7.

Multiplicative Inverse Property:

1 1 a   a 1; if a a

a 0 ;

1 a

reciprocal ______________________________________________________________________________________ 2/ 16

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Algebra

Topic 1

of a. 8.

Difference :

a  b a  (  b)

9.

Quotient:

1 a  a ; if b b

10.

Multiplication by zero:

a 0 0

11.

Division Properties:

0 0 ; a

12.

Rules of Signs:

a(  b)  ( ab) ;

(  a) b  ( ab) ;

a 1; a

Cancellation Properties:

14.

Product law:

15.

Arithmetic of Quotients:

if

a 0

(  a)(  b)  ab

 ( a ) a ;

13.

b 0

 a a a   ;  b b b

 a a   b b

ab bc implies a  c if b  0 ac a  if b 0 , c 0 bc b If ab 0 , then a = 0 or b = 0 or both. a b a b a b

c ad  bc  if b 0, d 0 d bd c ac   if b 0, d 0 d bd c a d ad     if b 0, c 0, d 0 d b c bc



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ALGEBRA ESSENTIALS

The Real Number Line - Every real number corresponds to point on the line and, conversely, each point on the line has a unique real number associated with it. - The number line divides the real numbers into three classes: Negative real numbers, Zero, and Positive real numbers. Constants - a fixed number; a letter that represents a fixed number. Variables - a letter used to represent any number from a given set of numbers.

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Inequalities < less than; than or equal to a b a0

or 

a 0



a 0



a 0



>

b a





greater than; b a

less than or equal to; 

greater

is positive.

a is positive a is negative a is nonnegative a is nonpositive

Absolute Value a a   a

Absolute value is defined by the rule: a

if a 0 if a  0

represents the distance to the origin from the point a.

Properties : a , b  R

a 0 ,

 a a ,

a 0 

a 0 ,

a a  , b b

a  b a  b

Distance Between P & Q If P and Q are 2 points on a real number line with coordinates a and b, respectively. The distance between P & Q, is d (P , Q ) b  a  a  b d  Q , P Exponents a n a a a ... a (n factors)

where a is a real number (base); n is a positive integer (exponent/power). 1 ; an

n Note: a 

a0 1 , if

Laws of Exponent: 1. a n a m  a m n 2. am  4.  am n an a 0 1 7.

a 

m n

a 0

amn

3.

 a b n

n

5.

an  a    n b  b

6.

a n b n  a    b

 n

a    b 

n

Square Roots b 2 a



b a

where a, b are nonnegative real numbers.

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b is the principal square root of a. Note:

a

2

a

1.

Properties: Example: 5 1. 24

2.

3

ab  a b

2.

a a  b b

10  50  25 2 5 2

 8  4 2 2 2

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POLYNOMIALS

Monomial and Polynomial. Monomial A monomial in one variable is the product of a constant times a variable raised to a nonnegative integer power, axk where a = a constant known as a coefficient of the monomial x = a variable k = an integer greater than zero known as the degree of the monomial if a 0 Example : 2

6x 

5x

8

Polynomial A Polynomial is an algebraic expression of the n n 1 form an x  an  1x  ....a1 x  a0 where a n , an 1 ,....a1 , a0 = constants called the

coefficient of the polynomials n  0 is an integer and known as the degree of

the polynomial if an 0 ( a n is called the leading coefficient)

Example :  6 x3 

14 x

26 x13 

11

5 x2 14 x 

1 2

3 8 x  11 x5  7 x  9 5

Definition of the terms 

Like terms - Two monomials with the same variable raised to the same power.



Binomial – The sum or difference of two monomials having different degree

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

Algebra

Topic 1

Trinomial- The sum or difference of three monomials having three different degree

Example : Identify the following expression either a monomial or polynomial or neither. Hence, determine its coefficient and degree. No 1 2

Expression

Type

Coefficient

Degree

5x3  2 x 2  3 x  5

5 x7 8

3

4 5z 

4

3

2z 3 z   3

3 7y6 3x  5 x 2

5

Adding and Subtracting Polynomials Find the sum and difference of the polynomials x 5  12 x3  8 x 2  7 and  7 x 8  3x 5  4 x 4  6 x 3  8 x  12

Using Horizontal Method. x 5  12 x 3  8 x 2  7 - (  7 x 8  3x 5  4 x 4  6 x 3  8 x  12 ) =

Using Vertical Method x5  0 x 4  12 x 3  8 x 2  0 x  7

minus  7 x  3 x  4 x 4  6 x 3  0 x 2  8 x  12 8

5

=

Multiplying Polynomials Find the product of 3 x 2  5  x  3 x  8



2



Using Horizontal Method.

3 x

2





2

 5 x  3 x  8 3 x4  9 x3  24 x 2  5 x 2  15 x  40

3 x  9 x  21x  15 x  40 4

3

2

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Using Vertical Method

x

3 x 2

2



 5

 3x  8



=

Special Products FOIL Method (First Outer Inner Last) (ax+b)(cx+d) = ax(cx+d)+b(cx+d) = axcx+adx+bcx+bd =acx2+ (ad+bc)x+bd Example : (2x+1) (3x-5) = Difference of Two Squares. (x-a)(x+a)=x2-a2

Example : (4x-3)(4x+3) =

Squares of Binomials or Perfect Squares. (x+a)2=x2+2ax+a2 (x-a)2=x2-2ax+a2

Example : a) (x+7)2 = b) (x-5)2 =

Cubes of Binomials, or Perfect Cubes. (x-a)3=x3-3ax2+3a2x-a3 ______________________________________________________________________________________ 7/ 16

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(x+a)3=x3+3ax2+3a2x+a3

Example : a) (x+2)3 = b) (x-5)3 =

Difference and Sum of Two Cubes (x-a)(x2+ax+a2) = x3-a3 (x+a)(x2-ax+a2) = x3+ a3 Example : (x+2)(x2-2x+4) =

Divide Polynomials Using Long Division To divide two polynomials, we first must write each polynomial in standard form. To check the answer obtained in a division problem, multiply the quotient by the divisor and add the remainder.The answer should be the dividend. (Quotient)(Divisor)+Remainder = Dividend

Example: Find the quotient and the remainder when 3x 3  4 x 2  x  7 is divided by x 2 1 Solution: Each polynomial is in standard form. The dividend is 3x 3  4 x 2  x  7 , and the divisor is x 2 1 STEP 1: Divide the leading term of the dividend, 3x 3 by the leading term of the divisor, x2 . Enter the result, 3x, over the term 3x 3 , as follows:

3x 3  4x 2  x 7

3x x2  1

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STEP 2: Multiply 3 x by x 2 1 and enter the result below the dividend.

3x 3  4x 2  x 7

3x 2 x 1 3x 3

 3x

STEP 3: Subtract and bring down the remaining terms.

3x 3  4x 2  x 7

3x x2  1 3 3x

 3x

4 x2  2 x  7

STEP 4: Repeat Steps 1–3 using

4x2  2x  7 as the dividend.

3x  4 3  4x 2  x 7

3x x2  1 3 3x

 3x

4 x2  2 x  7 4x 2

4

 2x  3

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Since

Algebra

Topic 1

x 2 does not divide  2 evenly (that is, the result is not a

monomial), the process ends. The quotient is

 2x  3

3x  4 and the remainder is

Check: (Quotient)(Divisor)+Remainder





(3 x  4) x 2  1    2 x  3 3 x3  3 x  4 x 2  4    2 x  3

3x 3  4 x 2  x  7 = Dividend  2x  3 3x 3  4x 2  x  7 = 3x  4 2 x 2 1 x 1

Then

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FACTORING POLYNOMIALS

Common Factors If a polynomial has a common factor, the common factor of highest degree is factored out. Example: 2a 2x  4ax  6a  2a (ax ) 2a (2x ) 2a (3) 2a( ax  2 x  3)

{ the greatest common factor is 2a.}

Factoring Formulas Formula a. Difference of two squares :

Examples x 2  16  (x  4 )(x  4)

a  b ( a  b)( a  b) 2

b.

2

Perfect Square Trinomial:

a  2ab  b ( a  b ) 2

2

x 2  8x  16  (x  4)2

2

4x 2  4xy  y 2  (2x )2  2(2 x )( y )  y 2

a 2  2ab  b2 ( a  b)2

2

 (2x  y ) z 3  27  z 3  (3)3

c. Sum of Two cubes a  b ( a  b)( a  ab  b ) 3

3

2

2

8 x3  y 6 (2 x) 3  ( y 2) 3

d. Difference of two cubes a  b ( a  b)( a  ab  b ) 3

3

2

2

 (z  3)(z  3z  9)

2

2

2

2

4

(2 x  y )(4 x  2 xy  y )

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Trial-and-Error Factorization The factors of the second-degree polynomial have the form (ax+b)(cx+d) where ac=p, ad+bc=q, and bd=r.

px2  qx  r

, where p,q, and r are integers,

We use trial-and-error method to factor polynomials having this form since there is a limited number of choices are possible. Example:

x 2  2x  3  (x  1)(x  3)

Factoring by Regrouping Sometimes a polynomials may be factored by regrouping and rearranging terms so that a common term can be factored out.





x 3  x  x 2  1  x 3  x 2   x  1

Example:

x ( x 1) 1  x 1  2

 (x  1)(x 2  1)

R.5

SYNTHETIC DIVISION

Synthetic Division A Synthetic Division is a shortened version of long division for a polynomial. We can use this method to i) find the quotient and remainder ii) verify a factor iii) find the value of a polynomial Examples: 1) Find the quotient and remainder when f (x) = 3x4 + 8x2 - 7x + 4 is divided by g(x) = x - 1 Solution: The divisor is x -1, so row 3 entries will be multiplied by 1, entered in row 2 and added to row 1. ______________________________________________________________________________________ 11/ 16

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 7

Topic 1

row 1 3 3 11 4 row 2 (add) --------------------------3 3 11 4 8 row 3 (quotient) (remainder)

1 3

0

8

4

Then the quotient = 3x3 + 3x2 +11x + 4, and remainder = 8 Note: The quotient has degree 1 less than that of the dividend. 2)

Use synthetic division to show that g (x) = x + 3 is a factor of f (x) = 2x5 + 5x4 - 2x3 + 2x2 -2x +3. Solution: The divisor is x + 3 = x - (-3)  3 2 5  2 2  2 3 row 1 Then -6 3 -3 3 -3 row 2 (add) ------------------------------------2 -1 1 -1 1 0 row 3 Remainder = 0  f (-3) = 0, then by factor theorem g (x) = x + 3 is a factor of f (x). 3) Use synthetic division to find the value of f(x) = -3x4 + 2x3 - x + 1 at x = -2. ie. f (-2) Solution: Remainder Theorem says that the value of the polynomial function at x = -2 equals the remainder when the function is divided by x - (-2).

Then,  2  3

2

0

 1

1

6 -16 32 -62 -----------------------------3 8 -16 31 -61 Then, by Remainder Theorem, f (-2) = - 61 Note that there are 3 ways to find the value of a polynomial function f (x) at a number c: 1) Substitution - replace x by the number c to find f (c). 2) Remainder Theorem. - use synthetic division to divide f (x) by x - c. The remainder is f (c). 3) Use a calculator to find f(c).

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R.6

Algebra

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RATIONAL EXPRESSIONS

Quotients of polynomials are called rational expressions. Examples: 2x  1 5x  3

4 x 2 y 5  2 xy 5 xy  y

and

Simplifying Rational Expressions A rational expression is simplified, or reduced to lowest terms, if the numerator and the denominator have no common factors other than 1 and –1.

Example: The rational expression x2  5x  6 ( x  2)( x  3)

is first been factor then been simplified by canceling the common factor ( x + 3 ) and writing x2  5 x  6 ( x  2)(x  3)  (x  2)( x  3) ( x  2)(x  3)



( x  2) ( x  2)

Multiplication and Division The operations of multiplication and division are performed with rational expressions in the same way they are with arithmetic fractions.

Example: 1. 2.

2 x ( x  1) 2 x( x  1)   y( y  1) y ( y  1) 2 x  3 y  1 x 3 x    2 y x y y 1  x ( x 3)  y ( y 2 1)

Addition and Subtraction For rational expressions, the operations of addition and subtraction are performed by finding a common denominator for the fractions and then adding or subtracting the fractions. ______________________________________________________________________________________ 13/ 16

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Example: 2x  1 x  3  x 2 x 1 2 x  1 x  1 x 3 x  2     x2 x 1 x 1 x 2 (2 x  1)( x  1)  ( x  3)( x  2)  ( x  2)( x  1) 

(2x 2  3x 1)  (x 2  5x  6) (x  2)(x  1)



x 2  8x  5 (x  2)(x  1)

[Common denominator is (x + 2...