Topic 1 Fundamental Concept of Algebra PDF

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Gives a fundamental idea about mathematics...

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Topic 1: Fundamental Concepts of Algebra Sub topics: 1.1 Real Numbers and Algebraic Expressions 1.2 Exponents 1.3 Radicals and Rational Exponents 1.4 Polynomials 1.5 Factoring Polynomial 1.6 Rational Expressions -----------------------------------------------------------------------------------------------------------Objectives: 1. Know the classification of numbers. 2. Graph sets of numbers, equations, inequalities, and absolute value. 3. Properties of real numbers. 4. Understand and use integer exponents and properties of exponents. 5. Evaluate square roots and radicals, and their properties. 6. Understand the vocabulary of polynomials and its properties and operations. 7. Know how to use different methods of factoring polynomials. 8. Specify numbers that must be excluded from the domain of rational expression and its operations and simplify complex rational expression. -----------------------------------------------------------------------------------------------------------1.1 Real Numbers and Algebraic Expressions The Set of Real Numbers Basic ideas: • • • •

A set is a collection of objects whose contents can be clearly determined. The objects in a set are called the elements of the set. We use braces to indicate a set and commas to separate the elements of that set. For example: {1, 3, 5, 7, 9}

Important Subsets of the Real Numbers Name Natural numbers, ℕ Whole Numbers, 𝕎 Integers

ℤ Rational numbers

ℚ

Description These numbers are used for counting.

Examples {1, 2, 3, 4, 5, …}

Adding 0 to the set of natural numbers.

{0, 1, 2, 3, 4, 5, …}

Consist of: - positive integers {1, 2, 3, . . . } - negative integers { . . , -3, -2, -1} - zero {0} Numbers that can be expressed as a quotient of two integers.  a a and b are integers, b  0  Q b  - may be represented as decimals: terminating, or non-terminating with repeating digits

{. . ., -3, -2, -1, 0, 1, 2, 3, . . .}

2 3  0.375 8 3  0.272727...  0.27 11

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Irrational numbers

𝕀 Real Numbers (R ) Complex Numbers (C )

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This is the set of numbers whose decimal representations are neither terminating nor repeating. Irrational numbers cannot be expressed as a quotient of integers. all rational numbers together with all irrational numbers

2  1.414213...   3.14159...

in the form a + bi, where a, b are real numbers

0, 2i, 3 - 4i, i

Notes: Real Numbers = Rational Numbers + Irrational Numbers

The Real Number Line

-3

-2

-1

Negative

0

1

2

Origin

3

Positive

Ordering the Real Numbers  On the real number line, the real numbers increase from left to right.  The lesser of two real numbers is the one farther to the left on a number line.  The greater of two real numbers is the one farther to the right on a number line. Inequalities

Symbols

ab ab

 

less than less than or equal to Meaning a is less than or equal to b

b is greater than or equal to a

 

greater than greater than or equal to

Example

58 88 85  4  4

Explanation Because 5 < 8 Because 8 = 8 Because 8 > 5 Because – 4 = – 4 2

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Absolute Value

 x x  x

x

x0 if x  0

if

represents the distance to the origin from the point x.

Properties of Absolute Value For all real number a and b 1.

a 0

4.

2.

a  a

5.

3.

a a

6.

ab  a b a a  ,b  0 b b ab  a  b

Algebraic Expressions A combination of variables and numbers using the operations of addition, subtraction, multiplication, or division, as well as powers or roots, is called an algebraic expression. Examples of algebraic expressions:

x + 6, x – 6, 6x, x/6, 3x + 5.

Evaluating Algebraic Expressions  find the value of the expression for a given value of the variable The order of Operations Agreement 1. Perform operations within the innermost parentheses and work outward. If the algebraic expression involves division, treat the numerator and the denominator as if they were each enclosed in parentheses. 2. Evaluate all exponential expressions. 3. Perform multiplication or division as they occur, working from left to right. 4. Perform addition or subtraction as they occur, working from left to right. Example: Evaluate the algebraic expression 2.35x + 179.5 when i) x = 20 ii) x = 30

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Properties of Real Numbers and Algebraic Expressions Name Commutative Property of Addition Commutative Property of Multiplication Associative Property of Addition

Meaning Examples Two real numbers can be added in any  13 + 7 = 7 + 13 order.  13x + 7 = 7 + 13x a+b=b+a Two real numbers can be multiplied in  x · 6 = 6x any order. ab = ba If 3 real numbers are added, it makes  3 + ( 8 + x) = (3 + 8) + x no difference which 2 are added first. = 11 + x (a + b) + c = a + (b + c)

Associative Property of Multiplication

If 3 real numbers are multiplied, it makes no difference which 2 are multiplied first. (a · b) · c = a · (b · c) Multiplication distributes over addition. a · (b + c) = a · b + a · c

 -2(3x) = (-2·3)x = -6x

Zero can be deleted from a sum. a+0=a 0+a=a One can be deleted from a product. a · 1 = a and 1 · a = a

 0 + 6x = 6x

The sum of a real number and its additive inverse gives 0, the additive identity. a + (-a) = 0 and (-a) + a = 0 The product of a nonzero real number and its multiplicative inverse gives 1, the multiplicative identity. a · 1/a = 1 and 1/a · a = 1

 (-6x) + 6x = 0

Distributive Property of Multiplication over Addition Identity Property of Addition Identity Property of Multiplication Inverse Property of Addition

Inverse Property of Multiplication

 5 · (3x + 7) = 5 · 3x + 5 · 7 = 15x + 35

 1 · 2x = 2x

 2 · 1/2 = 1

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Simplifying Algebraic Expression An algebraic expression is simplified when parentheses have been removed and like terms have been combined. Simplify: 1. 4(2 y  6)  3(5 y  10)

2.  (5 x 13 y  1)

Properties of Negatives Let a and b represent real numbers, variables, or algebraic expressions.

1. ( 1)a   a 2.  ( a )  a 3. (a )b  ab

4. a( b)  ab 5.  (a  b )  a  b 6.  (a  b )  a  b  b  a

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Exponents

bn  b  b  b  ...  b

(n factors)

where b is a real number (base); n is a positive integer (exponent/power). Example: Find a.

2  2  2  23 , base = 2, exponent = 3

24

b.

 24

c.

(2) 4

d. (4) .(2) 2

3

Properties of Exponents

1. b n 

1 bn

2. b0  1

3. b .b  b m

n

bm 4. n  b m n b

m n

5. (b m )n  b mn

6. (ab) n  a nb n

n

an  a 7.    n b  b An exponential expression is simplified when  no parentheses appear ;  no powers are raised to powers  each base occurs only once ;  no negative exponents appear Examples: Simplify

1. xy

3

 6 12

2. x x

4 2

3. (6x )

20 x24 4. 10 x6

 3x4 5.   y

  

6

3

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Radicals and Rational Exponents

Principal Square Root If a is a nonnegative real number, the nonnegative number b such that by

b 2  a, denoted

b  a , is the principal square root of a. a Radicand

Radical Sign

Square roots of perfect squares:

a2  a

The Product Rule for Square Roots Simplified - when it radicand has no factor others than 1 that are perfect squares. If a & b represent non-negative real numbers, then

ab  a b and a b  ab Examples:

1. 125 x2

2.

6 x. 3 x2

Quotient Rule for Square Roots If a & b represent non-negative real numbers and

a a  and b b

Examples:

121 1. 9

b  0 then

a a  b b

2.

24x 4 3x

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Adding and Subtracting Square Roots Examples:

1. 8 5  11 5

2. 4 12  2 75

Rationalizing Denominator

2 10

1.

Examples:

2.

3 3 7

Other Kinds of Roots Principal nth Root of a Real Number n

a b

If n is even, If n is odd,

 a  bn ; a  0, b  0

a, b  R

Examples: 1. 2. 3.

3

5

and

n

a is the radicand; n is the index. n

and

an  a

an a

64   32  4

16 

Perfect nth Power A number that is the nth power of a rational number is called a perfect nth power. If n is odd, n a n  a If n is even, n a n  a 8

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Properties: 1.

3.

n

ab  n a n b

n

am 

2.

  n

a

m

m

 an

4.

n

a na  ,b  0 b nb

m n

a 

mn

a

Simplifying, Multiplying and Dividing Higher Roots Examples:

1. 3 8

2. 3 -125

3 3. x5

4 4.

162 x 5 4 2x

5.

3.3 81  4.3 3

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Rational Exponents If

n

a

represents a real number and n  2 is an integer, then 1 n

a  a ; n

a



1 n

1



a If

n

1 n



1 n a

m represent a real number, is a rational number reduced to lowest terms, and n  n

a

m n

a  (n a )m  n a m

2 is an integer, the

m consist of two parts: n 1. the denominator n is the root. 2. the numerator m is the exponent.

The exponent

a



m n



1 a

m

n

Examples:

1. 8

2

3

2. 16

5

2

2

3

3. 3 x .4 x

3

4

4.

72 x 9x

3

1

4

3

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1.4 Polynomials A polynomial is a single term or the sum of two or more terms containing variables with whole number exponents. e.g.

3x4 –3x3 + 4x2 –7x +5

Standard form – writing the terms in the order of descending powers of the variables. The Degree of axn: If a  0, the degree of

ax n

is n. The degree of a nonzero constant is 0. 3x3 + 4x2 –7x +5 Deg. 0

Deg. 3

Deg. 1

Deg. 2

Monomial – one term e.g. 7x Binomial – two terms, each with different exponents. Trinomial –three term, each with different exponents. Degree of a Polynomial – the highest degree of all the terms of the polynomial. Example:

3x3 + 5x  binomial with degree 3.

Definition of a Polynomial in x A polynomial in x is an algebraic expression of the form

an xn  a xn1  a x n2  ...  a x  a 1 n n2 1 0 where an, an-1, an-2, …, a1 and a0 are real numbers, an  0, and n is a nonnegative integer. The polynomial is of degree n, an is the leading coefficient and a0 is the constant term. Adding and Subtracting Polynomials Polynomials are added and subtracted by combining like terms. Example: 1. (–7x3+6x2-11x+13) + (19x3-11x2+7x-17)



 



2. 13 x  9 x  7 x 1   7 x  2 x  5 x  9 3

2

3

2

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Multiplying Polynomials Example: 1. (x+8) (x+5)

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2. (x+5) (x2-5x+25)

Special Products Let A and B represent real numbers, variables or algebraic expression. Sum and Difference of Two Terms Squaring a Binomial

Cubing a Binomial

(A + B)(A – B) = A2 – B2 (A + B)2 = A2 + 2AB + B2 (A – B)2 = A2 – 2AB + B2 (A + B)3 = A3 + 3A2B + 3AB2 + B3 (A – B)3 = A3 – 3A2B + 3AB2 – B3

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1.5 Factoring Polynomial - the process of writing a polynomial as the product of two or more polynomials. - use one or more factoring techniques until each of the polynomials factor is prime or irreducible  factored completely. Common Factor Examples:

1. 16x – 24

2.

x2 (2x  5)  17(2 x  5)

2.

x3  5 x2  2 x 10

Factoring by Grouping Examples:

1.

x3  3 x2  4 x 12

Factoring Trinomials In the forms of ax2 + bx + c A Strategy for Factoring ax2 + bx + c (Assume, for the moment, that there is no greatest common factor.) 1. Find two First terms whose product is ax2:

2. Find two Last terms whose product is c:

3. By trial and error, perform steps 1 and 2 until the sum of the Outside product and Inside product is bx:

If no such combinations exist, the polynomial is prime. Examples: 1. x 13 x  40 2

2.

x 2  5x  14

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Factoring the Difference of Two Squares

A2  B 2  ( A  B)( A  B) Examples: 1.

x 2 144

2.

81x 4  25

3.

36 x 2  49 y 2

Factoring Perfect Square Trinomials 1.

A2  2 AB  B2  ( A  B)2

Examples: 1.

x  4x  4 2

2. 2.

A2  2 AB  B 2  ( A  B)2

4x 2 16x 1

Factoring the Sum and Difference of Two Cubes 1.

A 3  B 3  ( A  B)( A2  AB  B2 )

2.

A3  B 3  ( A  B)( A2  AB  B2 )

Examples: 1. x

3

 64

2.

27 x3  1

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A Strategy for Factoring a Polynomial 1. If there is a common factor, factor out the GCF. 2. Determine the number of terms in the polynomial and try factoring as follows: a) If there are two terms, can the binomial be factored by one of the special forms including difference of two squares, sum of two cubes, or difference of two cubes? b) If there are three terms, is the trinomial a perfect square trinomial? If the trinomial is not a perfects square trinomial, try factoring by trial and error. c) If there are four or more terms, try factoring by grouping. 3. Check to see if any factors with more than one term in the factored polynomial can be factored further. If so, factor completely. Example: 1. 2x3 + 8x2 + 8x

2. x3 – 5x2 – 4x + 20

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1.6 Rational Expressions - quotient of two polynomials. -

Examples: 1.

3x x 5

2.

x2  2 2x 2  4x  6

Excluding Numbers from the Domain - must exclude numbers from a rational expression’s domain that make the denominator zero -

Examples: 1.

13 x9

2.

x 3 x 2  4 x  45

Simplifying Rational Expressions 1. Factor the numerator and denominator completely. 2. Divide both the numerator and denominator by common factors.

4x  8 Examples: 1. 2 x  4x  4

x  14 x  49 2. 2 x  49 2

Multiplying Rational Expressions 1. Factor all numerators and denominators completely. 2. Divide both the numerator and denominator by common factors. 3. Multiply the remaining factors in the numerator and multiply the remaining factors in the denominator.

6x  9 x  5  Examples: 1. 3 x 15 4x  6

2.

x 2  5x  6 x 2  9  x2  x  6 x 2  x  6

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Dividing Rational Expressions Examples: 1.

x  5 4 x  20  7 9

2.

x2  x x2  1  x 2  4 x2  5 x  6

Adding and Subtracting Rational Expressions with Different Denominator

a c ad  bc   b d bd

;

a c ad  bc   b d bd ,

b  0, d  0

Examples: 1.

2. 3.

3x  2 3 x  6   3x  4 3 x  4 8 2     3 x 2 x 3 5x   5 x  2 25 x2  4

Complex Rational Expression - numerators and denominators containing one or more rational expression.

- Examples:

1.

x 1 4 x 4

2.

x 1 x 2 3 1 x2 4

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