Copyright © 2000 by the McGraw-Hill Companies, Inc.

Barnett/Ziegler/ByleenPrecalculus: A Graphing Approach

Appendix A

Basic Algebra Review

Copyright © 2000 by the McGraw-Hill Companies, Inc.

N Natural Numbers 1, 2, 3, . . .

Z Integers . . . , –2, –1, 0, 1, 2, . . .

Q Rational Numbers –4, 0, 8, –35 ,

23 , 3.14, –5.2727

__

I Irrational Numbers 2 , 3

7 , 1.414213 . . .

R Real Numbers –7, 0, 3

5 , –23 , 3.14, 0.333

– ,

The Set of Real Numbers

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Subsets of the Set of Real Numbers

Natural numbers (N)

Negatives of naturalnumbers

ZeroIntegers (Z)

Noninteger ratios

of integers

Rational numbers (Q)

Irrational numbers (I)

Real numbers (R)

N Z Q R

A-1-114

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Basic Real Number Properties

Let R be the set of real numbers and let x, y, and z be arbitraryelements of R.

Addition Properties

Closure: x + y is a unique element in R.

Associative: (x + y ) + z = x + ( y + z )

Commutative: x + y = y + x

Identity: 0 + x = x + 0 = x

Inverse: x + (– x ) = (– x ) + x = 0

A-1-115(a)

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Basic Real Number Properties

Multiplication Properties

Closure: xy is a unique element in R .

Associative:

Commutative: xy = yx

Identity: (1) x = x (1) = x

Inverse: X

1

x =

1

x x = 1 x 0

Combined Property

Distributive: x (y + z ) = xy + xz(x + y) z = xz + yz

(xy)z = x(yz)

A-1-115(b)

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Foil Method

F O I LFirst Outer Inner LastProduct Product Product Product

(2x

– 1)(3x + 2)

=6x2 + 4x – 3x – 2

1. (a – b)(a + b) = a2 – b2

2. (a + b)2 = a2 + 2ab + b2

3. (a – b)2 = a2 – 2ab + b2

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Special Products

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1. Perfect Square

2. u2 – 2uv + v 2 = (u – v) 2 Perfect Square

3. u2 – v 2 = (u – v)(u + v) Difference of Squares

4. u3 – v

3 = (u – v)(u 2 + uv + v

2) Difference of Cubes

5. u3 + v

3 = (u + v)(u2 – uv + v

2) Sum of Cubes

u 2 + 2uv + v 2 = ( u + v)2

Special Factoring Formulas

A-3-117

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The Least Common Denominator (LCD)

The LCD of two or more rational expressions is found as follows:

1. Factor each denominator completely.

2. Identify each different prime factor from all the denominators.

3. Form a product using each different factor to the highest power that occurs in any one denominator. This product is the LCD.

A-4-118

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1. For n a positive integer:

an = a · a · … · an factors of a

2. For n = 0 ,

a0 = 1 a 000 is not defined

3. For n a negative integer,

an = 1

a–n a 0

1. am an = am+ n

2. ( )an m = amn

3. (ab)m = am bm

4.

a

bm

= am

bm b 0

5.am

an = am–n = 1

an–m a 0

A-5-119

Definition of an Exponent Properties

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For n a natural number and b a real number,

b1/n is the principal nth root of bdefined as follows:

1. If n is even and b is positive, then b1/n represents the positive nth root of b.

2. If n is even and b is negative, then b1/n does not represent a real number.

3. If n is odd, then b1/n represents the real nth root of b (there is only one).

4. 01/n = 0

For m and n natural numbers and b any real number (except b cannot be negative when n is even): b

m/n = ()

()

/

/

b

b

nm

mn

1

1

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Definition of b1/n

Rational Exponents

For n a natural number greater than 1 and b a real number, we define n

b to be the principal nth root of b; that is,

nb = b1/n

If n = 2, we write b in place of 2

b .

nb , nth-Root Radical

For m and n positive integers (n > 1), and b not negative when n is even,

bm/n =

(bm)1/n = n

bm

(b1/n)m = (n

b)m

Rational Exponent/Radical Conversions

1. n

xn = x

2. n

xy = n

x n

y

3. n x

y = n

x

ny

Properties of Radicals

A-7-121Copyright © 2000 by the McGraw-Hill Companies, Inc.

Copyright © 2000 by the McGraw-Hill Companies, Inc.

1. No radicand (the expression within the radical sign) contains afactor to a power greater than or equal to the index of the radical.

(For example, x5 violates this condition.)

2. No power of the radicand and the index of the radical have acommon factor other than 1.

(For example, 6

x4 violates this condition.)

3. No radical appears in a denominator.

(For example, yx violates this condition.)

4. No fraction appears within a radical.

(For example, 35 violates this condition.)

Simplified (Radical) Form

A-7-122

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[a, b] a x b [ ]a b

x Closed

[a, b) a x < bb

[a

) x Half-open

(a, b] a < x b ]a b

x( Half-open

(a, b) a < x < ba b

x( ) Open

Interval InequalityNotation Notation Line Graph Type

Interval Notation

A-8-123(a)

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[b , ) x bb

x[Closed

(b, ) x > b bx(

Open

( –, a] x a ax]

Closed

( –, a) x < a ax)

Open

Interval InequalityNotation Notation Line Graph Type

Interval Notation

A-8-123(b)

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1. If a < b and b < c, then a < c. Transitive Property

2. If a < b, then a + c < b + c. Addition Property

3. If a < b, then a – c < b – c. Subtraction Property

4. If a < b and c is positive, then ca < cb .

5. If a < b and c is negative, then ca > cb .

Multiplication Property(Note difference between4 and 5.)

6. If a < b and c is positive, then ac <

bc .

7. If a < b and c is negative, then ac >

bc .

Division Property(Note difference between6 and 7.)

For a, b, and c any real numbers:

Inequality Properties

A-8-124

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Any proper fraction P(x)/D(x) reduced to lowest terms can be decomposed in the sum of partial fractions as follows:

1. If D(x) has a nonrepeating linear factor of the form ax + b, then the partial fraction decomposition of P(x)/D(x) contains a term of the form

A a constant

2. If D(x) has a k-repeating linear factor of the form (ax + b)k, then the partial fraction decomposition of P(x)/D(x) contains k terms of the form

3. If D(x) has a nonrepeating quadratic factor of the form ax2 + bx + c, which is prime relative to the real numbers, then the partial fraction decomposition of P(x)/D(x) contains a term of the form

4. If D(x) has a k-repeating quadratic factor of the form (ax2 + bx + c)k, where ax2 + bx + c is prime relative to the real numbers, then the partial fraction decomposition of P(x)/D(x) contains k terms of the form

Aax + b

A1ax + b +

A2

(ax + b)2 + … + Ak

(ax + b)k A1 , A2 , …, Ak constants

Ax + B

ax2 + bx + c A, B constants

A1x + B1

ax2 + bx + c +

A2x + B2(ax2 +bx + c)2

+ … + Akx + Bk

(ax2 + bx + c)k

A1 , …, Ak , B1 , …, Bk constants

Partial Fraction Decomposition

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Significant Digits

If a number x is written in scientific notation as

x = a 10n 1 a < 10 , n an integer

then the number of significant digits in x is the number of digits in a.

The number of significant digits in a number with no decimal point if found by counting the digits from left to right, starting with the first digit and ending with the last nonzero digit.

The number of significant digits in a number containing a decimal point is found by counting the digits from left to right, starting with the first nonzero digit and ending with the last digit.  

Rounding Calculated Values

The result of a calculation is rounded to the same number of significant digits as the number used in the calculation that has the least number of significant digits.

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