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Appendix A

Basic Algebra

Review

Appendix A Review

Important Terms, Symbols, Concepts

 _{A-1 Real Numbers}

• Real Number - Any number that has a decimal representation

• Natural Numbers N - Counting numbers

• Integers Z - Natural numbers, their negatives, and 0 • Rational Numbers Q - can be represented by a/b,

repeating or terminating decimals

• Irrational Numbers I - represented by nonrepeating and nonterminating decimals

Appendix A Review

 A-1 Real Numbers (continued)

• Addition Properties

o Associative (a + b) + c = a + (b + c) o Commutative a + b = b + a

o Identity is 0

o Inverse of a is –a

• Multiplication Properties

o Associative (ab)c = a(bc) o Commutative ab = ba

o Identity is 1

Appendix A Review

 _{A-1 Real Numbers (continued)}

• Distributive Properties

5(3 + 4) = 5 • 3 + 5 • 4 = 15 + 20 = 35

9(m + n) = 9m + 9n

Appendix A Review

 _{A-1 Real Numbers (continued)}

• Subtraction and Division Properties

For all real numbers a and b,

Subtraction: a – b = a + (–b)

Appendix A Review

 _{A-1 Real Numbers (continued)}

• For all real numbers a and b,

1. − −( ) =a a

2. ( )−a b = −( )ab

3. ( )−a ( ) =−b ab

4. ( )−1 a= −a

5. −a

b = − a b =

a

−b, b ≠0

6. −a

−b = − −a

b = − a

−b =

a

Appendix A Review

 _{A-1 Real Numbers (continued)}

• For all real numbers a and b,

1. a ⋅0 = 0

Appendix A Review

 _{A-1 Real Numbers (continued)}

The quotient a ÷ b (b  0) written as a/b is called a fraction. The quantity a is called the numerator, and the quantity b is called the denominator.

For all real numbers a, b, c, d, and k, (division by 0 excluded)

1. a

b = c

d if and only if ad = bc

2. ka

kb = a b

5. a

b + c b = a+ c b 6. a b − c b =

a−c

b 7.

a b +

c d =

ad + bc bd

3. a

Appendix A Review

 _{A-2 Operations on Polynomials}

• Natural Number Exponent

First Property of Exponents

For any natural numbers m and n, and any real number b, bn _{= b}

⋅b⋅...⋅b n factors of b

bm

Appendix A Review

 _{A-2 Operations on Polynomials}

• Algebraic Expressions

• Polynomial in one variable x: axn

• Polynomial in two variables x and y: axmyn • Degree of the term

• Degree of the term with two or more variables • Degree of the polynomial

• Polynomial of degree 0

Appendix A Review

 _{A-2 Operations on Polynomials}

• Distributive Properties

• Addition and Subtraction remove parentheses and combine like terms

• Multiplication involves distributive property - multiply each term of one by each term of other

1. a b( + c) = (b+ c)a= ab+ ac 2. a b( −c) = (b−c)a= ab−ac

Appendix A Review

 _{A-2 Operations on Polynomials}

Note that in simplifying, we usually remove grouping symbols starting from the inside. That is, we remove

parentheses ( ) first, then brackets [ ], and finally braces { }, if present.

Order of Operations

Multiplication and division precede addition and subtraction, and taking powers precedes multiplication and division._{3x −5 −3 x−x(3−x)}

⎡⎣ ⎤⎦

{ } = 3x−{5 −3⎡⎣x−3x+ x2 ⎤⎦}

= 3x−{5 −3x+ 9x−3x2}

Appendix A Review

 _{A-3 Factoring Polynomials}

• Common Factors

• Factoring by Grouping

A. 3x3_y

−6x2y2 −3xy3 = (3xy)x2 −(3xy)2xy−(3xy)y2

= 3xy x2

−2xy−y2

( )

A. 3x2

−3x−x+1 = 3x( 2 −3x) −(x−1)

= 3x(x−1)−(x−1)

Appendix A Review

 A-3 Factoring Polynomials ac Test for Factorability

If in polynomials of the form

the product ac has two integer factors p and q whose sum is the coefficient b of the middle term; that is, if integers p and q exist so that

then the polynomials have first-degree factors with integer coefficients. If no integers p and q exist that satisfy equations (2), then the polynomials in equations (1) will not have first-degree factors with integer coefficients.

ax2 _{+ bx+ c or ax}2 _{+ bxy+ cy}2 ₍₁₎

Appendix A Review

 A-3 Factoring Polynomials

Special Factoring Formulas Perfect square:

Perfect square:

Difference of squares: Difference of cubes: Sum of cubes:

u2 _{+ 2uv+ v}2 _{= (u+ v)}2

u2

−2uv+ v2 = (u−v)2

u2

−v2 = (u−v)(u+ v)

u3

−v3 = (u−v)(u2 + uv+ v2)

u3 _{+ v}3 _{= (u+ v) u}2

−uv+ v2

Appendix A Review

 A-3 Factoring Polynomials

Step 1

Take out any factors common to all terms.

Step 2

Use any of the special factoring formulas.

Step 3

Apply the

ac

test to any remaining

second-degree polynomial factors.

Note:

It may be necessary to perform some of these

steps more than once. Furthermore, the order of

Appendix A Review

 _{A-4 Operations on Rational Expressions}

AGREEMENT Variable Restriction

Even though not always explicitly stated, we always

assume that variables are restricted so that division by 0 is excluded.

Fundamental Property of Fractions

If a, b, and k are real numbers with b, k  0, then

ka kb =

Appendix A Review

 _{A-4 Operations on Rational Expressions}

Multiplication and Division

If a, b, c, and d are real numbers, then

1. a

b ⋅ c d =

ac

bd , b, d ≠0 2. a

b ÷ c d =

a b ⋅

d

Appendix A Review

 _{A-4 Operations on Rational Expressions}

Addition and Subtraction

For a, b, and c real numbers,

1. a

b + c b =

a+ c

b , b ≠0 2. a

b − c b =

a−c

Appendix A Review

 A-4 Operations on Rational Expressions

Least Common Denominator

The least common denominator (LCD) of two or more rational expressions is found as follows:

1. Factor each denominator completely, including integer factors.

2. Identify each different factor from all the denominators. 3. Form a product using each different factor to the highest

Appendix A Review

 A-4 Operations on Rational Expressions

Compound Fractions

A fractional expression with fractions in its numerator, denominator, or both is called a compound fraction. It is often necessary to represent a compound fraction as a

simple fraction–that is (in all cases we will consider), as the quotient of two polynomials.

We will use the two different methods. Use division of rational forms.

Appendix A Review

 _{A-5 Integer Exponents and Scientific Notation}

For n an integer and a a real number: 1. For n a positive integer,

2. For n = 0,

3. For n a negative integer,

an _{= a}

⋅a⋅...⋅a n factors of a

a0 _{= 1 a}

≠0

00 _{is not defined.}

an ₌ 1

a−n a≠0

a−n = 1

an and a

n ₌ 1

Appendix A Review

 _{A-5 Integer Exponents and Scientific Notation}

For n and m integers and a and b real numbers,

1. am_an _{= a}m+n

5. am

an = a

m−n = 1

an−m a≠0

2. _{( )}an m _{= a}m n

3. ( )ab m = am_bm

4. a

b

⎛ ⎝⎜

⎞ ⎠⎟

m

Appendix A Review

 _{A-5 Integer Exponents and Scientific Notation}

A number in the following form is said to be in scientific notation: a × 10n ₁

≤a< 10, a in decim al form , n an integer

Write in standard decimal form. Write in scientific notation.

7,320,000 = 7.32 × 106 _{0.00000054 = 5.4 × 10}−7

Appendix A Review

 A-6 Rational Exponents and Radicals

For any natural number

n

,

r

is the

n

th root of

b

if

r

= b

Real

n

th root of

b

b1 n _or n _b

b

n

n

th root radical

Appendix A Review

 A-6 Rational Exponents and Radicals

If m and n are natural numbers without common prime factors, b is a real number, and b is nonnegative when n is even, then

bm n ₌ b

1 n

( )m = (n b)m

bm

( )1 n = n bm

⎧ ⎨ ⎪

⎩ ⎪

and

b−m n = 1

Appendix A Review

 A-6 Rational Exponents and Radicals

If c, n, and m are natural numbers greater than or equal to 2, and if x and y are positive real numbers, then

1. n xn _{= x}

2. n xy = n x n y

3. x

y

n = x

n

x

Appendix A Review

 A-7 Quadratic Equations

• A quadratic equation in one variable is any equation that can be written in the form

ax2 _{+ bx+ c= 0 a}

≠0

where x is a variable and a, b, and c are constants.

5x2

Appendix A Review

 A-7 Quadratic Equations

•

Square Root Property

•

Recall:

ab

= 0 if and only if

a

= 0 or

b

= 0

Use when solving a polynomial equation that can

be factored

Appendix A Review

 A-7 Quadratic Equations

• If ax2 + bx + c = 0, a  0, then

x = −b± b2 −4ac 2a

b2 – 4ac ax2 + bx + c = 0

Positive Two real solutions Zero One real solution Negative No real solutions

Appendix A Review

 A-7 Quadratic Equations

Factor Theorem

If r₁ and r₂ are solutions to the second-degree ax2 + bx + c = 0, then

ax2 _{+ bx+ c= a x}

−r₁

Appendix A Review

 A-7 Quadratic Equations

Supply-and-demand analysis is a very important part of

business and economics. Producers are willing to supply more of an item as the price of an item increases and less of an item as the price decreases. Buyers are willing to buy less of an item as the price increases, and more of an item as the price

decreases. We have a dynamic situation where the price, supply, and demand fluctuate until a price is reached at which the