1

Date: ____________________

Section P.2: Exponents and Radicals

Properties of Exponents:

���� � ���� � �� �

���� � ���� � � �� �

Example #1: Simplify.

a.) �3����4���� � b.) 2����� � c.) 3��4���� �

d.) ���� �� �

Example #2: Simplify.

a.) �� � b.) �

���� �

c.) ���� ������ � d.) ���

� �� �

2

Square Root: Principal nth Root:

√��

Example #3: Simplify.

a.) √49 =

b.) �√49 �

c.) "���#�

� =

d.) √�32$ =

e.) √�8� �

3

Properties of Radicals:

√��� �

" √��� � Example #4: Simplify.

a.) √8 • √2 =

b.) √5� �� �

c.) √��� �

4

Simplifying Radicals: An expression involving radicals is in simplest form when the following conditions are satisfied:

1.) All possible factors have been removed from the radical. 2.) All fractions have radical-free denominators (rationalizing the

denominator – not in this class!) 3.) The index of the radical is reduced.

Example #5: Simplifying Even Roots

a.) 484

b.) 75x 3

c.) (5x)44

5

Example #6: Simplifying Odd Roots

a.) 243

b.) 24a43

c.) −40x 63

Rational Exponents: Definition of Rational Exponents: If a is a real number and n is a positive integer such that the principal nth root of a exists, we define to be

a1

n

a1

n =

6

If m is a positive integer that has no common factor with n, then

and

Example #7: Changing from Radical to Exponential Form

a.) 3

b.) 3xy( )5

c.) 2x x 34

Example #8: Changing from Exponential to Radical Form

a.) x 2 + y 2( )3

2

am

n = a1

n

m

= amn( ) am

n = am( )1

n = amn

7

b.) 2y3

4 z1

4

c.) a−3

2

d.) x 0.2

Example #9: Simplifying with Rational Exponents

a.) 27( )2

6

b.) −32( )−4

5

8

c.) −5x

5

3

3x

−3

4

d.) a39

e.) 1253

f.) 2x −1( )4

3 2x −1( )−1

3

9

Example #10: Combining Radicals

a.) 2 48 − 3 27

b.) 16x3 − 54x 43

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Date: ____________________

Section P.3: Polynomials and Factoring

Polynomials: In standard form, a polynomial is written with descending powers of x. The highest exponent in the polynomial is the degree, and the number in front of that term is the leading coefficient. The number in the polynomial without a variable is called the constant term. Example #1: Writing Polynomials in Standard form.

4x2 − 5x7 − 2 + 3x Example #2: Sums and Differences of Polynomials

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

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Example #3: Multiplying Polynomials – The FOIL Method

(3x − 2)(5x + 7) Example #4: The Product of Two Trinomials

(x + y − 2)(x + y + 2) Example #5: Removing Common Factors

a.) 3x 3 + 9x 2

b.) (x − 2)(2x) + (x − 2)(3)

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Example #6: Removing a Common Factor First

3 −12x2

Example #7: Factoring the Difference of Two Squares

a.) (x + 2)2 − y 2

b.) 16x4 − 81

Example #8: Factoring Perfect Square Trinomials

a.) 16x 2 + 8x +1

b.) x2 −10x + 25

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Example #11: Factoring a Trinomial: Leading Coefficient Is 1

x 2 − 7x +12 Example #12: Factoring a Trinomial: Leading Coefficient Is Not 1

2x 2 + x −15 Example #13: Factoring by Grouping

x3 − 2x2 − 3x + 6

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Date: ____________________

Section P.4: Fractional Expressions Domain of an Algebraic Expression: The set of real numbers for which an algebraic expression is defined is the domain. Example #1: Finding the Domain of an Algebraic Expression

a.) The domain of the polynomial: 2x3 + 3x + 4 is…

b.) The domain of the radical expression x − 2 is…

c.) The domain of the expression

x + 2x − 3 is…

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Example #2: Reducing a Rational Expression

Write

x 2 + 4x −12

3x − 6 in reduced form.

Simplifying Rational Expressions: Example #3: Reducing Rational Expressions

a.)

x 3 − 4 x

x 2 + x − 2

b.)

12 + x − x 2

2x 2 − 9x + 4

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Operations with Rational Expressions: Example #4: Multiplying Rational Expressions

2x 2 + x − 6x 2 + 4 x − 5

•x 3 − 3x 2 + 2x

4 x 2 − 6x

Example #5: Dividing Rational Expressions

x 3 − 8x 2 − 4

÷x 2 + 2x + 4

x 3 + 8

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Example #6: Subtracting Rational Expressions

x

x − 3−

2

3x + 4

Example #7: Combining Rational Expressions: The LCD Method

3

x −1−

2

x+

x + 3

x 2 −1

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Compound Fractions: Example #8: Simplifying a Compound Fraction

2x

− 3

1 −1

x −1

Example #9: Simplifying an Expression with Negative Exponents

x(1 − 2x)−3

2 + (1 − 2x)−1

2

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Example #10: Simplifying a Compound Fraction

(4 − x 2)1

2 + x 2(4 − x 2)−1

2

4 − x 2

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Date: ____________________

Section P.5: Solving Equations

Linear Equations: Example #1: Solving a Linear Equation

Solve 3x − 6 = 0 Example #2: An Equation Involving Fractional Expressions

Solve

x

3+

3x

4= 2

Example #3: An Equation with an Extraneous Solution

Solve

1

x − 2=

3

x + 2−

6x

x 2 − 4

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Example #4: Solving Quadratic Equations by Factoring

a.) 2x 2 + 9x + 7 = 3

b.) 6x 2 − 3x = 0

Example #5: Extracting Square Roots

a.) 4 x 2 = 12

b.) x − 3( )2 = 7

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Example #6: The Quadratic Formula: Two Distinct Solutions

Use the Quadratic Formula to solve: x 2 + 3x = 9

Example #7: The Quadratic Formula: One Repeated Solution

Use the Quadratic Formula to solve: 8x 2 − 24 x +18 = 0

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Polynomial Equations of Higher Degree: Example #8: Solving a Polynomial Equation by Factoring

Solve 3x 4 = 48 x 2

Example #9: Solving a Polynomial Equation by Factoring

Solve x3 − 3x 2 − 3x + 9 = 0

Radical Equations: Example #10: Solving an Equation Involving a Rational Exponent

Solve 4 x 3 2 − 8 = 0

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Example #11: Solving an Equation Involving a Radical

Solve 2x + 7 − x = 2

Absolute Value Equations: Example #12: Solving an Equation Involving Absolute Value

Solve x 2 − 3x = −4 x + 6