Chapter 0 Review of Algebra

INTRODUCTORY MATHEMATICAL ANALYSIS 0. Review of Algebra Applications and More Algebra Functions and Graphs Lines, Parabolas, and Systems Exponential and Logarithmic Functions Mathematics of Finance Matrix Algebra Linear Programming Introduction to Probability and Statistics

INTRODUCTORY MATHEMATICAL ANALYSIS Additional Topics in Probability Limits and Continuity Differentiation Additional Differentiation Topics Curve Sketching Integration Methods and Applications of Integration Continuous Random Variables Multivariable Calculus

Chapter 0: Review of Algebra Chapter Objectives To be familiar with sets, real numbers, real-number line. To relate properties of real numbers in terms of their operations. To review the procedure of rationalizing the denominator. To perform operations of algebraic expressions. To state basic rules for factoring. To rationalize the denominator of a fraction. To solve linear equations. To solve quadratic equations.

Chapter Outline 0.1) Sets of Real Numbers Chapter 0: Review of Algebra Chapter Outline 0.1) Sets of Real Numbers Some Properties of Real Numbers Exponents and Radicals Operations with Algebraic Expressions Factoring Fractions Equations, in Particular Linear Equations Quadratic Equations 0.2) 0.3) 0.4) 0.5) 0.6) 0.7) 0.8)

0.1 Sets of Real Numbers A set is a collection of objects. Chapter 0: Review of Algebra 0.1 Sets of Real Numbers A set is a collection of objects. An object in a set is called an element of that set. Different type of integers: The real-number line is shown as

0.2 Some Properties of Real Numbers Chapter 0: Review of Algebra 0.2 Some Properties of Real Numbers Important properties of real numbers The Transitive Property of Equality The Closure Properties of Addition and Multiplication The Commutative Properties of Addition and Multiplication

The Commutative Properties of Addition and Multiplication Chapter 0: Review of Algebra 0.2 Some Properties of Real Numbers The Commutative Properties of Addition and Multiplication The Identity Properties The Inverse Properties The Distributive Properties

Example 3 – Applying Properties of Real Numbers Solution: Chapter 0: Review of Algebra 0.2 Some Properties of Real Numbers Example 1 – Applying Properties of Real Numbers Example 3 – Applying Properties of Real Numbers Solution: Show that Solution:

Show that Solution: Chapter 0: Review of Algebra 0.2 Some Properties of Real Numbers Example 3 – Applying Properties of Real Numbers Show that Solution:

0.3 Exponents and Radicals Chapter 0: Review of Algebra 0.3 Exponents and Radicals Properties: exponent base

Example 1 – Exponents Chapter 0: Review of Algebra 0.3 Exponents and Radicals Example 1 – Exponents

The symbol is called a radical. Chapter 0: Review of Algebra 0.3 Exponents and Radicals The symbol is called a radical. n is the index, x is the radicand, and is the radical sign.

Eliminate negative exponents in and simplify. Solution: Chapter 0: Review of Algebra 0.3 Exponents and Radicals Example 3 – Rationalizing Denominators Solution: Example 5 – Exponents Eliminate negative exponents in and simplify. Solution:

b. Simplify by using the distributive law. Solution: Chapter 0: Review of Algebra 0.3 Exponents and Radicals Example 5 – Exponents b. Simplify by using the distributive law. Solution: Eliminate negative exponents in

d. Eliminate negative exponents in Solution: Chapter 0: Review of Algebra 0.3 Exponents and Radicals Example 5 – Exponents d. Eliminate negative exponents in Solution: e. Apply the distributive law to

a. Simplify Solution: Simplify Example 7 – Radicals Chapter 0: Review of Algebra 0.3 Exponents and Radicals Example 7 – Radicals a. Simplify Solution: Simplify

If x is any real number, simplify Chapter 0: Review of Algebra 0.3 Exponents and Radicals Example 7 – Radicals c. Simplify Solution: If x is any real number, simplify Thus, and

0.4 Operations with Algebraic Expressions Chapter 0: Review of Algebra 0.4 Operations with Algebraic Expressions If symbols are combined by any or all of the operations, the resulting expression is called an algebraic expression. A polynomial in x is an algebraic expression of the form: where n = non-negative integer cn = constants

is an algebraic expression in the variable x. Chapter 0: Review of Algebra 0.4 Operations with Algebraic Expressions Example 1 – Algebraic Expressions is an algebraic expression in the variable x. is an algebraic expression in the variable y. is an algebraic expression in the variables x and y.

Simplify Solution: Example 3 – Subtracting Algebraic Expressions Chapter 0: Review of Algebra 0.4 Operations with Algebraic Expressions Example 3 – Subtracting Algebraic Expressions Simplify Solution:

A list of products may be obtained from the distributive property: Chapter 0: Review of Algebra 0.4 Operations with Algebraic Expressions A list of products may be obtained from the distributive property:

By Rule 2, By Rule 3, Example 5 – Special Products Chapter 0: Review of Algebra 0.4 Operations with Algebraic Expressions Example 5 – Special Products By Rule 2, By Rule 3,

c. By Rule 5, By Rule 6, By Rule 7, Chapter 0: Review of Algebra 0.4 Operations with Algebraic Expressions Example 5 – Special Products c. By Rule 5, By Rule 6, By Rule 7,

Example 7 – Dividing a Multinomial by a Monomial Chapter 0: Review of Algebra 0.4 Operations with Algebraic Expressions Example 7 – Dividing a Multinomial by a Monomial

Chapter 0: Review of Algebra 0.5 Factoring If two or more expressions are multiplied together, the expressions are called the factors of the product.

Solution: b. Factor completely. Example 1 – Common Factors Chapter 0: Review of Algebra 0.5 Factoring Example 1 – Common Factors a. Factor completely. Solution: b. Factor completely.

Chapter 0: Review of Algebra 0.5 Factoring Example 3 – Factoring

0.6 Fractions Simplifying Fractions Chapter 0: Review of Algebra 0.6 Fractions Simplifying Fractions Allows us to multiply/divide the numerator and denominator by the same nonzero quantity. Multiplication and Division of Fractions The rule for multiplying and dividing is

Rationalizing the Denominator Chapter 0: Review of Algebra 0.6 Fractions Rationalizing the Denominator For a denominator with square roots, it may be rationalized by multiplying an expression that makes the denominator a difference of two squares. Addition and Subtraction of Fractions If we add two fractions having the same denominator, we get a fraction whose denominator is the common denominator.

Solution: b. Simplify Example 1 – Simplifying Fractions a. Simplify Chapter 0: Review of Algebra 0.6 Fractions Example 1 – Simplifying Fractions a. Simplify Solution: b. Simplify

Example 3 – Dividing Fractions Chapter 0: Review of Algebra 0.6 Fractions Example 3 – Dividing Fractions

Example 5 – Adding and Subtracting Fractions Chapter 0: Review of Algebra 0.6 Fractions Example 5 – Adding and Subtracting Fractions

Example 7 – Subtracting Fractions Chapter 0: Review of Algebra 0.6 Fractions Example 7 – Subtracting Fractions

0.7 Equations, in Particular Linear Equations Chapter 0: Review of Algebra 0.7 Equations, in Particular Linear Equations Equations An equation is a statement that two expressions are equal. The two expressions that make up an equation are called its sides. They are separated by the equality sign, =.

Chapter 0: Review of Algebra 0.7 Equations, in Particular Linear Equations Example 1 – Examples of Equations A variable (e.g. x, y) is a symbol that can be replaced by any one of a set of different numbers.

There are three operations that guarantee equivalence: Chapter 0: Review of Algebra 0.7 Equations, in Particular Linear Equations Equivalent Equations Two equations are said to be equivalent if they have exactly the same solutions. There are three operations that guarantee equivalence: Adding/subtracting the same polynomial to/from both sides of an equation. Multiplying/dividing both sides of an equation by the same nonzero constant. Replacing either side of an equation by an equal expression.

Operations That May Not Produce Equivalent Equations Chapter 0: Review of Algebra 0.7 Equations, in Particular Linear Equations Operations That May Not Produce Equivalent Equations Multiplying both sides of an equation by an expression involving the variable. Dividing both sides of an equation by an expression involving the variable. Raising both sides of an equation to equal powers.

A linear equation in the variable x can be written in the form Chapter 0: Review of Algebra 0.7 Equations, in Particular Linear Equations Linear Equations A linear equation in the variable x can be written in the form where a and b are constants and . A linear equation is also called a first-degree equation or an equation of degree one.

Solve Solution: Example 3 – Solving a Linear Equation Chapter 0: Review of Algebra 0.7 Equations, in Particular Linear Equations Example 3 – Solving a Linear Equation Solve Solution:

Solve Solution: Example 5 – Solving a Linear Equations Chapter 0: Review of Algebra 0.7 Equations, in Particular Linear Equations Example 5 – Solving a Linear Equations Solve Solution:

The letters are called literal constants. Chapter 0: Review of Algebra 0.7 Equations, in Particular Linear Equations Literal Equations Equations where constants are not specified, but are represented as a, b, c, d, etc. are called literal equations. The letters are called literal constants.

Solve for x. Solution: Example 7 – Solving a Literal Equation Chapter 0: Review of Algebra 0.7 Equations, in Particular Linear Equations Example 7 – Solving a Literal Equation Solve for x. Solution:

Chapter 0: Review of Algebra 0 Chapter 0: Review of Algebra 0.7 Equations, in Particular Linear Equations Example 9 – Solving a Fractional Equation Solve Solution: Fractional Equations A fractional equation is an equation in which an unknown is in a denominator.

If express u in terms of the remaining letters; that is, solve for u. Chapter 0: Review of Algebra 0.7 Equations, in Particular Linear Equations Example 11 – Literal Equation If express u in terms of the remaining letters; that is, solve for u. Solution:

A radical equation is one in which an unknown occurs in a radicand. Chapter 0: Review of Algebra 0.7 Equations, in Particular Linear Equations Radical Equations A radical equation is one in which an unknown occurs in a radicand. Example 13 – Solving a Radical Equation Solve Solution:

where a, b, and c are constants and Chapter 0: Review of Algebra 0.8 Quadratic Equations A quadratic equation in the variable x is an equation that can be written in the form where a, b, and c are constants and A quadratic equation is also called a second-degree equation or an equation of degree two.

Solution: Factor the left side factor: Chapter 0: Review of Algebra 0.8 Quadratic Equations Example 1 – Solving a Quadratic Equation by Factoring a. Solve Solution: Factor the left side factor: Whenever the product of two or more quantities is zero, at least one of the quantities must be zero.

Chapter 0: Review of Algebra 0.8 Quadratic Equations Example 1 – Solving a Quadratic Equation by Factoring b. Solve Solution:

a. Solve Solution: b. Solve Chapter 0: Review of Algebra 0.8 Quadratic Equations Example 3 – Solving a Higher-Degree Equation by Factoring a. Solve Solution: b. Solve

Solve Solution: Example 5 – Solution by Factoring Chapter 0: Review of Algebra 0.8 Quadratic Equations Example 5 – Solution by Factoring Solve Solution:

The roots of the quadratic equation Chapter 0: Review of Algebra 0.8 Quadratic Equations Quadratic Formula The roots of the quadratic equation can be given as

Solve by the quadratic formula. Chapter 0: Review of Algebra 0.8 Quadratic Equations Example 7 – A Quadratic Equation with One Real Root Solve by the quadratic formula. Solution: Here a = 9, b = 6√2, and c = 2. The roots are

Quadratic-Form Equation Chapter 0: Review of Algebra 0.8 Quadratic Equations Quadratic-Form Equation When a non-quadratic equation can be transformed into a quadratic equation by an appropriate substitution, the given equation is said to have quadratic-form.

This equation can be written as Substituting w =1/x3, we have Chapter 0: Review of Algebra 0.8 Quadratic Equations Example 9 – Solving a Quadratic-Form Equation Solve Solution: This equation can be written as Substituting w =1/x3, we have Thus, the roots are