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Intermediate Algebra 5th Edition
by Elayn Martin-Gay
SUMMARY

Chapter 5: Exponents, Polynomials, and Polynomial Functions

Section 5.1: Exponents and Scientific Notation

Textbook Objectives
  1. Use the product rule for exponents.
  2. Evaluate expressions raised to the zero power.
  3. Use the quotient rule for exponents.
  4. Evaluate expressions raised to the nth power.
  5. Convert between scientific notation and standard notation.
Additional Goals
  1. Be able to identify an exponential expression.
  2. Be able to identify the base in an exponential expression.
  3. Be able to identify the exponent in an exponential expression.
  4. Know the meaning of a zero exponent.
  5. Know the meaning of a negative exponent.
  6. Know the definition of scientific notation.

Exponents: Introduction: Scientific Notation

Exponents may be any real number. The previous formulas (and more in the next section) show how to manipulate exponential expressions regardless of the type of number in the exponent. These formulas also give meaning to negative exponents. The same rules formulas) for manipulating exponentials with natural number exponents apply to exponentials with integer exponents, non-integer rational exponents, and irrational exponents.

Changing from scientific notation to standard notation:

A number in scientific notation like xx.xxx X 10n can be changed to standard notation simply by moving the decimal point in xx.xxx n places and omitting the power of 10. If n is positive the decimal is moved to the right and if n is negative the decimal is moved to the left.

Changing from standard notation to scientific notation:

A number in standard notation like xxx.xxx can be changed to scientific notation simply by moving the decimal point in xxx.xxx so that there is one significant digit to the left of the decimal. Count how many places the decimal was moved. If the decimal was moved n places to the right, multiply by 10-n. If the decimal was moved n places to the left, multiply by 10n.

About Exponents and Scientific Notation

Remember that on the index page for Intermediate Algebra there are links to worked out examples.
To see a few worked out exercises involving scientific notation go HERE.
If you want worked out examples of using the laws of exponents to simplify (or simply change) exponential expressions, try these HERE and HERE and HERE and HERE.

Minimal List of Exercises Page 262.

If you understand the previous material you should be able to answer the following questions.
Each of the individually listed exercises should be done. In each of the lists (like 5 - 12) work as many as needed to insure that you understand the concept being illustrated or utilized. Most of these questions or question types are included in the MyMathLab homework requirement. If a particular concept is difficult for you, you should study the related text material and then try to answer some additional questions from the list provided by the department. In each case you are expected to make an honest adult evaluation of your understanding of the concept. Your ability to answer these questions is one tool to help you make that evaluation.

Vocabulary and Concept Check: 1 - 12
Section 5.1 Exercise Set: 1 - 10, 11, 13, 15, 16, 17, 18, 19 - 28, 29 - 48, 49 - 82, 83, 85, 87, 91, 93 - 98, 111 - 120.

 

Section 5.2: More Work with Exponents and Scientific Notation

Textbook Objectives

  1. Use the power rules for exponents.
  2. Use exponent rules and definitions to simplify exponential expressions.
  3. Compute, using scientific notation.

Additional Goals

  1. Know the rule for an exponential of an exponential.
  2. Know the rule for an exponential of a product.
  3. Know the rule for an exponential of a quotient.

Exponents: Powers, Products, and Quotients

About Exponents: Powers, Products, and Quotients

Minimal List of Exercises Page 269.

If you understand the previous material you should be able to answer the following questions.
Each of the individually listed exercises should be done. In each of the lists (like 5 - 12) work as many as needed to insure that you understand the concept being illustrated or utilized. Most of these questions or question types are included in the MyMathLab homework requirement. If a particular concept is difficult for you, you should study the related text material and then try to answer some additional questions from the list provided by the department. In each case you are expected to make an honest adult evaluation of your understanding of the concept. Your ability to answer these questions is one tool to help you make that evaluation.

Vocabulary and Concept Check: 1 - 10
Section 5.2 Exercise Set: 1 - 15, 16 - 27, 57 - 68, 69, 71, 73, 75, 79, 83. Do not use a calculator for the scientific notation problems.

 

Section 5.3: Polynomials and Polynomial Functions Equations

Textbook Objectives

  1. Identify
    • term
    • constant
    • polynomial
    • monomial
    • binomial
    • trinomial
    • degree of a term
    • degree of a polynomial
  2. Define polynomial functions equations.
  3. Review combining adding like terms.
  4. Add polynomials.
  5. Subtract polynomials.
  6. Recognize the graph of a polynomial functions equation from the degree of the polynomial.

Additional Goals

  1. Know the definitions of:
    • term
    • coefficient and numerical coefficient
    • constant and constant term
    • polynomial, polynomial in one variable, and polynomial in x
    • monomial, binomial, and trinomial
    • degree of a term and degree of a polynomial
    • equality of polynomials
    • polynomial equation
    • like terms
    • opposite of a polynomial
  2. Be able to add (subtract) like terms.
  3. Be able to add (subtract) polynomials.

 

Introduction to Polynomials

About Introduction to Polynomials

Minimal List of Exercises Page 279.

If you understand the previous material you should be able to answer the following questions.
Each of the individually listed exercises should be done. In each of the lists (like 5 - 12) work as many as needed to insure that you understand the concept being illustrated or utilized. Most of these questions or question types are included in the MyMathLab homework requirement. If a particular concept is difficult for you, you should study the related text material and then try to answer some additional questions from the list provided by the department. In each case you are expected to make an honest adult evaluation of your understanding of the concept. Your ability to answer these questions is one tool to help you make that evaluation.

Vocabulary and Readiness Check: 1 - 14
Section 5.3 Exercise Set: 1 - 10, 11 - 18, 29 - 36, 37, 38, 39, 43, 44, 45, 49, 55, 61, 63, 71, 97, 98, 109, 110.

 

Section 5.4: Multiplying Polynomials

Textbook Objectives

  1. Multiply two polynomials.
  2. Multiply binomials.
  3. Square binomials.
  4. Multiply the sum and difference of two terms.
  5. Multiply three or more polynomials.
  6. Evaluate polynomial functions.

Additional Goals

  1. Recognize the square of a sum.
  2. Recognize the square of a difference.
  3. Recognize the difference of two squares.
  4. Divide one polynomial by another polynomial.

 

Multiplication of Polynomials

About Multiplication of Polynomials

HERE are some examples of Long Division of Polynomials.

Minimal List of Exercises Page 289.

If you understand the previous material you should be able to answer the following questions.
Each of the individually listed exercises should be done. In each of the lists (like 5 - 12) work as many as needed to insure that you understand the concept being illustrated or utilized. Most of these questions or question types are included in the MyMathLab homework requirement. If a particular concept is difficult for you, you should study the related text material and then try to answer some additional questions from the list provided by the department. In each case you are expected to make an honest adult evaluation of your understanding of the concept. Your ability to answer these questions is one tool to help you make that evaluation.

Vocabulary and Readiness Check: 1 - 6
Section 5.4 Exercise Set: 1 - 12, 17 - 26, 27 - 34, 43, 45, 49, 51, 53, 54, 59, 61, 62, 63, 69, 71, 73, 75, 76, 77, 103, 104, 105.

Division of polynomials is not included in this section of the textbook, so there are some exercises listed HERE. Solutions are on the same page.

 

Section 5.5: The Greatest Common Factor and Factoring by Grouping

Textbook Objectives

  1. Identify the GCF.
  2. Factor out the GCF of a polynomial's terms.
  3. Factor polynomials by grouping.

Additional Goals

  1. Know the definition of factor.
    • As a noun.
    • As a verb.

 

GCF and Factoring by Grouping

When used as a noun the word factor refers to the individual parts of a product. When used as a verb (to factor) it means to write something as a product.

For example, in many arithmetic situations recognizing the number 136 as the product of 8 and 17 is a benefit. In such situations it is customary to claim that to factor 136 as (8)(17) is a simplification. In the same way factorizations of polynomials is usually a simplification. There are several methods for factoring polynomials. No single one of these methods is sufficient for all occassions, but in combination they can yield a factored (simplified) form of seemingly complex polynomials.

The very first step when factoring any polynomial is to identify the GCF of the terms of the polynomial and to write the polynomial as a product with the GCF as one of the factors. The Distributive Property is used to verify that the factorization is correct. A common phrase used to refer to this procedure is "to factor out the GCF".

About GCF and Factoring by Grouping

Minimal List of Exercises Page 295.

If you understand the previous material you should be able to answer the following questions.
Each of the individually listed exercises should be done. In each of the lists (like 5 - 12) work as many as needed to insure that you understand the concept being illustrated or utilized. Most of these questions or question types are included in the MyMathLab homework requirement. If a particular concept is difficult for you, you should study the related text material and then try to answer some additional questions from the list provided by the department. In each case you are expected to make an honest adult evaluation of your understanding of the concept. Your ability to answer these questions is one tool to help you make that evaluation.

Vocabulary and Readiness Check: 1 - 16
Section 5.5 Exercise Set: 1, 3, 5, 7, 9 - 23 (As many as you have time for), 25, 27, 29, 30, 31, 32.

 

Section 5.6: Factoring Trinomials

Textbook Objectives

  1. Factor trinomials of the form x2 + bx + c.
  2. Factor trinomials of the form ax2 + bx + c.
    • Trial and Error Method
    • Grouping method
  3. Factor by substitution.

 

Factoring Trinomials

About Factoring Trinomials

Factoring is a SKILL that can be quite helpful. This is one of the few topics in mathematics where practice is really beneficial. Practice by working as many exercises as time permits.

Minimal List of Exercises Page 303.

If you understand the previous material you should be able to answer the following questions.
Each of the individually listed exercises should be done. In each of the lists (like 5 - 12) work as many as needed to insure that you understand the concept being illustrated or utilized. Most of these questions or question types are included in the MyMathLab homework requirement. If a particular concept is difficult for you, you should study the related text material and then try to answer some additional questions from the list provided by the department. In each case you are expected to make an honest adult evaluation of your understanding of the concept. Your ability to answer these questions is one tool to help you make that evaluation.

Vocabulary and Readiness Check: 1 - 4
Section 5.6 Exercise Set: 1 - 14 (As many as you have time for.), 15 - 34 (As many as you have time for.), 35 - 42 (As many as you have time for.). Use Method 1 and substitution methods. Don't bother with method 2.

 

Section 5.7: Factoring by Special Products

Textbook Objectives

  1. Factor a perfect square trinomial.
  2. Factor the difference of two squares.
  3. Factor the sum or difference of two cubes.

Additional Goals

  1. Recognize certain sums of squares as prime polynomials.

 

Factoring by Special Products

About Factoring by Special Products

Minimal List of Exercises Page 310.

If you understand the previous material you should be able to answer the following questions.
Each of the individually listed exercises should be done. In each of the lists (like 5 - 12) work as many as needed to insure that you understand the concept being illustrated or utilized. Most of these questions or question types are included in the MyMathLab homework requirement. If a particular concept is difficult for you, you should study the related text material and then try to answer some additional questions from the list provided by the department. In each case you are expected to make an honest adult evaluation of your understanding of the concept. Your ability to answer these questions is one tool to help you make that evaluation.

Vocabulary and Readiness Check: 1 - 10
Section 5.7 Exercise Set: 1 - 8, 9 - 16, 17 - 28, 29, 31, 33.

 

Section 5.8: Solving Equations by Factoring and Problem Solving

Textbook Objectives

  1. Solve polynomial equations by factoring.
  2. Solve problems that can be modeled by polynomial equations.
  3. Find the x-intercepts of a polynomial function equation in two variables.

Additional Goals

  1. Know the zero factor property.
  2. Know the definition of quadratic equation in one variable.
  3. Find the y-intercepts of polynomial equations in two variables.
  4. Know the definition of a right triangle.
  5. Identify the legs and hypotenuse of a right triangle.
  6. Know the Pythagorean Theorem.
  7. Use the Pythagorean Theorem to solve problems.

 

Solving Equations by Factoring

About Solving Equations by Factoring

Minimal List of Exercises Page 324.

If you understand the previous material you should be able to answer the following questions.
Each of the individually listed exercises should be done. In each of the lists (like 5 - 12) work as many as needed to insure that you understand the concept being illustrated or utilized. Most of these questions or question types are included in the MyMathLab homework requirement. If a particular concept is difficult for you, you should study the related text material and then try to answer some additional questions from the list provided by the department. In each case you are expected to make an honest adult evaluation of your understanding of the concept. Your ability to answer these questions is one tool to help you make that evaluation.

Vocabulary and Readiness Check: 1 - 16
Section 5.8 Exercise Set: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 15, 17, 21, 22, 23, 24, 73, 74, 77.