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Elementary Algebra

Key Concepts

Elementary AlgebraKey Concepts
Table of contents
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Introduction to Whole Numbers
    3. 1.2 Use the Language of Algebra
    4. 1.3 Add and Subtract Integers
    5. 1.4 Multiply and Divide Integers
    6. 1.5 Visualize Fractions
    7. 1.6 Add and Subtract Fractions
    8. 1.7 Decimals
    9. 1.8 The Real Numbers
    10. 1.9 Properties of Real Numbers
    11. 1.10 Systems of Measurement
    12. Chapter Review
      1. Key Terms
      2. Key Concepts
    13. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations and Inequalities
    1. Introduction
    2. 2.1 Solve Equations Using the Subtraction and Addition Properties of Equality
    3. 2.2 Solve Equations using the Division and Multiplication Properties of Equality
    4. 2.3 Solve Equations with Variables and Constants on Both Sides
    5. 2.4 Use a General Strategy to Solve Linear Equations
    6. 2.5 Solve Equations with Fractions or Decimals
    7. 2.6 Solve a Formula for a Specific Variable
    8. 2.7 Solve Linear Inequalities
    9. Chapter Review
      1. Key Terms
      2. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Math Models
    1. Introduction
    2. 3.1 Use a Problem-Solving Strategy
    3. 3.2 Solve Percent Applications
    4. 3.3 Solve Mixture Applications
    5. 3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem
    6. 3.5 Solve Uniform Motion Applications
    7. 3.6 Solve Applications with Linear Inequalities
    8. Chapter Review
      1. Key Terms
      2. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Graphs
    1. Introduction
    2. 4.1 Use the Rectangular Coordinate System
    3. 4.2 Graph Linear Equations in Two Variables
    4. 4.3 Graph with Intercepts
    5. 4.4 Understand Slope of a Line
    6. 4.5 Use the Slope–Intercept Form of an Equation of a Line
    7. 4.6 Find the Equation of a Line
    8. 4.7 Graphs of Linear Inequalities
    9. Chapter Review
      1. Key Terms
      2. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Systems of Linear Equations
    1. Introduction
    2. 5.1 Solve Systems of Equations by Graphing
    3. 5.2 Solve Systems of Equations by Substitution
    4. 5.3 Solve Systems of Equations by Elimination
    5. 5.4 Solve Applications with Systems of Equations
    6. 5.5 Solve Mixture Applications with Systems of Equations
    7. 5.6 Graphing Systems of Linear Inequalities
    8. Chapter Review
      1. Key Terms
      2. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Polynomials
    1. Introduction
    2. 6.1 Add and Subtract Polynomials
    3. 6.2 Use Multiplication Properties of Exponents
    4. 6.3 Multiply Polynomials
    5. 6.4 Special Products
    6. 6.5 Divide Monomials
    7. 6.6 Divide Polynomials
    8. 6.7 Integer Exponents and Scientific Notation
    9. Chapter Review
      1. Key Terms
      2. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Factoring
    1. Introduction
    2. 7.1 Greatest Common Factor and Factor by Grouping
    3. 7.2 Factor Quadratic Trinomials with Leading Coefficient 1
    4. 7.3 Factor Quadratic Trinomials with Leading Coefficient Other than 1
    5. 7.4 Factor Special Products
    6. 7.5 General Strategy for Factoring Polynomials
    7. 7.6 Quadratic Equations
    8. Chapter Review
      1. Key Terms
      2. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Rational Expressions and Equations
    1. Introduction
    2. 8.1 Simplify Rational Expressions
    3. 8.2 Multiply and Divide Rational Expressions
    4. 8.3 Add and Subtract Rational Expressions with a Common Denominator
    5. 8.4 Add and Subtract Rational Expressions with Unlike Denominators
    6. 8.5 Simplify Complex Rational Expressions
    7. 8.6 Solve Rational Equations
    8. 8.7 Solve Proportion and Similar Figure Applications
    9. 8.8 Solve Uniform Motion and Work Applications
    10. 8.9 Use Direct and Inverse Variation
    11. Chapter Review
      1. Key Terms
      2. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Roots and Radicals
    1. Introduction
    2. 9.1 Simplify and Use Square Roots
    3. 9.2 Simplify Square Roots
    4. 9.3 Add and Subtract Square Roots
    5. 9.4 Multiply Square Roots
    6. 9.5 Divide Square Roots
    7. 9.6 Solve Equations with Square Roots
    8. 9.7 Higher Roots
    9. 9.8 Rational Exponents
    10. Chapter Review
      1. Key Terms
      2. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Quadratic Equations
    1. Introduction
    2. 10.1 Solve Quadratic Equations Using the Square Root Property
    3. 10.2 Solve Quadratic Equations by Completing the Square
    4. 10.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 10.4 Solve Applications Modeled by Quadratic Equations
    6. 10.5 Graphing Quadratic Equations
    7. Chapter Review
      1. Key Terms
      2. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  12. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
  13. Index

Key Concepts

7.1 Greatest Common Factor and Factor by Grouping

  • Finding the Greatest Common Factor (GCF): To find the GCF of two expressions:
    1. Step 1. Factor each coefficient into primes. Write all variables with exponents in expanded form.
    2. Step 2. List all factors—matching common factors in a column. In each column, circle the common factors.
    3. Step 3. Bring down the common factors that all expressions share.
    4. Step 4. Multiply the factors as in Example 7.2.
  • Factor the Greatest Common Factor from a Polynomial: To factor a greatest common factor from a polynomial:
    1. Step 1. Find the GCF of all the terms of the polynomial.
    2. Step 2. Rewrite each term as a product using the GCF.
    3. Step 3. Use the ‘reverse’ Distributive Property to factor the expression.
    4. Step 4. Check by multiplying the factors as in Example 7.5.
  • Factor by Grouping: To factor a polynomial with 4 four or more terms
    1. Step 1. Group terms with common factors.
    2. Step 2. Factor out the common factor in each group.
    3. Step 3. Factor the common factor from the expression.
    4. Step 4. Check by multiplying the factors as in Example 7.15.

7.2 Factor Quadratic Trinomials with Leading Coefficient 1

  • Factor trinomials of the form x2+bx+cx2+bx+c
    1. Step 1. Write the factors as two binomials with first terms x: (x)(x)(x)(x).
    2. Step 2. Find two numbers m and n that
       Multiply to c, m·n=cm·n=c
       Add to b, m+n=bm+n=b
    3. Step 3. Use m and n as the last terms of the factors: (x+m)(x+n)(x+m)(x+n).
    4. Step 4. Check by multiplying the factors.

7.3 Factor Quadratic Trinomials with Leading Coefficient Other than 1

  • Factor Trinomials of the Form ax2+bx+cax2+bx+c using Trial and Error: See Example 7.33.
    1. Step 1. Write the trinomial in descending order of degrees.
    2. Step 2. Find all the factor pairs of the first term.
    3. Step 3. Find all the factor pairs of the third term.
    4. Step 4. Test all the possible combinations of the factors until the correct product is found.
    5. Step 5. Check by multiplying.
  • Factor Trinomials of the Form ax2+bx+cax2+bx+c Using the “ac” Method: See Example 7.38.
    1. Step 1. Factor any GCF.
    2. Step 2. Find the product ac.
    3. Step 3. Find two numbers m and n that:
       Multiply toacm·n=a·cAdd tobm+n=bMultiply toacm·n=a·cAdd tobm+n=b
    4. Step 4. Split the middle term using m and n:
       This figure shows two equations. The top equation reads a times x squared plus b times x plus c. Under this, is the equation a times x squared plus m times x plus n times x plus c. Above the m times x plus n times x is a bracket with b times x above it.
    5. Step 5. Factor by grouping.
    6. Step 6. Check by multiplying the factors.
  • Choose a strategy to factor polynomials completely (updated):
    1. Step 1. Is there a greatest common factor? Factor it.
    2. Step 2. Is the polynomial a binomial, trinomial, or are there more than three terms?
       If it is a binomial, right now we have no method to factor it.
       If it is a trinomial of the form x2+bx+cx2+bx+c
          Undo FOIL (x)(x)(x)(x).
       If it is a trinomial of the form ax2+bx+cax2+bx+c
          Use Trial and Error or the “ac” method.
       If it has more than three terms
          Use the grouping method.
    3. Step 3. Check by multiplying the factors.

7.4 Factor Special Products

  • Factor perfect square trinomials See Example 7.42.
     Step 1.Does the trinomial fit the pattern?a2+2ab+b2a2−2ab+b2Is the first term a perfect square?(a)2(a)2Write it as a square.Is the last term a perfect square?(a)2(b)2(a)2(b)2Write it as a square.Check the middle term. Is it2ab?(a)2↘2·a·b↙(b)2(a)2↘2·a·b↙(b)2Step 2.Write the square of the binomial.(a+b)2(a−b)2Step 3.Check by multiplying.Step 1.Does the trinomial fit the pattern?a2+2ab+b2a2−2ab+b2Is the first term a perfect square?(a)2(a)2Write it as a square.Is the last term a perfect square?(a)2(b)2(a)2(b)2Write it as a square.Check the middle term. Is it2ab?(a)2↘2·a·b↙(b)2(a)2↘2·a·b↙(b)2Step 2.Write the square of the binomial.(a+b)2(a−b)2Step 3.Check by multiplying.
  • Factor differences of squares See Example 7.47.
     Step 1.Does the binomial fit the pattern?a2−b2Is this a difference?____−____Are the first and last terms perfect squares?Step 2.Write them as squares.(a)2−(b)2Step 3.Write the product of conjugates.(a−b)(a+b)Step 4.Check by multiplying.Step 1.Does the binomial fit the pattern?a2−b2Is this a difference?____−____Are the first and last terms perfect squares?Step 2.Write them as squares.(a)2−(b)2Step 3.Write the product of conjugates.(a−b)(a+b)Step 4.Check by multiplying.
  • Factor sum and difference of cubes To factor the sum or difference of cubes: See Example 7.54.
    1. Step 1. Does the binomial fit the sum or difference of cubes pattern? Is it a sum or difference? Are the first and last terms perfect cubes?
    2. Step 2. Write them as cubes.
    3. Step 3. Use either the sum or difference of cubes pattern.
    4. Step 4. Simplify inside the parentheses
    5. Step 5. Check by multiplying the factors.

7.5 General Strategy for Factoring Polynomials

  • General Strategy for Factoring Polynomials See Figure 7.4.
  • How to Factor Polynomials
    1. Step 1. Is there a greatest common factor? Factor it out.
    2. Step 2.
      Is the polynomial a binomial, trinomial, or are there more than three terms?
      • If it is a binomial:
         Is it a sum?
        • Of squares? Sums of squares do not factor.
        • Of cubes? Use the sum of cubes pattern.
        Is it a difference?
        • Of squares? Factor as the product of conjugates.
        • Of cubes? Use the difference of cubes pattern.
      • If it is a trinomial:
         Is it of the form x2+bx+cx2+bx+c? Undo FOIL.
         Is it of the form ax2+bx+cax2+bx+c?
        • If ‘a’ and ‘c’ are squares, check if it fits the trinomial square pattern.
        • Use the trial and error or ‘ac’ method.
      • If it has more than three terms:
         Use the grouping method.
    3. Step 3. Check. Is it factored completely? Do the factors multiply back to the original polynomial?

7.6 Quadratic Equations

  • Zero Product Property If a·b=0a·b=0, then either a=0a=0 or b=0b=0 or both. See Example 7.69.
  • Solve a quadratic equation by factoring To solve a quadratic equation by factoring: See Example 7.73.
    1. Step 1. Write the quadratic equation in standard form, ax2+bx+c=0ax2+bx+c=0.
    2. Step 2. Factor the quadratic expression.
    3. Step 3. Use the Zero Product Property.
    4. Step 4. Solve the linear equations.
    5. Step 5. Check.
  • Use a problem solving strategy to solve word problems See Example 7.80.
    1. Step 1. Read the problem. Make sure all the words and ideas are understood.
    2. Step 2. Identify what we are looking for.
    3. Step 3. Name what we are looking for. Choose a variable to represent that quantity.
    4. Step 4. Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebra equation.
    5. Step 5. Solve the equation using good algebra techniques.
    6. Step 6. Check the answer in the problem and make sure it makes sense.
    7. Step 7. Answer the question with a complete sentence.
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