Skip to Content
OpenStax Logo
Intermediate Algebra

Key Concepts

Intermediate AlgebraKey Concepts
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Use the Language of Algebra
    3. 1.2 Integers
    4. 1.3 Fractions
    5. 1.4 Decimals
    6. 1.5 Properties of Real Numbers
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations
    1. Introduction
    2. 2.1 Use a General Strategy to Solve Linear Equations
    3. 2.2 Use a Problem Solving Strategy
    4. 2.3 Solve a Formula for a Specific Variable
    5. 2.4 Solve Mixture and Uniform Motion Applications
    6. 2.5 Solve Linear Inequalities
    7. 2.6 Solve Compound Inequalities
    8. 2.7 Solve Absolute Value Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Graphs and Functions
    1. Introduction
    2. 3.1 Graph Linear Equations in Two Variables
    3. 3.2 Slope of a Line
    4. 3.3 Find the Equation of a Line
    5. 3.4 Graph Linear Inequalities in Two Variables
    6. 3.5 Relations and Functions
    7. 3.6 Graphs of Functions
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Systems of Linear Equations
    1. Introduction
    2. 4.1 Solve Systems of Linear Equations with Two Variables
    3. 4.2 Solve Applications with Systems of Equations
    4. 4.3 Solve Mixture Applications with Systems of Equations
    5. 4.4 Solve Systems of Equations with Three Variables
    6. 4.5 Solve Systems of Equations Using Matrices
    7. 4.6 Solve Systems of Equations Using Determinants
    8. 4.7 Graphing Systems of Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Polynomials and Polynomial Functions
    1. Introduction
    2. 5.1 Add and Subtract Polynomials
    3. 5.2 Properties of Exponents and Scientific Notation
    4. 5.3 Multiply Polynomials
    5. 5.4 Dividing Polynomials
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Factoring
    1. Introduction to Factoring
    2. 6.1 Greatest Common Factor and Factor by Grouping
    3. 6.2 Factor Trinomials
    4. 6.3 Factor Special Products
    5. 6.4 General Strategy for Factoring Polynomials
    6. 6.5 Polynomial Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Rational Expressions and Functions
    1. Introduction
    2. 7.1 Multiply and Divide Rational Expressions
    3. 7.2 Add and Subtract Rational Expressions
    4. 7.3 Simplify Complex Rational Expressions
    5. 7.4 Solve Rational Equations
    6. 7.5 Solve Applications with Rational Equations
    7. 7.6 Solve Rational Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Roots and Radicals
    1. Introduction
    2. 8.1 Simplify Expressions with Roots
    3. 8.2 Simplify Radical Expressions
    4. 8.3 Simplify Rational Exponents
    5. 8.4 Add, Subtract, and Multiply Radical Expressions
    6. 8.5 Divide Radical Expressions
    7. 8.6 Solve Radical Equations
    8. 8.7 Use Radicals in Functions
    9. 8.8 Use the Complex Number System
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Quadratic Equations and Functions
    1. Introduction
    2. 9.1 Solve Quadratic Equations Using the Square Root Property
    3. 9.2 Solve Quadratic Equations by Completing the Square
    4. 9.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 9.4 Solve Quadratic Equations in Quadratic Form
    6. 9.5 Solve Applications of Quadratic Equations
    7. 9.6 Graph Quadratic Functions Using Properties
    8. 9.7 Graph Quadratic Functions Using Transformations
    9. 9.8 Solve Quadratic Inequalities
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Exponential and Logarithmic Functions
    1. Introduction
    2. 10.1 Finding Composite and Inverse Functions
    3. 10.2 Evaluate and Graph Exponential Functions
    4. 10.3 Evaluate and Graph Logarithmic Functions
    5. 10.4 Use the Properties of Logarithms
    6. 10.5 Solve Exponential and Logarithmic Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Conics
    1. Introduction
    2. 11.1 Distance and Midpoint Formulas; Circles
    3. 11.2 Parabolas
    4. 11.3 Ellipses
    5. 11.4 Hyperbolas
    6. 11.5 Solve Systems of Nonlinear Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  13. 12 Sequences, Series and Binomial Theorem
    1. Introduction
    2. 12.1 Sequences
    3. 12.2 Arithmetic Sequences
    4. 12.3 Geometric Sequences and Series
    5. 12.4 Binomial Theorem
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  14. 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
    11. Chapter 11
    12. Chapter 12
  15. Index

6.1 Greatest Common Factor and Factor by Grouping

  • How to find the greatest common factor (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.
  • Distributive Property: If a, b, and c are real numbers, then
    a(b+c)=ab+acandab+ac=a(b+c)a(b+c)=ab+acandab+ac=a(b+c)

    The form on the left is used to multiply. The form on the right is used to factor.
  • How to factor the 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.
  • Factor as a Noun and a Verb: We use “factor” as both a noun and a verb.
    Noun:7 is afactorof 14Verb:factor3 from3a+3Noun:7 is afactorof 14Verb:factor3 from3a+3
  • How to factor by grouping.
    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.

6.2 Factor Trinomials

  • How to factor trinomials of the form x2+bx+c.x2+bx+c.
    1. Step 1. Write the factors as two binomials with first terms x. x2+bx+c(x)(x)x2+bx+c(x)(x)
    2. Step 2. Find two numbers m and n that
      multiply toc,m·n=cadd tob,m+n=bmultiply toc,m·n=cadd tob,m+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.
  • Strategy for Factoring Trinomials of the Form x2+bx+cx2+bx+c: When we factor a trinomial, we look at the signs of its terms first to determine the signs of the binomial factors.
    x2+bx+c(x+m)(x+n)Whencis positive,mandnhave the same sign.bpositivebnegativem,npositivem,nnegativex2+5x+6x26x+8(x+2)(x+3)(x4)(x2)same signssame signsWhencis negative,mandnhave opposite signs.x2+x12x22x15(x+4)(x3)(x5)(x+3)opposite signsopposite signsx2+bx+c(x+m)(x+n)Whencis positive,mandnhave the same sign.bpositivebnegativem,npositivem,nnegativex2+5x+6x26x+8(x+2)(x+3)(x4)(x2)same signssame signsWhencis negative,mandnhave opposite signs.x2+x12x22x15(x+4)(x3)(x5)(x+3)opposite signsopposite signs
    Notice that, in the case when m and n have opposite signs, the sign of the one with the larger absolute value matches the sign of b.
  • How to factor trinomials of the form ax2+bx+cax2+bx+c using trial and error.
    1. Step 1. Write the trinomial in descending order of degrees as needed.
    2. Step 2. Factor any GCF.
    3. Step 3. Find all the factor pairs of the first term.
    4. Step 4. Find all the factor pairs of the third term.
    5. Step 5. Test all the possible combinations of the factors until the correct product is found.
    6. Step 6. Check by multiplying.
  • How to factor trinomials of the form ax2+bx+cax2+bx+c using the “ac” method.
    1. Step 1. Factor any GCF.
    2. Step 2. Find the product ac.
    3. Step 3. Find two numbers m and n that:
      Multiply toac.m·n=a·cAdd tob.m+n=bax2+bx+cMultiply toac.m·n=a·cAdd tob.m+n=bax2+bx+c
    4. Step 4. Split the middle term using m and n. ax2+mx+nx+cax2+mx+nx+c
    5. Step 5. Factor by grouping.
    6. Step 6. Check by multiplying the factors.

6.3 Factor Special Products

  • Perfect Square Trinomials Pattern: If a and b are real numbers,
    a2+2ab+b2=(a+b)2a22ab+b2=(ab)2a2+2ab+b2=(a+b)2a22ab+b2=(ab)2
  • How to factor perfect square trinomials.
    Step 1.Does the trinomial fit the pattern?a2+2ab+b2a22ab+b2 Is the first term a perfect square?(a)2(a)2 Write it as a square. Is the last term a perfect square?(a)2(b)2(a)2(b)2 Write it as a square. Check the middle term. Is it2ab?(a)22·a·b(b)2(a)22·a·b(b)2 Step 2.Write the square of the binomial.(a+b)2(ab)2 Step 3.Check by multiplying.Step 1.Does the trinomial fit the pattern?a2+2ab+b2a22ab+b2 Is the first term a perfect square?(a)2(a)2 Write it as a square. Is the last term a perfect square?(a)2(b)2(a)2(b)2 Write it as a square. Check the middle term. Is it2ab?(a)22·a·b(b)2(a)22·a·b(b)2 Step 2.Write the square of the binomial.(a+b)2(ab)2 Step 3.Check by multiplying.
  • Difference of Squares Pattern: If a,ba,b are real numbers,
    a squared minus b squared is a minus b, a plus b. Here, a squared minus b squared is the difference of squares and a minus b, a plus b are conjugates.
  • How to factor differences of squares.
    Step 1.Does the binomial fit the pattern?a2b2Is 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.(ab)(a+b)Step 4.Check by multiplying.Step 1.Does the binomial fit the pattern?a2b2Is 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.(ab)(a+b)Step 4.Check by multiplying.
  • Sum and Difference of Cubes Pattern
    a3+b3=(a+b)(a2ab+b2)a3b3=(ab)(a2+ab+b2)a3+b3=(a+b)(a2ab+b2)a3b3=(ab)(a2+ab+b2)
  • How to factor the sum or difference of cubes.
    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.

6.4 General Strategy for Factoring Polynomials

This chart shows the general strategies for factoring polynomials. It shows ways to find GCF of binomials, trinomials and polynomials with more than 3 terms. For binomials, we have difference of squares: a squared minus b squared equals a minus b, a plus b; sum of squares do not factor; sub of cubes: a cubed plus b cubed equals open parentheses a plus b close parentheses open parentheses a squared minus ab plus b squared close parentheses; difference of cubes: a cubed minus b cubed equals open parentheses a minus b close parentheses open parentheses a squared plus ab plus b squared close parentheses. For trinomials, we have x squared plus bx plus c where we put x as a term in each factor and we have a squared plus bx plus c. Here, if a and c are squares, we have a plus b whole squared equals a squared plus 2 ab plus b squared and a minus b whole squared equals a squared minus 2 ab plus b squared. If a and c are not squares, we use the ac method. For polynomials with more than 3 terms, we use grouping.
  • How to use a general strategy for factoring 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+c?x2+bx+c? Undo FOIL.
      Is it of the form ax2+bx+c?ax2+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?

6.5 Polynomial Equations

  • Polynomial Equation: A polynomial equation is an equation that contains a polynomial expression. The degree of the polynomial equation is the degree of the polynomial.
  • Quadratic Equation: An equation of the form ax2+bx+c=0ax2+bx+c=0 is called a quadratic equation.
    a,b,care real numbers anda0a,b,care real numbers anda0
  • Zero Product Property: If a·b=0,a·b=0, then either a=0a=0 or b=0b=0 or both.
  • How to use the Zero Product Property
    1. Step 1. Set each factor equal to zero.
    2. Step 2. Solve the linear equations.
    3. Step 3. Check.
  • How to solve a quadratic equation by factoring.
    1. Step 1. Write the quadratic equation in standard form, ax2+bx+c=0.ax2+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. Substitute each solution separately into the original equation.
  • Zero of a Function: For any function f, if f(x)=0,f(x)=0, then x is a zero of the function.
  • How to use a problem solving strategy to solve word problems.
    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 algebraic equation.
    5. Step 5. Solve the equation using appropriate 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.
Citation/Attribution

Want to cite, share, or modify this book? This book is Creative Commons Attribution License 4.0 and you must attribute OpenStax.

Attribution information
  • If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
    Access for free at https://openstax.org/books/intermediate-algebra/pages/1-introduction
  • If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:
    Access for free at https://openstax.org/books/intermediate-algebra/pages/1-introduction
Citation information

© Sep 16, 2020 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License 4.0 license. The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.