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Intermediate Algebra

Key Concepts

Intermediate AlgebraKey Concepts
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Key Concepts

5.1 Add and Subtract Polynomials

  • Monomial
    • A monomial is an algebraic expression with one term.
    • A monomial in one variable is a term of the form axm,axm, where a is a constant and m is a whole number.
  • Polynomials
    • Polynomial—A monomial, or two or more algebraic terms combined by addition or subtraction is a polynomial.
    • monomial —A polynomial with exactly one term is called a monomial.
    • binomial — A polynomial with exactly two terms is called a binomial.
    • trinomial —A polynomial with exactly three terms is called a trinomial.
  • Degree of a Polynomial
    • The degree of a term is the sum of the exponents of its variables.
    • The degree of a constant is 0.
    • The degree of a polynomial is the highest degree of all its terms.

5.2 Properties of Exponents and Scientific Notation

  • Exponential Notation
     The figure shows the letter a in a normal font with the label base and the letter m in a superscript font with the label exponent. This means we multiply the number a with itself, m times.
    This is read a to the mthmth power.
     In the expression amam, the exponent m tells us how many times we use the base a as a factor.
  • Product Property for Exponents
     If a is a real number and m and n are integers, then
    am·an=am+nam·an=am+n

     To multiply with like bases, add the exponents.
  • Quotient Property for Exponents
     If aa is a real number, a≠0,a≠0, and m and n are integers, then
    aman=am−n,m>nandaman=1an−m,n>maman=am−n,m>nandaman=1an−m,n>m
  • Zero Exponent
    • If a is a non-zero number, then a0=1.a0=1.
    • If a is a non-zero number, then a to the power of zero equals 1.
    • Any non-zero number raised to the zero power is 1.
  • Negative Exponent
    • If n is an integer and a≠0,a≠0, then a−n=1ana−n=1an or 1a−n=an.1a−n=an.
  • Quotient to a Negative Exponent Property
     If a,ba,b are real numbers, a≠0,b≠0a≠0,b≠0 and nn is an integer, then
    (ab)−n=(ba)n(ab)−n=(ba)n
  • Power Property for Exponents
     If aa is a real number and m,nm,n are integers, then
    (am)n=am·n(am)n=am·n

     To raise a power to a power, multiply the exponents.
  • Product to a Power Property for Exponents
     If a and b are real numbers and m is a whole number, then
    (ab)m=ambm(ab)m=ambm

     To raise a product to a power, raise each factor to that power.
  • Quotient to a Power Property for Exponents
     If aa and are real numbers, b≠0,b≠0, and mm is an integer, then
    (ab)m=ambm(ab)m=ambm

     To raise a fraction to a power, raise the numerator and denominator to that power.
  • Summary of Exponent Properties
     If a and b are real numbers, and m and n are integers, then
    Property Description
    Product Property am·an=am+nam·an=am+n
    Power Property (am)n=am·n(am)n=am·n
    Product to a Power (ab)n=ambm(ab)n=ambm
    Quotient Property aman=am−n,a≠0aman=am−n,a≠0
    Zero Exponent Property a0=1,a≠0a0=1,a≠0
    Quotient to a Power Property: (ab)m=ambm,b≠0(ab)m=ambm,b≠0
    Properties of Negative Exponents a−n=1ana−n=1an and 1a−n=an1a−n=an
    Quotient to a Negative Exponent (ab)−n=(ba)n(ab)−n=(ba)n
  • Scientific Notation
     A number is expressed in scientific notation when it is of the form
    a×10nwhere1≤a<10andnis an integer.a×10nwhere1≤a<10andnis an integer.
  • How to convert a decimal to scientific notation.
    1. Step 1. Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.
    2. Step 2. Count the number of decimal places, n,n, that the decimal point was moved.
    3. Step 3.
      Write the number as a product with a power of 10. If the original number is.
      • greater than 1, the power of 10 will be 10n.10n.
      • between 0 and 1, the power of 10 will be 10−n.10−n.
    4. Step 4. Check.
  • How to convert scientific notation to decimal form.
    1. Step 1. Determine the exponent, n,n, on the factor 10.
    2. Step 2.
      Move the decimal nn places, adding zeros if needed.
      • If the exponent is positive, move the decimal point nn places to the right.
      • If the exponent is negative, move the decimal point |n||n| places to the left.
    3. Step 3. Check.

5.3 Multiply Polynomials

  • How to use the FOIL method to multiply two binomials.
     The figure shows how to use the FOIL method to multiply two binomials. The example is the quantity a plus b in parentheses times the quantity c plus d in parentheses. The numbers a and c are labeled first and the numbers b and d are labeled last. The numbers b and c are labeled inner and the numbers a and d are labeled outer. A note on the side of the expression tells you to Say it as you multiply! FOIL First Outer Inner Last. The directions are then given in numbered steps. Step 1. Multiply the First terms. Step 2. Multiply the Outer terms. Step 3. Multiply the Inner terms. Step 4. Multiply the Last Terms. Step 5. Combine like terms when possible.
  • Multiplying Two Binomials: To multiply binomials, use the:
    • Distributive Property
    • FOIL Method
  • Multiplying a Polynomial by a Polynomial: To multiply a trinomial by a binomial, use the:
    • Distributive Property
    • Vertical Method
  • Binomial Squares Pattern
     If a and b are real numbers, The figure shows the result of squaring two binomials. The first example is a plus b squared equals a squared plus 2 a b plus b squared. The equation is written out again with each part labeled. The quantity a plus b squared is labeled binomial squared. The terms a squared is labeled first term squared. The term 2 a b is labeled 2 times product of terms. The term b squared is labeled last term squared. The second example is a minus b squared equals a squared minus 2 a b plus b squared. The equation is written out again with each part labeled. The quantity a minus b squared is labeled binomial squared. The terms a squared is labeled first term squared. The term negative 2 a b is labeled 2 times product of terms. The term b squared is labeled last term squared.
  • Product of Conjugates Pattern
     If a,ba,b are real numbers
     The figure shows the result of multiplying a binomial with its conjugate. The formula is a plus b times a minus b equals a squared minus b squared. The equation is written out again with labels. The product a plus b times a minus b is labeled conjugates. The result a squared minus b squared is labeled difference of squares.
    The product is called a difference of squares.
     To multiply conjugates, square the first term, square the last term, write it as a difference of squares.
  • Comparing the Special Product Patterns
    Binomial Squares Product of Conjugates
    (a+b)2=a2+2ab+b2(a+b)2=a2+2ab+b2 (a−b)2=a2−2ab+b2(a−b)2=a2−2ab+b2
    (a−b)(a+b)=a2−b2(a−b)(a+b)=a2−b2
    •  Squaring a binomial •  Multiplying conjugates
    •  Product is a trinomial •  Product is a binomial.
    •  Inner and outer terms with FOIL are the same. •  Inner and outer terms with FOIL are opposites.
    •  Middle term is double the product of the terms •  There is no middle term.
  • Multiplication of Polynomial Functions:
    • For functions f(x)f(x) and g(x),g(x),
      (f·g)(x)=f(x)·g(x)(f·g)(x)=f(x)·g(x)

5.4 Dividing Polynomials

  • Division of a Polynomial by a Monomial
    • To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.
  • Division of Polynomial Functions
    • For functions f(x)f(x) and g(x),g(x), where g(x)≠0,g(x)≠0,
       (fg)(x)=f(x)g(x)(fg)(x)=f(x)g(x)
  • Remainder Theorem
    • If the polynomial function f(x)f(x) is divided by x−c,x−c, then the remainder is f(c).f(c).
  • Factor Theorem: For any polynomial function f(x),f(x),
    • if x−cx−c is a factor of f(x),f(x), then f(c)=0f(c)=0
    • if f(c)=0,f(c)=0, then x−cx−c is a factor of f(x)f(x)
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