Key Concepts

6.1 Add and Subtract Polynomials

  • Monomials
    • A monomial is a term of the form axm, where a is a constant and m is a whole number
  • Polynomials
    • polynomial—A monomial, or two or more monomials 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

6.2 Use Multiplication Properties of Exponents

  • Exponential Notation
    This figure has two columns. In the left column is a to the m power. The m is labeled in blue as an exponent. The a is labeled in red as the base. In the right column is the text “a to the m powder means multiply m factors of a.” Below this is a to the m power equals a times a times a times a, followed by an ellipsis, with “m factors” written below in blue.
  • Properties of Exponents
    • If a,b are real numbers and m,n are whole numbers, then

6.3 Multiply Polynomials

  • FOIL Method for Multiplying Two Binomials—To multiply two binomials:
    • Step 1. Multiply the First terms.
    • Step 2. Multiply the Outer terms.
    • Step 3. Multiply the Inner terms.
    • Step 4. Multiply the Last terms.
  • Multiplying Two Binomials—To multiply binomials, use the:
    • Distributive Property (Example 6.34)
    • FOIL Method (Example 6.39)
    • Vertical Method (Example 6.44)
  • Multiplying a Trinomial by a Binomial—To multiply a trinomial by a binomial, use the:
    • Distributive Property (Example 6.45)
    • Vertical Method (Example 6.46)

6.4 Special Products

  • Binomial Squares Pattern
    • If a,b are real numbers,
  • Product of Conjugates Pattern
    • If a,b are real numbers,
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    • (ab)(a+b)=a2b2
    • 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

6.5 Divide Monomials

  • Quotient Property for Exponents:
  • Zero Exponent
    • If a is a non-zero number, then a0=1.
  • Quotient to a Power Property for Exponents:
    • If a and b are real numbers, b≠0, and m is a counting number, then:
    • To raise a fraction to a power, raise the numerator and denominator to that power.
  • Summary of Exponent Properties
    • If a,b are real numbers and m,n are whole numbers, then

6.6 Divide Polynomials

  • Fraction Addition
    • If a,b,andc are numbers where c≠0, then

Division of a Polynomial by a Monomial

  • To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.

6.7 Integer Exponents and Scientific Notation

  • Property of Negative Exponents
    • If n is a positive integer and a≠0, then 1/an=a/n
  • Quotient to a Negative Exponent
    • If a,b are real numbers, b≠0 and n is an integer , then (a/b)−n=(b/a)n
  • To convert a decimal to scientific notation:
    • Step 1. Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.
    • Step 2. Count the number of decimal places, n, that the decimal point was moved.
    • 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
      • between 0 and 1, the power of 10 will be 10−n
    • Step 4. Check.
  • To convert scientific notation to decimal form:
    • Step 1. Determine the exponent, n, on the factor 10.
    • Step 2. Move the decimal nplaces, adding zeros if needed.
    • If the exponent is positive, move the decimal point n places to the right.
    • If the exponent is negative, move the decimal point |n| places to the left.
    • Step 3. Check.