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Key Concepts

6.1 Add and Subtract Polynomials

  • Monomials
    • A monomial is a term of the form axmaxm, where aa is a constant and mm 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,ba,b are real numbers and m,nm,n are whole numbers, then
      Product Propertyam·an=am+nPower Property(am)n=am·nProduct to a Power(ab)m=ambmProduct Propertyam·an=am+nPower Property(am)n=am·nProduct to a Power(ab)m=ambm

6.3 Multiply Polynomials

  • FOIL Method for Multiplying Two Binomials—To multiply two binomials:
    1. Step 1. Multiply the First terms.
    2. Step 2. Multiply the Outer terms.
    3. Step 3. Multiply the Inner terms.
    4. Step 4. Multiply the Last terms.

  • Multiplying Two Binomials—To multiply binomials, use the:
  • Multiplying a Trinomial by a Binomial—To multiply a trinomial by a binomial, use the:

6.4 Special Products

  • Binomial Squares Pattern
    • If a,ba,b are real numbers,
      No Alt Text
    • (a+b)2=a2+2ab+b2(a+b)2=a2+2ab+b2
    • (ab)2=a22ab+b2(ab)2=a22ab+b2
    • To square a binomial: square the first term, square the last term, double their product.

  • Product of Conjugates Pattern
    • If a,ba,b are real numbers,
      No Alt Text
    • (ab)(a+b)=a2b2(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:
    • If aa is a real number, a0a0, and m,nm,n are whole numbers, then:
      aman=amn,m>nandaman=1amn,n>maman=amn,m>nandaman=1amn,n>m
  • Zero Exponent
    • If aa is a non-zero number, then a0=1a0=1.

  • Quotient to a Power Property for Exponents:
    • If aa and bb are real numbers, b0,b0, and mm is a counting number, 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,ba,b are real numbers and m,nm,n are whole numbers, then
      Product Propertyam·an=am+nPower Property(am)n=am·nProduct to a Power(ab)m=ambmQuotient Propertyaman=amn,a0,m>naman=1anm,a0,n>mZero Exponent Definitionao=1,a0Quotient to a Power Property(ab)m=ambm,b0Product Propertyam·an=am+nPower Property(am)n=am·nProduct to a Power(ab)m=ambmQuotient Propertyaman=amn,a0,m>naman=1anm,a0,n>mZero Exponent Definitionao=1,a0Quotient to a Power Property(ab)m=ambm,b0

6.6 Divide Polynomials

  • Fraction Addition
    • If a,b,andca,b,andc are numbers where c0c0, then
      ac+bc=a+bcanda+bc=ac+bcac+bc=a+bcanda+bc=ac+bc

  • 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 nn is a positive integer and a0a0, then 1an=an1an=an
  • Quotient to a Negative Exponent
    • If a,ba,b are real numbers, b0b0 and nn is an integer , then (ab)n=(ba)n(ab)n=(ba)n
  • 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, nn, 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 10n10n
      • between 0 and 1, the power of 10 will be 10n10n
    4. Step 4. Check.

  • To convert scientific notation to decimal form:
    1. Step 1. Determine the exponent, nn, on the factor 10.
    2. Step 2.
      Move the decimal nnplaces, 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.
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