10.8.2: Key Concepts

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    From LibreTexts, under CC BY 4.0, as the source states it; its equations are written in TeX, which the PDF shows typeset. This page at LibreTexts.

    Key Concepts

    10.2 Use Multiplication Properties of Exponents

    • This is read to the power.
    • Product Property of Exponents
      • If is a real number and are counting numbers, then

      • To multiply with like bases, add the exponents.
    • Power Property for Exponents
      • If is a real number and are counting numbers, then

    • Product to a Power Property for Exponents
      • If and are real numbers and is a whole number, then

    10.3 Multiply Polynomials

    • Use the FOIL method for multiplying 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.
      Step 5. Combine like terms, when possible.
  1. Multiplying Two Binomials: To multiply binomials, use the:
    • Distributive Property
    • FOIL Method
    • Vertical Method
  2. Multiplying a Trinomial by a Binomial: To multiply a trinomial by a binomial, use the:
    • Distributive Property
    • Vertical Method

    3. 10.4 Divide Monomials

    • Equivalent Fractions Property
      • If are whole numbers where then

    • Zero Exponent
      • If is a non-zero number, then
      • Any nonzero number raised to the zero power is
    • Quotient Property for Exponents
      • If is a real number, and are whole numbers, then

    • Quotient to a Power Property for Exponents
      • If and are real numbers, and is a counting number, then

      • To raise a fraction to a power, raise the numerator and denominator to that power.

    10.5 Integer Exponents and Scientific Notation

    • Summary of Exponent Properties
      • If are real numbers and are integers, then

    • Convert from Decimal Notation to Scientific Notation: 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, , 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 .
        • If the original number is between 0 and 1, the power of 10 will be .
      4. Step 4. Check.
    • Convert Scientific Notation to Decimal Form: To convert scientific notation to decimal form:
      1. Step 1. Determine the exponent, , on the factor 10.
      2. Step 2. Move the decimal places, adding zeros if needed.
        • If the exponent is positive, move the decimal point places to the right.
        • If the exponent is negative, move the decimal point places to the left.
      3. Step 3. Check.

    10.6 Introduction to Factoring Polynomials

    • Find the greatest common factor.
      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 , , are real numbers, then
        and
    • 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 Distributive Property ‘in reverse’ to factor the expression.
      4. Step 4. Check by multiplying the factors.

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