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Mathematics LibreTexts

10.S: Polynomials (Summary)

  • Page ID
    7277
  • [ "article:topic", "authorname:openstax" ]

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    Key Terms

    binomial A polynomial with exactly two terms
    degree of a constant The degree of a constant is 0.
    degree of a polynomial The degree of a polynomial is the highest degree of all its terms.
    degree of a term The degree of a term of a polynomial is the exponent of its variable.
    greatest common factor The greatest common factor (GCF) of two or more expressions is the largest expression that is a factor of all the expressions.
    monomial A term of the form axm, where a is a constant and m is a whole number, is called a monomial.
    negative exponent If n is a positive integer and a ≠ 0, then \(a^{-n} = \frac{1}{a^{n}}\).
    polynomial A polynomial is a monomial, or two or more monomials, combined by addition or subtraction.
    scientific notation A number expressed in scientific notation when it is of the form a × 10n, where a ≥ 1 and a < 10, and n is an integer.
    trinomial A trinomial is a polynomial with exactly three terms.
    zero exponent If a is a non-zero number, then a0 = 1. Any nonzero number raised to the zero power is 1.

    Key Concepts

    10.2 - Use Multiplication Properties of Exponents

    • Exponential Notation

    CNX_BMath_Figure_10_02_013_img.jpg

    This is read a to the mth power.

    • Product Property of Exponents
      • If a is a real number and m, n are counting numbers, then am • an = am + n
      • To multiply with like bases, add the exponents.
    • Power Property for Exponents
      • If a is a real number and m, n are counting numbers, then (am)n = am • n
    • Product to a Power Property for Exponents
      • If a and b are real numbers and m is a whole number, then (ab)m = ambm

    10.3 - Multiply Polynomials

    • Use the FOIL method for multiplying two binomials.

    Step 1. Multiply the First terms.

     

    CNX_BMath_Figure_10_03_025_img.jpg

    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
      • Vertical Method
    • Multiplying a Trinomial by a Binomial: To multiply a trinomial by a binomial, use the:
      • Distributive Property
      • Vertical Method

    10.4 - Divide Monomials

    • Equivalent Fractions Property
      • If a, b, c are whole numbers where b ≠ 0, c ≠ 0, then$$\frac{a}{b} = \frac{a \cdot c}{b \cdot c} \quad and \quad \frac{a \cdot c}{b \cdot c} = \frac{a}{b}$$
    • Zero Exponent
      • If a is a non-zero number, then a0 = 1.
      • Any nonzero number raised to the zero power is 1.
    • Quotient Property for Exponents
      • If a is a real number, a ≠ 0, and m, n are whole numbers, then$$\frac{a^{m}}{a^{n}} = a^{m-n},\; m>n \quad and \quad \frac{a^{m}}{a^{n}} = \frac{1}{a^{n-m}},\; n>m$$
    • Quotient to a Power Property for Exponents
      • If a and b are real numbers, b ≠ 0, and m is a counting number, then$$\left(\dfrac{a}{b}\right)^{m} = \frac{a^{m}}{b^{m}}$$
      • 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 a, b are real numbers and m, n are integers, then
    Product Property am • an = am + n
    Power Property (am)n = am • n
    Product to a Power Property (ab)m = ambm
    Quotient Property \(\frac{a^{m}}{a^{n}}\) = am − n, a ≠ 0, m > n
      \(\frac{a^{m}}{a^{n}} = \frac{1}{a^{n-m}}\), a ≠ 0, n > m
    Zero Exponent Property a0 = 1, a ≠ 0
    Quotient to a Power Property \(\left(\dfrac{a}{b}\right)^{m} = \frac{a^{m}}{b^{m}}\), b ≠ 0
    Definition of a Negative Exponent \(a^{-n} = \frac{1}{a^{n}}\)
    • Convert from Decimal Notation to Scientific Notation: To convert a decimal to scientific notation:
      1. Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.
      2. Count the number of decimal places, n, that the decimal point was moved. 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.
        • If the original number is between 0 and 1, the power of 10 will be 10n.
      3. Check.
    • Convert Scientific Notation to Decimal Form: To convert scientific notation to decimal form:
      1. Determine the exponent, n, on the factor 10.
      2. Move the decimal n places, 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.
      3. Check.

    10.6 - Introduction to Factoring Polynomials

    • Find the greatest common factor.
      1. Factor each coefficient into primes. Write all variables with exponents in expanded form.
      2. List all factors—matching common factors in a column. In each column, circle the common factors.
      3. Bring down the common factors that all expressions share.
      4. Multiply the factors.
    • Distributive Property
      • If a , b , c are real numbers, then a(b + c) = ab + ac and ab + ac = a(b + c).
    • Factor the greatest common factor from a polynomial.
      1. Find the GCF of all the terms of the polynomial.
      2. Rewrite each term as a product using the GCF.
      3. Use the Distributive Property ‘in reverse’ to factor the expression.
      4. Check by multiplying the factors.

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