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Mathematics LibreTexts

10.S: Polynomials (Summary)

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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