10.S: Polynomials (Summary)
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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 ax^{m}, 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 × 10^{n}, 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 nonzero number, then a^{0} = 1. Any nonzero number raised to the zero power is 1. 
Key Concepts
10.2  Use Multiplication Properties of Exponents
 Exponential Notation
This is read a to the m^{th }power.
 Product Property of Exponents
 If a is a real number and m, n are counting numbers, then a^{m} • a^{n} = a^{m + 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 (a^{m})^{n} = a^{m • n}
 Product to a Power Property for Exponents
 If a and b are real numbers and m is a whole number, then (ab)^{m} = a^{m}b^{m}
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. 
 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 nonzero number, then a^{0} = 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$$\frac{a^{m}}{a^{n}} = a^{mn},\; m>n \quad and \quad \frac{a^{m}}{a^{n}} = \frac{1}{a^{nm}},\; n>m$$
 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  a^{m} • a^{n} = a^{m + n} 
Power Property  (a^{m})^{n} = a^{m • n} 
Product to a Power Property  (ab)^{m} = a^{m}b^{m} 
Quotient Property  \(\frac{a^{m}}{a^{n}}\) = a^{m − n}, a ≠ 0, m > n 
\(\frac{a^{m}}{a^{n}} = \frac{1}{a^{nm}}\), a ≠ 0, n > m  
Zero Exponent Property  a^{0} = 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:
 Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.
 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 10^{n}.
 If the original number is between 0 and 1, the power of 10 will be 10^{n}.
 Check.
 Convert Scientific Notation to Decimal Form: To convert scientific notation to decimal form:
 Determine the exponent, n, on the factor 10.
 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.
 Check.
10.6  Introduction to Factoring Polynomials
 Find the greatest common factor.
 Factor each coefficient into primes. Write all variables with exponents in expanded form.
 List all factors—matching common factors in a column. In each column, circle the common factors.
 Bring down the common factors that all expressions share.
 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.
 Find the GCF of all the terms of the polynomial.
 Rewrite each term as a product using the GCF.
 Use the Distributive Property ‘in reverse’ to factor the expression.
 Check by multiplying the factors.
Contributors
Lynn Marecek (Santa Ana College) and MaryAnne AnthonySmith (Formerly of Santa Ana College). This content is licensed under Creative Commons Attribution License v4.0 "Download for free at http://cnx.org/contents/fd53eae1fa2...49835c3c@5.191."