Algebra 2
Unit 4
Notes: Laws of Exponents
Polynomials and Real Number
Operations
Monomials: Exponents are used in expressions called monomials.
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A monomial is an expression that is a number, variable, or product of a number and one
or more variables.
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Examples of monomials are 5c, - a, 17, 𝑥 3 , and
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Monomials cannot contain variables whose exponents cannot be written as whole
numbers. (Exponents cannot be negative or fractions)
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Examples of expressions with exponents that are not monomials are
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Constants are monomials that contain no variables.
The numerical factor of a monomial is the coefficient of the variable. For example the
coefficient in -6m is -6.
The degree of a monomial is the sum of the exponents of its variables. For example the
degree of 12 𝑔7 ℎ4 is 7+4 or 11. The degree of a nonzero constant is 0.
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1
2
𝑥4𝑦2.
1
1
𝑛
and √𝑛 = 𝑛2
2
Exponent Properties:
Negative Exponents
For any real number a, and any integer n, where a≠ 0, 𝑎−𝑛 =
1
1
𝑎𝑛𝑑 𝑎−𝑛 = 𝑎𝑛 .
𝑛
Multiplying Powers
Dividing Powers
For any real number a and integers m and n, 𝑎𝑚 ∙ 𝑎𝑛 = 𝑎𝑚+𝑛 .
Properties of Powers
Suppose m and n are integers and a and b are real numbers. Then the
following properties hold.
Power of a Power: (𝑎𝑚 )𝑛 = 𝑎𝑚𝑛
Power of a Product: (𝑎𝑏)𝑚 = 𝑎𝑚 𝑏 𝑚
For any real number a except a=0, and integers m and n,
𝑎
Product Property of
Radicals
Quotient Property of
Radicals


𝑎𝑚
𝑎𝑛
= 𝑎𝑚−𝑛 .
𝑎𝑛
Power of a Quotient: (𝑏)𝑛 = 𝑏𝑛 , b≠ 0
For any real numbers a and b, and any integer n, n> 1,
𝑛
𝑛
𝑛
1. If n is even, then √𝑎𝑏 = √𝑎 ∙ √𝑏 when a and b are both
nonnegative, and
𝑛
𝑛
𝑛
2. If n is odd, then √𝑎𝑏 = √𝑎 ∙ √𝑏
For real numbers a and b, b≠ 0, and any integer n, n> 1,
𝑛
𝑎
√𝑏 =
𝑛
√𝑎
𝑛
√𝑏
, if all roots are defined.
To simplify monomials apply exponent properties.
To simplify radicals apply radical exponent properties.
Scientific Notation: Use exponents to express very large or very small numbers in base 10
using exponents. 𝑎 𝑥 10𝑛 , where 1≤ 𝑎 < 10 , and n is an integer. Number in scientific
notation can be added, subtracted, multiplied, and divided. If 𝑛 ≥ 0, then number is greater
than or equal to 1. If 𝑛 < 0, then the number is less than 1 but greater than 0.
Polynomial:
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A polynomial is a monomial or a sum of monomials.
The monomials that make up a polynomial are called the terms of the polynomial.
A polynomial with three monomial terms is a trinomial.
A polynomial with two monomial terms is a binomial.
The degree of a polynomial is the degree of the monomial term with the largest degree.
Polynomials can be added or subtracted by combining like terms.
Polynomial
Like Terms
A monomial or a sum of monomials
Terms that have the same variable(s) raised to the same power(s)
Example:
(2𝑥 2 + 3𝑥 − 4) + ( −3𝑥 3 + 4𝑥 2 − 6) =
Use associative property to group like terms
−3𝑥 3 + 2𝑥 2 + 4𝑥 2 + 3𝑥 − 4 − 6 =
Combine like terms using Addition/Subtraction
−3𝑥 3 + 6𝑥 2 + 3𝑥 − 10
Substitute simplified polynomial sum
Example:
(3𝑦 3 + 4𝑦 − 6) − (−4𝑦 4 + 2𝑦 3 − 5𝑦 2 + 2𝑦 − 8) = Change signs for subtraction and group
4𝑦 4 + 3𝑦 3 − 2𝑦 3 + 5𝑦 2 + 4𝑦 − 2𝑦 − 6 + 8 =
Combine like terms
4𝑦 4 + 𝑦 3 + 5𝑦 2 + 2𝑦 + 2
Substitute simplified polynomial sum
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Polynomials can be multiplied by applying the distributive property or if you are
multiplying two binomials, the FOIL method.
FOIL Method of
Multiplying
Polynomials
The product of two binomials is the sum of the products of
F: the first terms.
O: the outer terms.
I: the inner terms.
L: the last terms.
Example:
(𝑥 2 + 2𝑥 + 3)(𝑥 − 4) =
Rewrite using distribution
𝑥(𝑥 2 + 2𝑥 + 3) − 4(𝑥 2 + 2𝑥 + 3) =
Distribute the terms of the binomial to the trinomial
𝑥(𝑥 2 ) + 𝑥(2𝑥) + 𝑥(3) + (−4)𝑥 2 + (−4)2𝑥 + (−4)3 = Simplify
𝑥 3 + 2𝑥 2 + 3𝑥 − 4𝑥 2 − 8𝑥 − 12 =
Substitute simplified polynomial
𝑥 3 + 2𝑥 2 − 4𝑥 2 + 3𝑥 − 8𝑥 − 12 =
Group like terms using the associative property
𝑥 3 − 2𝑥 2 − 5𝑥 − 12
Substitute simplified polynomial sum
Division of Polynomials:

To divide a polynomial by a monomial, use the properties of exponents for quotients.

To divide a polynomial by a polynomial, use a long division pattern. Remember that
only like terms can be added or subtracted.
If numerator is missing a term in descending order by degree, it must be replaced by
zero to use long division.
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Synthetic Division:
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Synthetic division can also be used if the divisor is a binomial and the coefficient of the
variable term is one and the exponent on the variable is one. Otherwise long division
must be used.
If the dividend is missing a term in descending order then zero must replace that term
as a place holder.
The following is an example of synthetic division.
Pascal’s Triangle:

Pascal’s triangle can be use for binomial expansion. The triangle follows the following
pattern.