Unit 5-1: I will multiply, divide and simplify monomials and expressions involving powers so I can add,
subtract and multiply polynomials.
Unit 5-1 Guided Notes
When dealing with polynomials, we must always keep in mind that we are simplifying an expression
every time. To simplify an expression containing powers means to rewrite the expression without
parentheses or negative exponents.
Before we start we will need to review some properties of Exponents. Go to Page 303 and fill in the
following table. When filling in the information, make sure you stop and ask yourself “Why?” or “How?”.
Look at the examples that are given for each property.
Properties of Exponents
Definition
Property
Examples
Product of Powers
Quotient of Powers
Negative Exponent
Power of a Power
Power of a Product
Power of a Quotient
Zero Power
Note: We always go ahead and simplify numbers with exponents. For example, 34 = 81.
Simplifying Monomials- A monomial (a number, variable or an expression) is in simplified form when:




There are no powers of powers
Each base appears exactly once
All fractions are in simplest form, and
There are no negative exponents
Unit 5-1: I will multiply, divide and simplify monomials and expressions involving powers so I can add,
subtract and multiply polynomials.
Simplify each expression. Assume that no variable equals 0. (More examples on page 304)
Example 1.
(2𝑥 −3 𝑦 3 )(−7𝑥 5 𝑦 −6 )
Solution:
(2𝑥 −3 𝑦 3 )(−7𝑥 5 𝑦 −6 )
= −14𝑥 2 𝑦 −3
= −
14𝑥 2
−14𝑥 2
𝑜𝑟
𝑦3
𝑦3
Example 2:
15𝑐 5 𝑑3
−3𝑐 2𝑑
7
Solution:
15𝑐 5 𝑑3
−3𝑐 2𝑑
7
= −5𝑐 3 𝑑−4
=−
5𝑐 3
−5𝑐 3
𝑜𝑟
𝑑4
𝑑4
Example3:
(−2𝑥 3 𝑦 2 )5
Solution:
(−2𝑥 3 𝑦 2 )5
= (−2)5 𝑥 15 𝑦10
= −32𝑥 15 𝑦10
NOTE: Since we are talking about polynomials, remember that polynomials are composed of one or
more monomials.
Polynomials
3𝑥 2 + 6
Non-Polynomials
2
+5
𝑥
𝑥 4 𝑦 3 − 3𝑥 3 𝑦 2 + 10𝑥𝑦 3 + 𝑦 4 − 2
1 3 𝑥3
𝑥 +
5
4
√𝑥 + 𝑥 − 3
𝑥 −3 − 2𝑦 3
Unit 5-1: I will multiply, divide and simplify monomials and expressions involving powers so I can add,
subtract and multiply polynomials.
The degree of a polynomial is the degree of the monomial with the greatest degree.
3
4
For example, let the following polynomial be given −16𝑝5 + 𝑝2 𝑡 2 . The degree of the first term is 5
and the second is 2+2 = 4. So the degree of this polynomial is 5.
Simplify Polynomial Expressions (More examples 305-306)
Example:
(−𝑥 2 − 3𝑥 + 4) − (𝑥 2 + 2𝑥 + 5)
Solution:
(−𝑥 2 − 3𝑥 + 4) − (𝑥 2 + 2𝑥 + 5)
= −𝑥 2 − 3𝑥 + 4 − 𝑥 2 − 2𝑥 − 5
= −2𝑥 2 − 5𝑥 − 1
Example:
(𝑥 2 + 4𝑥 + 16)(𝑥 − 4)
Solution
(𝑥 2 + 4𝑥 + 16)(𝑥 − 4)
= 𝑥 2 (𝑥 − 4) + 4𝑥(𝑥 − 4) + 16(𝑥 − 4)
= 𝑥 3 − 4𝑥 2 + 4𝑥 2 − 16𝑥 + 16𝑥 − 64
= 𝑥 3 − 64
(𝑥 − 4)(𝑥 2 + 4𝑥 + 16)
Or
= 𝑥(𝑥 2 + 4𝑥 + 16) − 4(𝑥 2 + 4𝑥 + 16)
= 𝑥 3 + 4𝑥 2 + 16𝑥 − 4𝑥 2 − 16𝑥 − 64
= 𝑥 3 − 64
Assignment:
Pg. 307
#16-39 odd, #40, #66, #68