This document covers the fundamentals of polynomials, including definitions, degree, leading coefficients, and types of polynomials such as monomials, binomials, and trinomials. It details operations on polynomials, including addition, subtraction, and multiplication, along with special product patterns and methods like the 'box' method for simplifying polynomial expressions. Additionally, it highlights working with polynomials of multiple variables, emphasizing the importance of combining like terms.

1. 1.4 Polynomials
Chapter 1 Prerequisites

2. Concepts & Objectives
⚫ Objectives of this section:
⚫ Identify the degree and leading coefficient of
polynomials.
⚫ Add and subtract polynomials.
⚫ Multiply polynomials.
⚫ Identify special product patterns.
⚫ Perform operations with polynomials of several
variables.

3. Polynomials
⚫ A polynomial is defined as a term or a finite sum of
terms, with only positive or zero integer exponents
permitted on the variables.
⚫ The degree of a term with one variable is the largest
exponent in the expression. The degree of a term
containing more than one variable has degree equal to
the sum of all of the variables’ exponents.
⚫ The degree of the polynomial is the greatest degree of
any term in the polynomial.

4. Polynomials (cont.)
⚫ A polynomial containing exactly three term is called a
trinomial. If it contains exactly two terms, it is called a
binomial, and a single-term polynomial is called a
monomial.
⚫ Since the variables used in polynomials represent real
numbers, a polynomial represents a real number. This
means that all of the properties of real numbers hold for
polynomials.

5. Polynomials (cont.)
⚫ Using the distributive property, we can simplify the
following:
3m5 ‒ 7m5 + 2m2
3 ‒ 7m5 + 2m2
‒4m5 + 2m2
⚫ Thus, polynomials are added/subtracted by adding or
subtracting coefficients of like terms.
⚫ Polynomials in one variable are usually written with
their terms in descending order of degree.

6. Multiplying Polynomials
⚫ To multiply polynomials, multiply each term by all of the
other terms.
Example: ( )( )
( )( ) ( )( ) ( )( )
( )( ) ( )( ) ( )( )
2
2
2
3 2 2
3 2
2 3 5
2 3 5
2 3 5
6 9 15 8 12 20
6 17 27 20
3
3 3
4
4 4 4
3
x x
x x
x x
x x x x x
x x
x
x
x
x x
− +
= + − +
+ + − +
= − + − + −
−
− +
−
−
−
−
=

7. Multiplying Polynomials
⚫ You can also use tools such as the “box” method:
2x2 ‒3x 5
3x
‒4

8. Multiplying Polynomials
⚫ You can also use tools such as the “box” method:
2x2 ‒3x 5
3x 6x3 ‒9x2 15x
‒4

9. Multiplying Polynomials
⚫ You can also use tools such as the “box” method:
2x2 ‒3x 5
3x 6x3 ‒9x2 15x
‒4 ‒8x2 12x ‒20

10. Multiplying Polynomials
⚫ You can also use tools such as the “box” method:
Combine like terms:
2x2 ‒3x 5
3x 6x3 ‒9x2 15x
‒4 ‒8x2 12x ‒20
3 2
6 17 27 20
x x x
− + −

11. Multiplying Polynomials
⚫ There are a couple of special patterns to be aware of
when multiplying two binomials:
⚫ Product of Sum and Difference/Difference of Squares
⚫ Square of a Binomial
( )( ) 2 2
a b a b a b
+ − = −
( )
 
= +
2 2 2
2
a b a ab b

12. Multiplying Polynomials
Examples: Simplify the following
⚫
⚫
( )( )
2 3 2 3
x x
+ −
( )
2
7 5
x −

13. Multiplying Polynomials
Examples: Simplify the following
⚫
⚫
( )( ) ( )
+ − = −
= −
2 2
2
2 3 2 3 2 3
4 9
x x x
x
( ) ( ) ( )( )
− = − +
= − +
2 2 2
2
7 5 7 2 7 5 5
49 70 25
x x x
x x

14. Multiplying Polynomials
EVERY TIME YOU DO THIS:
A KITTEN DIES
( )
2 2 2
x y x y
+ = +
Fair Warning!

15. Polynomials of Several Variables
⚫ A polynomial can contain several variables. All of the
same rules apply when working with polynomials
containing several variables.
⚫ When adding/subtracting, remember that you can only
combine terms with like variables and powers.
⚫ Example: cannot be simplified any
further because there are no other like terms.
2
4 2 3
a ab b
− +

16. Polynomials of Several Variables
Example: Multiply
The only terms we can combine are 12x + 5x, so the
eventual answer is
( )( )
4 3 2 5
x x y
+ − +
3x ‒2y 5
x 3x2 ‒2xy 5x
4 12x ‒8y 20
2
3 2 17 8 20
x xy x y
− + − +

17. Classwork
⚫ College Algebra 2e
⚫ 1.4: 6-16 (even); 1.3: 36-50 (even); 1.2: 44-58 (even)
(omit 52)
⚫ 1.4 Classwork Check
⚫ Quiz 1.3