1. Polynomials
Sec 9.1.1

2. Learning Targets Vocabulary Operations between polynomials
Introduction to graphs of polynomials

3. Definitions
Polynomial comes from poly- (meaning "many") and -nomial (in this case meaning "term") ... so it means “many terms”
Term: A number, a variable, or the product/quotient of numbers/variables.

4. Polynomial
Example of Polynomial
𝑦= 5𝑥 𝑦 2 − 19𝑥𝑦 3 −7
Terms

5. 5 𝑥 3 A Term has 3 Components:
Exponent:
Can only be positive integers: 0,1,2, 3,
Coefficient: can be any real number… including zero.
Variable
These components are very important!!!

6. NOT ALLOWED Negative exponents: 𝑥 −2 Variables in the denominator:
4 3−𝑥

7. Check In Which of the following is a polynomial: −6𝑦2 − 7 9 𝑥
−6𝑦2 − 𝑥
4𝑥− 𝑦+ 𝑥𝑦 −3 −41𝑥𝑦
3𝑥𝑦𝑧 + 3𝑥 𝑦2𝑧 − 0.1𝑥𝑧 − 200𝑦 + 0.5 5

8. Naming a Polynomial 7 5x + 2 4x2 + 3x - 4 6x3 - 18 1 2 3 monomial
We can classify a polynomial based on how many terms it has:
Polynomial 7
5x + 2
4x2 + 3x - 4
6x3 - 18
# Terms 1 2 3
# Terms Name
monomial
binomial
trinomial

9. Naming Cont.
Quadrinomial (4 term) and quintinomial (5 term) also exist, but those names are not often used.
Polynomials Can Have Lots and Lots of Terms
Polynomials can have as many terms as needed, but not an infinite number of terms.
For more than 3 terms say:
“a polynomial with n terms” or “an n-term polynomial”
11x8 + x5 + x4 - 3x3 + 5x2 - 3
“a polynomial with 6 terms” – or – “a 6-term polynomial”

10. The degree of a term is determined by the exponent of the variable.3 4x
-5x2
18x5
Degree of Term 1 2 5
New vocabulary!
It is a simple concept. It is just the exponent of the variable. You can find the degree of multiply variable terms, but we don’t deal with them in algebra 1.
Now these degrees are very important. They tell you which family of functions a term belongs to and will have a big impact on polynomials.

11. Naming a Polynomial 7 5x + 2 4x2 + 3x - 4 6x3 - 18 1 2 3 Constant
We can also classify a polynomial based on its highest degree:
Polynomial 7
5x + 2
4x2 + 3x - 4
6x3 - 18
Degree 1 2 3
# Degree Name
Constant
Linear
Quadratic
Cubic

12. Putting it All Together
Polynomial
-14x3
-1.2x2 -1
7x - 2
3x3+ 2x - 8
2x2 - 4x + 8
x4 + 3
Name
cubic monomial
quadratic monomial
constant monomial
linear binomial
cubic trinomial
quadratic trinomial
4th degree binomial

13. Standard Form of a Polynomial
A polynomial written so that the degree of the terms decreases from left to right and no terms have the same degree.
Very much common sense. Basically Standard form is simplified (like terms are combined), you just have to make sure to write the terms in the correct order. The highest degree comes first… the rest follow in order of decreasing degree.

14. Not Standard 6x + 3x2 - 2 15 - 3x - x+ 5x4 x + 10 + x 1 + x2 + x + x3
Examples

15. Operations
Polynomials can be added, subtracted, multiplied and/or divided
The following slides will cover addition, subtraction and multiplication
We will learn about division later on in the unit

16. Adding and Subtracting Polynomials
To add or subtract polynomials, simply combine like terms.
(5x2 - 3x + 7) + (2x2 + 5x - 7)
= 7x2 + 2x
(3x3 + 6x - 8) + (4x2 + 2x - 5)
= 3x3 + 4x2 + 8x - 13
So, nothing really new here. We’ve been doing this for a long while. I’m just going to add more terms than you are used to and terms of higher degree.
(2x3 + 4x2 - 6) – (3x3 + 2x - 2)
(2x3 + 4x2 - 6) + (-3x3 + -2x - -2)
= -x3 + 4x2 - 2x - 4

17. Polynomial Multiplication
To multiply polynomials we must distribute all of the terms
Ex: 𝑥 3 + 4𝑥 2 +1 × − 3𝑥 2 −2𝑥
𝑥 3 + 4𝑥 2 +1
−3 𝑥 2
−3 𝑥 5 −12 𝑥 4 −3 𝑥 2
Combine Like Terms!
−2𝑥
−2 𝑥 4 −8 𝑥 3 −2𝑥
−3𝑥 5 − 14𝑥 4 − 8𝑥 3 − 3𝑥 2 −2𝑥

18. Polynomial Multiplication
Multiply the following polynomials:

19. Polynomial Multiplication
(x + 5)
x (2x + -1)
-x + -5
2x2 + 10x +
2x2 + 9x + -5
(3w + -2)
x (2w + -5)
-15w + 10 +
6w2 + -4w
6w w + 10

20. Polynomial Multiplication
(2a2 + a + -1)
x (2a2 + 1)
2a2 + a + -1 +
4a4 + 2a3 + -2a2
4a4 + 2a3 + a + -1

21. Investigating Graphs of Polynomials
Pg. 437
In your notes go through problem 9-1 silently…
Write down any conjectures, similarities or patterns you see
After 5 minutes we will discuss in our teams

22. For Tonight Homework: Pg. 440: 9-8 9-11, 9-13, 9-14 and 9-18
Answers to these questions will be posted online tonight