1. Introduction to Polynomials
Keeper 9
Honors Algebra II

2. What is a Polynomial???
A sum or difference of terms that can contain variables with whole number exponents.

3. Polynomials can be classified in two ways:
By Term
By Degree
Terms are the “chunks of numbers and/or variables separated by the + or – signs.
The degree of a polynomial is the largest exponent on a variable.

4. Classifying by Terms… Since polynomials consist of two or more terms,
a binomial is a polynomial
a trinomial is a polynomial

5. Classifying by Degree..
For degrees of 5 or higher, we just say ___ degree or degree of ___.

6. Does order matter? Leading Coefficient
Yes! We always write polynomials in DESCENDING ORDER.
This means from LARGEST exponent to SMALLEST exponent.
This is called "standard form."
Leading Coefficient
When a polynomial is written in standard form, the number in front of the first term is called the LEADING COEFFICIENT.

7. Classify the Polynomial
𝑓 𝑥 =3 𝑥 𝑥 5 − 1 6 𝑥+ 2 𝑥 7 Terms: Name Based on Terms: Degree: Lead Coefficient:

8. Classify the Polynomial
𝑓 𝑥 =𝜋 𝑥 4 +3 𝑥 3 − 1 3 𝑥 Terms: Name Based on Terms: Degree: Lead Coefficient:

9. Classify the Polynomial
𝑓 𝑥 =5 Terms: Name Based on Terms: Degree: Lead Coefficient:

10. Classify the Polynomial
𝑓 𝑥 = 2 𝑥−1 −3 𝑥 2.5 Terms: Name Based on Terms: Degree: Lead Coefficient:

11. Classify the Polynomial
𝑓 𝑥 = 2𝑥− Terms: Name Based on Terms: Degree: Lead Coefficient:

12. Classify the Polynomial
𝑓 𝑥 =2𝑥𝑦 𝑧 2 −3 𝑥 2 𝑦 𝑧 3 +6 𝑥 4 𝑦 7 𝑧−8𝑥+6𝑦 Terms: Name Based on Terms: Degree: Lead Coefficient:

13. Operations with Polynomials Adding Polynomials – Combine Like Terms Subtracting Polynomials – Distribute the Negative Then Combine Like Terms Multiplying Polynomials - Multiply the coefficients and add the exponents

14. Example: 𝑓 𝑥 =2 𝑥 4 −3 𝑥 3 +𝑥−1 𝑎𝑛𝑑 𝑔 𝑥 =2 𝑥 2 +4
𝑓 𝑥 +𝑔(𝑥)

15. Example: 𝑓 𝑥 =2 𝑥 4 −3 𝑥 3 +𝑥−1 𝑎𝑛𝑑 𝑔 𝑥 =2 𝑥 2 +4
𝑓 𝑥 −𝑔(𝑥)

16. Example: 𝑓 𝑥 =2 𝑥 4 −3 𝑥 3 +𝑥−1 𝑎𝑛𝑑 𝑔 𝑥 =2 𝑥 2 +4
3𝑓 𝑥 −2𝑔(𝑥)

17. Example: 𝑓 𝑥 =2 𝑥 4 −3 𝑥 3 +𝑥−1 𝑎𝑛𝑑 𝑔 𝑥 =2 𝑥 2 +4
𝑓 𝑥 ⋅𝑔(𝑥)

18. Example: 𝑓 𝑥 =2 𝑥 4 −3 𝑥 3 +𝑥−1 𝑎𝑛𝑑 𝑔 𝑥 =2 𝑥 2 +4
𝑓 2 +𝑔(1)

19. Example: 𝑓 𝑥 =2 𝑥 4 −3 𝑥 3 +𝑥−1 𝑎𝑛𝑑 𝑔 𝑥 =2 𝑥 2 +4
𝑓 1 𝑓 3 +𝑔(2)

20. Example: 𝑓 𝑥 =2 𝑥 4 −3 𝑥 3 +𝑥−1 𝑎𝑛𝑑 𝑔 𝑥 =2 𝑥 2 +4
𝑓 2 −𝑔 1 2

21. Graphs of Polynomials
Smooth Continuous Curve – Domain: ℝ (−∞, ∞) No Discontinuities: No Jumps No Holes No Asymptotes No Corners

22. Example: Does the Graph Represent a Polynomial???

23. Example: Does the Graph Represent a Polynomial???

24. Example: Does the Graph Represent a Polynomial???

25. Example: Does the Graph Represent a Polynomial???

26. Example: Does the Graph Represent a Polynomial???

27. Example: Does the Graph Represent a Polynomial???

28. Example: Does the Graph Represent a Polynomial???

29. Example: Does the Graph Represent a Polynomial???