1. Polynomials and Polynomial Functions
Definitions
Term: a number or a product of a number and variables raised
to a power.
Coefficient: the numerical factor of each term.
Constant: the term without a variable.
Polynomial: a finite sum of terms of the form axn, where a is a
real number and n is a whole number.

2. Polynomials and Polynomial Functions
Definitions
Monomial: a polynomial with exactly one term.
Binomial: a polynomial with exactly two terms.
Trinomial: a polynomial with exactly three terms.

3. A polynomial is a monomial or a sum or difference of monomials.
The degree of a polynomial is the degree of the term with the greatest degree.

4. Example 1: Finding the Degree of a Monomial
Find the degree of each monomial. A.
4p4q3
The degree is 7.
Add the exponents of the variables: = 7.
B. 7ed
The degree is 2.
Add the exponents of the variables: 1+ 1 = 2.
C. 3
The degree is 0.
Add the exponents of the variables: 0 = 0.

5. Example 2: Finding the Degree of a Polynomial
Find the degree of each polynomial.
A. 11x7 + 3x3
11x7: degree 7 3x3: degree 3
Find the degree of each term.
The degree of the polynomial is the greatest degree, 7. B.
:degree 3
:degree 4
–5: degree 0
Find the degree of each term.
The degree of the polynomial is the greatest degree, 4.

6. Check It Out! Example 2
Find the degree of each polynomial.
a. 5x – 6
5x: degree 1
–6: degree 0
Find the degree of each term.
The degree of the polynomial is the greatest degree, 1.
b. x3y2 + x2y3 – x4 + 2
Find the degree of each term.
x3y2: degree 5
x2y3: degree 5
–x4: degree 4
2: degree 0
The degree of the polynomial is the greatest degree, 5.

7. Polynomials and Polynomial Functions
Practice Problems
Identify the degrees of each term and the degree of the polynomial.
4 𝑎 2 𝑏 4 +3 𝑎 3 𝑏 5 −9 𝑏 4 +4 8 8
4 𝑥 5 𝑦 4 +5 𝑥 4 𝑦 6 −6 𝑥 3 𝑦 3 +2𝑥𝑦 10 10

8. Example 4: Classifying Polynomials
Classify each polynomial according to its degree and number of terms.
A. 5n3 + 4n
5n3 + 4n is a cubic binomial.
Degree 3 Terms 2
B. 4y6 – 5y3 + 2y – 9
4y6 – 5y3 + 2y – 9 is a
6th-degree polynomial.
Degree 6 Terms 4
C. –2x
–2x is a linear monomial.
Degree 1 Terms 1

9. Polynomials and Polynomial Functions
Practice Problems
Evaluate each polynomial function

10. Example 1: Adding and Subtracting Monomials
Add or subtract.
A. 12p3 + 11p2 + 8p3
12p3 + 11p2 + 8p3
Identify like terms.
Rearrange terms so that like terms are together.
12p3 + 8p3 + 11p2
20p3 + 11p2
Combine like terms.
B. 5x2 – 6 – 3x + 8
Identify like terms.
5x2 – 6 – 3x + 8
Rearrange terms so that like terms are together.
5x2 – 3x + 8 – 6
5x2 – 3x + 2
Combine like terms.

11. Example 1: Adding and Subtracting Monomials
Add or subtract.
C. t2 + 2s2 – 4t2 – s2
t2 + 2s2 – 4t2 – s2
Identify like terms.
Rearrange terms so that like terms are together.
t2 – 4t2 + 2s2 – s2
–3t2 + s2
Combine like terms.
D. 10m2n + 4m2n – 8m2n
10m2n + 4m2n – 8m2n
Identify like terms.
6m2n
Combine like terms.

12. Check It Out! Example 1
Add or subtract.
a. 2s2 + 3s2 + s
2s2 + 3s2 + s
Identify like terms.
5s2 + s
Combine like terms.
b. 4z4 – z4 + 2
4z4 – z4 + 2
Identify like terms.
Rearrange terms so that like terms are together.
4z4 + 16z4 – 8 + 2
20z4 – 6
Combine like terms.

13. Check It Out! Example 1
Add or subtract.
c. 2x8 + 7y8 – x8 – y8
Identify like terms.
2x8 + 7y8 – x8 – y8
Rearrange terms so that like terms are together.
2x8 – x8 + 7y8 – y8
x8 + 6y8
Combine like terms.
d. 9b3c2 + 5b3c2 – 13b3c2
9b3c2 + 5b3c2 – 13b3c2
Identify like terms.
b3c2
Combine like terms.

14. Example 2: Adding Polynomials
A. (4m2 + 5) + (m2 – m + 6)
(4m2 + 5) + (m2 – m + 6)
Identify like terms.
Group like terms together.
(4m2 + m2) + (–m) +(5 + 6)
5m2 – m + 11
Combine like terms.
B. (10xy + x) + (–3xy + y)
(10xy + x) + (–3xy + y)
Identify like terms.
Group like terms together.
(10xy – 3xy) + x + y
7xy + x + y
Combine like terms.

15. Example 2C: Adding Polynomials
(6x2 – 4y) + (3x2 + 3y – 8x2 – 2y)
(6x2 – 4y) + (3x2 + 3y – 8x2 – 2y)
Identify like terms.
Combine like terms in the second polynomial.
(6x2 – 4y) + (–5x2 + y)
(6x2 –5x2) + (–4y + y)
Combine like terms.
x2 – 3y
Simplify.

16. Example 2D: Adding Polynomials
Identify like terms.
Group like terms together.
Combine like terms.

17. Check It Out! Example 2
Add (5a3 + 3a2 – 6a + 12a2) + (7a3 – 10a).
(5a3 + 3a2 – 6a + 12a2) + (7a3 – 10a)
Identify like terms.
(5a3 + 7a3) + (3a2 + 12a2) + (–10a – 6a)
Group like terms together.
12a3 + 15a2 – 16a
Combine like terms.

18. Lesson Quiz: Part I
Add or subtract.
1. 7m2 + 3m + 4m2
2. (r2 + s2) – (5r2 + 4s2)
3. (10pq + 3p) + (2pq – 5p + 6pq)
4. (14d2 – 8) + (6d2 – 2d +1)
11m2 + 3m
(–4r2 – 3s2)
18pq – 2p
20d2 – 2d – 7
5. (2.5ab + 14b) – (–1.5ab + 4b)
4ab + 10b

19. Polynomials and Polynomial Functions
Multiplication
Multiplying Monomials by Monomials
Examples:

20. Polynomials and Polynomial Functions
Multiplication
Multiplying Monomials by Polynomials
Examples:

21. 5.4 – Polynomials and Polynomial Functions
Multiplication
Multiplying Two Polynomials
Examples:

22. 5.4 – Multiplying Polynomials
Special Products
Multiplying Two Binomials using FOIL
First terms
Outer terms
Inner terms
Last terms

23. 5.4 – Multiplying Polynomials
Special Products
Multiplying Two Binomials using FOIL
First terms
Outer terms
Inner terms
Last terms

24. 5.4 – Multiplying Polynomials
Special Products
Squaring Binomials

25. 5.4 – Multiplying Polynomials
Special Products
Multiplying the Sum and Difference of Two Binomials

26. 5.4 – Multiplying Polynomials
Special Products
Dividing by a Monomial

27. Extra Problems

28. 5.4 – Polynomials and Polynomial Functions
Multiplication
Multiplying Two Polynomials
Examples:

29. 5.4 – Polynomials and Polynomial Functions
Multiplication
Multiplying Two Polynomials
Examples:

30. 5.4 – Polynomials and Polynomial Functions
Multiplication
Multiplying Two Polynomials
Examples: