1. Objectives Identify, evaluate, add, and subtract polynomials.
Classify and graph polynomials.

2. Define the following (from the back of the textbook):
monomial
polynomial
degree of a monomial
degree of a polynomial
leading coefficient
binomial
trinomial
polynomial function

3. A monomial is a number or a product of numbers and variables with whole number exponents. A polynomial is a monomial or a sum or difference of monomials.
Polynomials have no variables in denominators or exponents, no roots or absolute values of variables, and all variables have whole number exponents. 1 2 a7
Polynomials:
3x4
2z12 + 9z3
0.15x101
3t2 – t3 8
5y2 1 2
Not polynomials: 3x
|2b3 – 6b|
m0.75 – m
The degree of a monomial is the sum of the exponents of the variables.

4. Example 1: Identifying the Degree of a Monomial
Identify the degree of each monomial.
A. z6
B. 5.6
C. 8xy3
D. a2bc3

5. Degree of a polynomial is given by the term with the greatest degree.
Standard Form Terms are written in descending order by degree.
-The degree of the first term indicates the
degree of the polynomial.
- Leading coefficient is the coefficient of
the first term.

6. A polynomial can be classified by its number of terms:
A polynomial with one term is a monomial.
A polynomial with two terms is called a binomial.
A polynomial with three terms is called a trinomial.

7. A polynomial can also be classified by its degree:

8. Example 2: Classifying Polynomials
Rewrite each polynomial in standard form. Then identify the leading coefficient, degree, and number of terms. Name the polynomial.
A. 3 – 5x2 + 4x
B. 3x2 – 4 + 8x4
Leading coefficient:
Leading coefficient:
Degree:
Degree:
Terms:
Terms:
Name:
Name:

9. Your Turn: Example 2
Rewrite each polynomial in standard form. Then identify the leading coefficient, degree, and number of terms. Name the polynomial.
c. 4x – 2x2 + 2
d. –18x2 + x3 – 5 + 2x
Leading coefficient:
Leading coefficient:
Degree:
Degree:
Terms:
Terms:
Name:
Name:

10. Exit Slip/Homework: Find the degree of the monomials.
1. 4 𝑎 2 𝑏 𝑐 𝑥 3
Find the degree of the polynomials.
3. 3𝑥 2 +4𝑥− 𝑥 6 −5 𝑥 8 +4 𝑥 7
Write in standard form.
5. 2𝑥− 4𝑥 4 −10+3 𝑥 2 −6 𝑥 5

11. To ADD or SUBTRACT polynomials, combine like terms!!

12. Example 3: Adding and Subtracting Polynomials
Add or subtract. Write your answer in standard form.
A. (2x3 + 9 – x) + (5x x + x3)

13. Example 3: Adding and Subtracting Polynomials
Add or subtract. Write your answer in standard form.
B. (3 – 2x2) – (x2 + 6 – x)

14. Your Turn: Example 3a
Add or subtract. Write your answer in standard form.
c. (–36x2 + 6x – 11) + (6x2 + 16x3 – 5)

15. Your Turn: Example 3b
Add or subtract. Write your answer in standard form.
d. (5x x2) – (15x2 + 3x – 2)

16. Exit Slip/Homework 1. 3 𝑥 2 +2𝑥−5 +( 6𝑥 2 −5𝑥+7)
1. 3 𝑥 2 +2𝑥−5 +( 6𝑥 2 −5𝑥+7)
2. 𝑥 5 + 4𝑥 3 +8𝑥−5 +( 4𝑥 4 +2 𝑥 3 −2𝑥−1)
3. 6𝑥−5 −( 3𝑥 2 +𝑥−3)
4. 5 𝑥 3 −7 𝑥 2 +3𝑥+9 −( 2𝑥 3 +4 𝑥 2 −𝑥)

17. 6-2 Multiplying Polynomials
Warm Up:
Use the properties of exponents to simplify
the following terms.
𝑎 4 ∙ 𝑎 5
(𝑓 3 ) 5
( 𝑝 3 𝑞 𝑟 4 ) 2

18. Objectives 6-2 Multiplying Polynomials Multiply polynomials.
Use binomial expansion to expand binomial expressions that are raised to positive integer powers.

19. 6-2 Multiplying Polynomials
To multiply a polynomial by a monomial, use the Distributive Property and the Properties of Exponents.

20. Example 1: Multiplying a Monomial and a Polynomial
6-2 Multiplying Polynomials
Example 1: Multiplying a Monomial and a Polynomial
Find each product.
A. 4y2(y2 + 3)
B. fg(f4 + 2f3g – 3f2g2 + fg3)

21. 6-2 Multiplying Polynomials
Your Turn: Example 1
Find each product.
a. 3cd2(4c2d – 6cd + 14cd2)
b. x2y(6y3 + y2 – 28y + 30)

22. Example 2A: Multiplying Polynomials
Find the product.
(a – 3)(2 – 5a + a2)

23. Example 2B: Multiplying Polynomials
Find the product.
(y2 – 7y + 5)(y2 – y – 3)
y –y –3 y2
–7y 5 y4
–y3
–3y2
–7y3
7y2
21y
5y2
–5y
–15

24. 6-2 Multiplying Polynomials
Check It Out! Example 2a
Find the product.
(3b – 2c)(3b2 – bc – 2c2)

25. 6-2 Multiplying Polynomials
Check It Out! Example 2b
Find the product.
(x2 – 4x + 1)(x2 + 5x – 2)
x –4x x2 5x –2 x4
–4x3 x2
5x3
–20x2 5x
–2x2 8x –2

26. Warm Up
Write the polynomial in standard form. Then identify the leading coefficient, the degree and number of terms. 2𝑥− 4𝑥 4 −10+3 𝑥 2 −6 𝑥 5
a. standard form: ______________
b. leading coefficient: ___________
c. degree of polynomial: _________
d. number of terms: ____________
e. name: _____________________
2. 6𝑥−5 − 3𝑥 2 +𝑥−3
3. ( 2𝑥 2 −4𝑥+3)( 𝑥 2 −3𝑥+7)

27. 6-2 Multiplying Polynomials
Check It Out! Example 4a
Find the product.
(x + 4)4

28. 6-2 Multiplying Polynomials
Check It Out! Example 4b
Find the product.
(2x – 1)3
8x3 – 12x2 + 6x – 1

29. Example 5: Using Pascal’s Triangle to Expand Binomial Expressions
6-2 Multiplying Polynomials
Example 5: Using Pascal’s Triangle to Expand Binomial Expressions
Expand each expression.
A. (k – 5)3
B. (6m – 8)3

30. 6-2 Multiplying Polynomials
Check It Out! Example 5
Expand each expression.
a. (x + 2)3
x3 + 6x2 + 12x + 8
b. (x – 4)5
x5 – 20x x3 – 640x x – 1024

31. 6-2 Multiplying Polynomials
Check It Out! Example 5
Expand the expression.
c. (3x + 1)4
81x x3 + 54x2 + 12x + 1

32. Exit Slip/Homework: Find the sum/difference. Find the product.
1. 2 𝑥 2 +4𝑥−6 +(3𝑥+7)
2. 𝑥 5 −3 𝑥 3 +2𝑥 −(7 𝑥 4 +5 𝑥 3 −10)
Find the product.
3. 5𝑥 3 ( 2𝑥 2 +3𝑥−1)
4. ( 2𝑥 2 −4𝑥+3)( 𝑥 2 −3𝑥+7)

33. 6-2 Multiplying Polynomials
Lesson Quiz
Find each product.
1. 5jk(k – 2j)
5jk2 – 10j2k
2. (2a3 – a + 3)(a2 + 3a – 5)
2a5 + 6a4 – 11a3 + 14a – 15
3. The number of items is modeled by
0.3x x + 2, and the cost per item is modeled by g(x) = –0.1x2 – 0.3x + 5. Write a polynomial c(x) that can be used to model the total cost.
–0.03x4 – 0.1x x2 – 0.1x + 10
4. Find the product.
(y – 5)4
y4 – 20y y2 – 500y + 625
5. Expand the expression.
(3a – b)3
27a3 – 27a2b + 9ab2 – b3