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Algebra I – Chapter 8 Day #1
Topic: Polynomials
Standards/Goals:
 A.APR.1./C.1.d.:
o I can determine the degree of a polynomial
o I can write a polynomial in standard form
o I can combine polynomials using addition and/or subtraction.
Today, we focus on learning about what a polynomial is and how to identify one.
Definitions:
1. Polynomial – is a monomial or a SUM of monomials.
Example:
Not an example:
2. Binomial – is the sum of TWO monomials.
Example:
3. Trinomial – is the sum of THREE monomials.
Example:
NOTE: When a polynomial has THREE OR MORE terms, it does NOT have a special name.
EXAMPLES:
State whether each expression is a polynomial. If the expression is a polynomial, identify it as a
MONOMIAL, BINOMIAL, or TRINOMIAL.
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#1. 36
#2. 𝑞2 + 5
#3. 7x – x + 5
#4. 8𝑔2 ℎ - 7gh + 2
#5.
1
4𝑦 2
+ 5𝑦 − 8
#6. 6x + 𝑥 2
#7. 7𝑎2 𝑏+ 3𝑏 2 − 𝑎2 𝑏
#8. 6𝑔2 ℎ3 𝑘
Definitions:
4. Degree – is the sum of the EXPONENTS of all its variables.
Example:
5. Degree of a polynomial – is the same as the degree of the monomial with the highest degree.
Example:
EXAMPLES: Find the degree of each polynomial.
#1. -2abc
#2. 15m
#3. s + 5t
#4. 22
#5. 4𝑥 2 𝑦 3 𝑧
#6. 18𝑥 2 + 4yz - 10y
#7. 𝑥 4 − 6𝑥 2 − 2𝑥 3 − 10
#8. 2𝑥 3 𝑦 2 − 4𝑥𝑦 3
#9. 9𝑥 2 + 𝑦𝑧 8
#10. −2𝑟 8 𝑠 4 + 7𝑟 2 𝑠 − 4𝑟 7 𝑠 6
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We also want to learn how to write a polynomial in both descending order (standard form)
and ascending order, as well.
EXAMPLES: Arrange the terms of each polynomial so that the powers of x are in descending
order. Descending order is typically what we consider: Standard Form.
#1. 2𝑥 + 𝑥 2 − 5
#2. 20𝑥 − 10𝑥 2 + 5𝑥 3
#3. 𝑥 2 + 4𝑦𝑥 − 10𝑥 5
#4. 9𝑏𝑥 + 3𝑏𝑥 2 − 6𝑥 3
#5. 𝑥 3 + 𝑥 5 − 𝑥 2
#6. 𝑎𝑥 2 + 8𝑎2 𝑥 5 − 4
Arrange the terms of each polynomial so that the powers of x are in ascending order.
Ascending order is simply another way to write a polynomial, but it is NOT in standard form.
#7. 2𝑥 + 𝑥 2 − 5
#8. 6𝑥 + 9 − 4𝑥 2
#9. 4𝑥𝑦 + 2𝑦 + 6𝑥 2
#10. 6𝑦 2 𝑥 − 6𝑥 2 𝑦 + 2
#11. 𝑥 4 + 𝑥 3 + 𝑥 2
#12. 2𝑥 3 − 𝑥 + 3𝑥 7
We finally want to be able to combine polynomials using either addition or subtraction.
Examples: What is the sum or difference of the two polynomials?
#1. 3𝑥 2 + 5𝑥 2
#2. 4𝑥 3 𝑦 − 𝑥 3 𝑦
#3. 2𝑥 2 + 7𝑥 2
#4. 8𝑥 2 𝑦 − 3𝑥 2 𝑦
Examples: What is the simplified form of the following?
#5. (2𝑥 3 + 4𝑥 2 − 3𝑥) − (6𝑥 3 + 5𝑥 2 − 4)
#6. (𝑥 3 − 3𝑥 2 + 5𝑥) − (7𝑥 3 + 5𝑥 2 − 12)
We also want to try a problem that incorporates a real-life situation using polynomials.
EXAMPLE:
The revenue generated by a company and the cost of producing ‘x’ units can be modeled by the
polynomials below:
REVENUE: 2𝑥 2 + 120𝑥
COST: −0.5𝑥 2 − 300𝑥 − 8000
Add the functions to determine the net profit or loss polynomial.
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HOMEWORK – Chapter 8 Day #1
Name ________________________________ Date _______
Find the degree of each monomial.
1. 2b2c2
2. 5x
3. 7y5
4. 19ab
5. 12
1 2
z
6. 2
7. t
8. 4d4e
Simplify.
9. 2a3b + 4a3b
10. 5x3 – 4x3
11. 3m6n3 – 5m6n3
12. –6ab + 3ab
13. 4c2d6 – 7c2d6
14. 315x2 – 30x2
Write each polynomial in standard form. Then name each polynomial based on its
degree and number of terms.
15. 15x – x3 + 3
16. 5x + 2x2 – x + 3x4
17. 9x3
18. 7b2 + 4b
19. –3x2 + 11 + 10x
20. 12t2 + 1 – 3x + 8 – 2x
Simplify. Write each answer in standard form.
21. (3k2 + 5) + (16x2 + 7)
22. (g4 – 4g2 + 11) + (–g3 + 8g)
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23. (3a2 + a + 5) – (2a – 5)
24. (6d – 10d3 + 3d2) – (5d3 + 3d – 4)
25. (–4s3 + 2s – 3) + (–2s2 + s + 7)
26. (8p3 – 6p + 2p2) + (9p2 – 5p – 11)
27. A local deli kept track of the sandwiches it sold for three months. The
polynomials below model the number of sandwiches sold, where s
represents days.
Ham and Cheese:
Pastrami:
4s3 – 28s2 + 33s + 250
–7.4s2 + 32s + 180
Write a polynomial that models the total number of these sandwiches that
were sold.
28. A pizza shop owner is monitoring the amount of cheese he uses each week. The
polynomials below model the pounds of cheese ordered in the past, where p
represents pounds.
Mozzarella: 3p3  6p2 + 14p + 125
Cheddar: 12.5p2 + 18p + 75
Write a polynomial that models the total number of pounds of cheese that were
ordered.