1. Section 2.3
Algebra 2
Polynomial Equations

2. Monomials A “Monomial” refers to.. A number A variable
Ex: ½ -4.93
A variable
Ex: a z -k x -r
The product of a number and one or more variables with whole number exponents.
Ex: 8p bc 12yz -4v4n -77h154w21d

3. Monomials
These are not monomials..
9 + x
2/x
100x
x-1
87x2 - 5

4. (#1)
Which of the following is a monomial?
A) 6a
B) 2/b
C) c-2
D) 3

5. (#2) Which of the following is not a monomial? A) f + 1 B) 4k C) 5a2b
D) 1

6. Degree The sum of the exponents of each variable. A number A variable
Ex: a z -k x -r
Multiple variables
degree = sum of exponents
Ex: 8p bc 12yz -4v4n -77h154w21d

7. (#3)
What is the degree of the monomial 13x ?
A) 0
B) 1
C) 2
D) 13

8. (#4)
What is the degree of the monomial 7 ?
A) 0
B) 1
C) 2
D) 7

9. (#5)
What is the degree of the monomial a3b2c ?
A) 0
B) 2
C) 3
D) 6

10. Polynomial Umbrella term Refers to a monomial or sum of monomials
Some polynomials have special names
Depend on the # of terms
Polynomials have a degree too

11. Binomial Type of polynomial Consists of two monomials
These are binomials
x + 1
a + b
15x2 – 15
(yz + 3)

12. Trinomial Type of polynomial Consists of three monomials
These are trinomials
x2 + x + 1
4m3 + h -6
a2 + ab + b2
uv + wx + yz

13. Types of Polynomials There are lots. . . You only need to remember
Monomial
Binomial
Trinomial

14. (#6) Which of the following is a binomial? A) 1 B) x + 1 C) x2 + x + 1
D) x3 + x2 + x + 1

15. (#7) Which of the following is a trinomial? A) 1 B) 2x + 1
C) 3x2 + 2x + 1
D) 4x3 + 3x2 + 2x + 1

16. Degrees of Polynomials
Equal to the greatest degree of its terms
Ex: 4 has a degree = 0
a has a degree = 1
x2 + x + 1 has a degree = 2
bc +7 has a degree =
p4q2r + 3b has a degree =
t t has a degree =

17. (#8)
What is the degree of -16t2 + 30t + 3 ?
A) 0
B) 1
C) 2
D) 3

18. (#9) Which of the following has a degree of 6 ? A) abcdefg
B) 21n3 – n + 7
C) x3y2z + 6xyz - 9
D) -t3u – tuv + 11

19. Standard Form Exponents of terms decrease from left to right
Coefficients in front of each term
Ex: 3x2 – 8x + 2
a5b4c3d2 – 1
-w2 + k + 123
2x3 + x2 – 5x + 12

20. Standard Form These are not in standard form Ex: 9 + x
h2 – 22r – 4 + r2h
yz2
-9a + ab + b

21. (#10) Which of the following is in standard form? A) 1 + 2x – 4x2
B) 2x + 1 – 4x2
C) 1 – 4x2 + 2x
D) -4x2 + 2x + 1

22. Lead Coefficients The coefficient of the first term in standard form.
Ex: has lead coefficient = 4
a + 2 has lead coefficient = 1
99x2 - x + 1 has lead coefficient =
bc +7 has lead coefficient =
4p4q2r + 3b has lead coefficient =
-6t t has lead coefficient =

23. (#11) Which of the following has a negative lead coefficient?
A) -3w + 7
B) 3w – 7
C) k3 – 2k
D) 6k10 – 3k5 – 1

24. Closed Polynomials Math operations apply to each term
Allows us to +, -, x, and ÷ polynomials

25. Closed Polynomials Ex: = (x + 1) + (x + 2) = -(x + 10) =
5(z2 – 3z + 1) =
2(x - 1) + 4 =

26. (#12) Simplify (x + 1) + (x + 1) and rewrite in standard form.
A) (x + 1) + (x + 1)
B) 2x + 2
C) 2(x + 1)
D) (x + 1)2

27. (#13) Simplify (2x + 3) – (x + 5) and rewrite in standard form. A) -1
B) 10
C) x + 8
D) x - 2

28. (#14) Simplify 7(b + 2) and rewrite in standard form. A) 14 + 7b
B) 7b + 14
C) (b + 2)7
D) b2 + 49

29. (#15) Simplify -2(m – v) and rewrite in standard form. A) m – v – 2
B) -m + v
C) -2m + 2v
D) -v + 2m

30. Skydiving Example
Two skydivers perform a stunt where they kick off of each other while in freefall. One skydiver’s altitude is modeled by the polynomial -16t t , and the other skydiver’s altitude is modeled by the polynomial -16t t A) Write a formula for the altitude between skydivers. B) How far apart are they after 3 seconds?

31. Skydiving Example -16t2 - 130t + 10000 - ( -16t2 - 150t + 10000 )
A) Write a formula for the altitude between skydivers.
-16t t
- ( -16t t )
B) How far apart are they after 3 seconds?

32. (#16) Simplify (-8x – 12) + (9x + 4) and rewrite in standard form.
A) x – 8
B) x + 8
C) -x – 8
D) -x + 8

33. (#17) Simplify (6x + 9) – (7x + 1) and rewrite in standard form.
A) x – 8
B) x + 8
C) -x – 8
D) -x + 8

34. (#18)
Simplify (x2 + 6x - 2) + (x3 + x + 4) and rewrite in standard form.
A) x3 + x2 + 7x + 2
B) 3x3 + 2x2 + x
C) x5 + 6x2 - 8
D) x3 - 2x2 + 7x + 8

35. (#19)
Simplify (x2 + 6x – 2) – (x3 + x + 4) and rewrite in standard form.
A) -x5 – 6x2 + 8
B) x5 + 6x2 – 8
C) -x3 + x2 + 5x – 6
D) x3 – x2 – 5x + 6

36. (#20)
Simplify (-3p3 + 5p2 – 2p) + (-p3 – 8p2 – 15p) and rewrite in standard form.
A) 4p3 + 3p2 + 17p
B) -4p3 – 3p2 – 17p
C) -2p3 + 13p2 + 13p
D) 2p3 – 13p2 – 13p

37. (#21)
Simplify (4m2 – m + 2) – (-3m2 + 10m + 4) and rewrite in standard form.
A) -7m2 – 11m + 2
B) 7m2 – 11m - 2
C) -7m2 + 11m - 2
D) 7m2 – 11m + 2

38. (#22)
Simplify (9r2 + 4r - 7) + (3r2 – 3r) and rewrite in standard form.
A) -3r2 + 12r + 4
B) 3r2 – 7r + 7
C) 12r2 + 7r – 4
D) 12r2 + r – 7

39. (#23)
Simplify (-r – 10) – (-4r3 + r2 + 7r) and rewrite in standard form.
A) 4r3 – r2 – 8r – 10
B) -4r3 + r2 – 8r + 10
C) 5r3 – r2 – 6r + 10
D) -5r3 + r2 + 6r – 10

40. (#24)
Simplify (s3 – 2s – 9) + (2s2 – 6s3 + s) and rewrite in standard form.
A) -7s3 + 2s2 – 3s – 9
B) 7s3 – 2s2 + 3s + 9
C) -5s3 + 2s2 – s – 9
D) 5s3 – 2s2 + s + 9

41. (#25)
Simplify (4d – 6d3 + 3d2) – (10d3 + 7d – 2) and rewrite in standard form.
A) -16d3 + 3d2 – 3d + 2
B) 16d3 – 3d2 + 3d – 2
C) -4d3 + 3d2 – 11d + 2
D) 4d3 – 3d2 + 11d – 2