1. Rational and Polynomial Relationships
Review

2. Use the following vocabulary to describe each
Expression, equation, term, factor, coefficient, variable, zero, function, domain, range
F(x) = 3(x + 2)(2x + 5)( 𝑥 )

3. Preform the following operations

4. Polynomial Division
Write 𝑎(𝑥) 𝑏(𝑥) as q x + 𝑟 𝑥 𝑏 𝑥
Divide

5. Remainder Theorem 𝒙 𝟑 +𝟑 𝒙 𝟐 +𝟓𝟓 𝒙 −𝟕
If   p(x) / (x – a)  =  q(x)  with remainder  r(x), then  p(x)  =  (x – a) q(x)  +  r(x)
Example:
(x^3 – 7x – 6) / (x – 4)   =   x2 + 4x + 9   with remainder 30,
So…  x3 – 7x – 6  =   (x – 4) (x2 + 4x + 9)  +  30.
Divide the following writing the answer in terms of the remainder theorem
𝒙 𝟑 +𝟑 𝒙 𝟐 +𝟓𝟓 𝒙 −𝟕

6. Re-write using remainder thrm.

7. Polynomial graphing techniques and Factorization

8. Strategies for visualizing polynomial graphs
Input / Output Table
End Behavior - even and odd degree functions
Y intercept
Descartes Sign change
Factoring / find zeroes
Remainder Theorem
Rational Zero Theorem (p/q)
Quadratic Techniques
Relative Minimums and Maximums by apprx.

9. Quick Sketch using end behavior
A positive quartic function
A negative quartic function
A positive cubic function
A negative cubic function

10. Explain the Fundamental Thrm. of Algebra

11. Descartes Sign Rule
The sign changes in f(x) gives the number of positive zeroes or an even increment of zeroes below that number The sign changes in f(-x) gives the number of negative zeroes or an even increment of zeroes below that number

12. Determine number of positive, negative, imaginary zeroes

13. Write and sketch a polynomial function given the roots

14. Write and sketch a polynomial function given the roots

15. Given the function and a root determine other roots

16. Factor

17. Remainder Theorem
If   p(x) / (x – a)  =  q(x)  with remainder  r(x), then  p(x)  =  (x – a) q(x)  +  r(x)
Example:
(x^3 – 7x – 6) / (x – 4)   =   x2 + 4x + 9   with remainder 30,
So…  x3 – 7x – 6  =   (x – 4) (x2 + 4x + 9)  +  30.
When is the remainder theorem a useful tool?

18. Rational Zero Theorem (p/q)
Determine all possible rational zeroes for the following polynomial function

19. Quadratic Techniques
Factor the following polynomial

20. Factor and graph the following polynomials

21. Squareroot functions
Graph the following squareroot functions

22. Rational expressions and functions

23. Preform the following operations

24. Strategies for visualizing rational graphs
Input / Output Table
Transformations of the parent function (1/x)
Holes and Asymptotes
Y intercept and X intercept(s)

25. Horizontal and vertical asymptote rules
If n < m , then the x axis is the horizontal asymptote
If n= m , then the line y = a/b is the horizontal asymptote
If n > m , then there is no horizontal, it is instead a slant or oblique
if n is greater than m by one degree then the quotient of the function is the slant asymptotes

26. Determine any holes or asymptotes
Why are some excluded values holes and others vertical asymptotes?

27. Graph