Polynomial Test 2 Review
2.1
- standard form and vertex form of a quadratic equation – identify vertex and graph
- convert from standard form to vertex form and from vertex form to standard form
- vertical free fall
2.3
- analyze a function
- determine the 0’s
- write a function based on the given 0’s
2.4
-Bounds: use synthetic division and look at the last line
Upper bound: if the last line is all “+”
Lower bound: if the last line alternates “+”, “-“
-Remainder Theorem: use synthetic division to find the remainder. Answers written in the form P(x) = r, where
r is the remainder (the remainder could be 0)
-Factor theorem: 2 parts: 1) factor, 2) find the zeros (or solutions or roots)
1) will be given 1 factor, remember to take the opposite of the number before doing synthetic division,
you MUST get a remainder of 0, rewrite your answer and factor that quadratic, must have 3 sets of
parenthesis when you are finished
2) do everything from above, except set your factors equal to 0 and solve, write your answers in set
notation from lowest to highest, the only thing different is that instead of giving you an expression (x 2), I give you the root (x= -2)
Extra credit: I’ll give you a polynomial function with a missing coefficient & the divisor, you need to
find the missing coefficient
-Rational Root theorem: similar to the factor theorem except you aren’t given any of the factors
1) find all the factors of your leading coefficient (a) and all your factors of your constant (c)
2) divide all of the factors of the constant by all the factors of the lead. Coefficient (±c/a)
3) choose one of your factors (it’s guess & check at this step) & do synthetic division, keep picking a factor
until you get a remainder of 0
4) rewrite your answer. If it’s a quadratic factor it and find your other zeros. If it’s not a quadratic
repeat steps 1-4
5) write your answer in set notation (remember the factor from step 3 is one of your roots or zeros)
(question may ask for rational roots, real zeros, or all zeros (real & non-real)
2.5
-Fundamental theorem of algebra: 2 parts: 1) find all roots, 2) given the roots find the polynomial function
1) how you do this depends on if you are given a real root or a complex root
a) given a real root: do synthetic division, rewrite your answer, find the remaining zeros by doing
quadratic formula (these will be complex numbers),write your answer in set notation.
b) given a complex root: find the conjugate, write both as (x – (a + bi))(x – (a – bi)) and foil to get
rid of the i’s, take that quadratic and the original problem and find the remaining roots by doing
long division. If need be factor or use quadratic formula to find the remaining roots or set equal
to 0 and solve, write your answer in set notation.
2) write your roots as (x - #)(x - #),etc. If given a complex root or a radical, DO NOT forget about it’s
conjugate. Multiply the parenthesis together either by FOIL or distributing. Write your final answer as
f(x) = …
2.6
-Graphing: find the domain, end behavior, EBA’s, VA’s, x-intercepts, y-intercepts, and graphs without a
calculator
 To determine domain: set denominator = 0 and solve. The domain will be all real #’s except for q (where
x = q), if the denominator is non-real then the domain is (-∞, ∞)
To write: (-∞, q) U (q, ∞)
**if the denominator is a quadratic, you may have 3 intervals
 End behavior: determine if the y values are going to ∞ or -∞ as x -∞ and x ∞
 end behavior asymptotes:
If n< m, the end behavior asymptote (EBA) is the HA y = 0
If n = m, the EBA is the HA y = a/b
If n >m, the EBA is the HA: there is no HA
 VA: these occur at real 0’s of the denominator provided that the 0’s aren’t also 0’s of the numerator
 x-intercepts: these occur at the real 0’s of the numerator, which aren’t also 0’s of the denominator, write
answer as an ordered pair w/ the y-coordinate 0
 y-intercepts: this is the value of f(0), if defined. Write the answer as an ordered pair w/ the x=coordinate
0
2.7
- you will be solving rational equations
- Don’t forget to check for extraneous solutions
1) If you have only 1 fraction
a) Eliminate the fraction by multiplying every term by the denominator
b) simplify your equation
c) solve for x by whatever method you prefer
2) if you have more than 1 fraction
a) find the LCD
b) multiply each term by the LCD (DO NOT multiply until you have cancelled the denominators)
c) simplify
d) solve for x by whatever method you prefer
**You must get a remainder of 0 for Factor thm, Rational Root thm, and Fundamental thm of algebra**
Determine if k is an upper or lower bound
1. f(x) = 3x3 – x2 – 5x – 3, k = -4
2. f(x) = 2x3 – 5x2 – 5x – 1, k = 5
Use the remainder theorem to find P(a)
3. P(a) = x3 + x2 – 4x – 4; a = -1
4. P(a) = 8x4 + 32x3 + x + 4; a = -4
5. P(a) = 2x3 + 2x2 – 2x – 2; x = 3
Use the factor theorem to completely factor the following
6. f(x) = x3 – 2x2 – 13x -10; (x + 1)
7. f(x) = x3 + x2 – 80x – 300; (x – 10)
8. f(x) = x3 – x2 – 34x – 56; (x + 4)
Use the factor theorem to find all zeros
9. f(x) = x3 – 6x2 + 11x – 6, x = 2
10. f(x) = 2x3 + 17x2 + 23x – 42, x = -6
11. f(x) = x3 – x2 – 5x – 3; x = 3
Find all zeros (real and non-real) using the rational root theorem
12. f(x) = 2x3 – 3x2 – 11x + 6
13. f(x) = x3 - 4x2 – 7x + 10
Use the fundamental theorem of algebra to find all zeros
14. f(x) = x3 + 2x2 - 3x – 10; -2 - i
15. f(x) = x3 + 2x2 + 4x – 7, 1
Use the fundamental theorem of algebra to write a polynomial function in simplest form
16. 2, 4 + i
17. 3, 2 – i
Graph: find the domain, end behavior, EBA’s, VA’s, x-intercepts, and y-intercepts, then sketch the graph
(you can complete a table of value if you want to but I don’t require it)
18. f(x) = 3
x3 – 4x
x
y
19. f(x) = x2 – x – 2
x2 – 2x - 8
x
y
Solving rational equations
20. x + = 7
21. 2 - 3 = 12
x+4 x2 + 4x
22. 4x + 3 = 15
x+4
x-1 x2 +3x-4
Finding the vertex of a parabola
23. f(x) = 2x2 – 4x + 10
24. f(x) = -2(x – 4)2 – 8
25. Analyze the function: f(x) =
26. Find the zeros of the function: f(x) = 2x4 – 8x2 + 5x – 10
27. Write the function for the given zeros: -2, 1, 4