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Pre-Calculus Unit 3 Review
Name ______________________ Pd: _______
Complete the square on the following quadratic equations to get them into Standard Vertex Format
y = a(x – h)2 + k , then find the vertex point, AOS, and opening direction:
1. y = x2 – 4x + 5
2. y = -4x2 – 24x + 30
Vertex: ___________
Vertex: ________________
AOS: ____________
AOS: _________________
Opening: _________
Opening: ______________
Find the Standard Vertex Equation of the parabola that passes through the given point and Vertex
Point:
3. Vertex (1, 1) and Point (3, 5)
4. Vertex (-4, 0) and Point (2, 6)
#5 - 8: Are the following polynomial functions in one variable? If yes, state the degree and leading
coefficient; if no, tell why not.
5.
4
3
1
1
f ( x)   x3  x 2  x 
3
2
9
99
6.
f ( x)  2 x 3 
5
x3
7.
f ( x)  x 2  3 y 2
8.
p( x)  3x 4  3x 
2x 7
 4x3  4
3
#9 - 10: Determine if the given possible root is in fact a root of the given polynomial equation. Yes or
No (Show work for credit!)
9.
x = 3; 2 x3  7 x 2  x  3  0
10.
11. Consider the polynomial F(x) = 13x9  4 x 2  2 x10  17 .
What is the degree of this polynomial?
What is the leading coefficient?
How many complex roots does it have?
x = -3 ; x 4  5 x 2  36  0
12.
If x = 5 is a solution of P(x), state a factor of P(x).
13.
If (x + 17) is a factor of P(x), state a root of P(x).
14.
If x = 7 is a root of P(x), state an x-intercept of the graph of P(x).
15.
If (3,0) is a point of the graph of P(x), state a factor of P(x).
16.
For each of the given polynomials, write the end behavior in limit notation, and give the yintercept.
a.
c.
f ( x)  2 x 6  3 x 3  2 x  7
f(x) = (2x + 3)(x – 4)(x + 1)
b.
p( x)  5x9  6 x8  3x7  2 x 4  3x 2  10
d.
p( x)  2 x 4  3x 2  10
17. Sketch each of the following types of polynomial functions.
7th degree; 3 real roots; negative lead coefficient
3rd degree; 2 x-intercepts; positive lead coefficient
18.
Write the end behavior of the polynomial in limit notation: y= x(2x + 6)(x + 2)(x - 1)
#19 - 21: Solve the polynomial equations over the set of complex numbers. Find the solutions:
19.
(x - 3)(x + 4)(7x + 2) = 0
22.
Is (x - 3) a factor of
(Show all work)
20. x 4  5 x 2  36  0
2x3 - 3x2 - 10x – 9 = 0?
23. Divide the (6x3 - 3x2 + 10x – 5) by (2x + 3) using LONG Division!
21. x3  x  0
24. Given the Linear Factorization of a polynomial below, find all the real zeros and their multiplicity.
What does the multiplicity tell you about the graph at each zero?
f(x) = (x + 3)(x - 2)2(3x - 2)3(x – 3)4
25. Divide using synthetic division: 3x4 – 4x2 + 6x – 9 divided by (x + 2)
26. Given the following polynomial and one of its factors; find the remaining factors and all zeros.
f ( x)  3x3  2 x 2  61x  20; ( x  5)
Use the values of A – F in the table below following for questions 27-32:
27. A + C
A = 2 + 9i
B = 3 – 4i
28. C - D
C = -7i + 4
D = -5i
29. AB
E=7
F=
30. Conjugate of B
31. Simplify F
32.
A
B
48
#33 – 34 Sketch the graph of the given polynomial function. Remember, your three keys: xintercepts, y-intercept, and end behavior. You may need to solve the equation first to find the xintercepts, by synthetic division, quadratic formula.
33. f(x) = x3 - 4x
34.
y = x3 - 3x2 - 9x + 27
Other review topics on test!!
1. Parent function characteristics, graphing with Transformations
2. Proving inverses by composition
3. Finding inverse functions algebraically
4. Function composition
5. Calculator portion: linear regression