Polynomial Functions Review for Check-in Quiz
Name: ________________________________________________
Date: _________
Simplify the functions by completing the operation.
1.
(2d3 – 5d2 + 1) + (3d3 + 2d2 – 7d)
2.
(x2 – 5x + 1) – (3x2 + x – 5)
3.
2 (x – 4)3
(2x + 7)2
4.
Function composition:
5. f(x) = 3x – 5
g(x) = 2x + 1
Find f(g(x)).
8.
f(x) = 2x2 + 11
Find g(f(x)).
6. f(x) = x – 2
g(x) = x2 + 3
Find g o f.
g(x) = 4x
9. f(x) = -4x + 6
Find f o g (–2)
Factor squares and cubes and trinomials
7. f(x) = x2 – 9
g(x) = 3x2
Find f o g (x)
g(x) = 5x – 1
10. f(x) = 2x2
Find f(g(–1)).
g(x) = x2 + 7
11.
x3 + 27
12.
8x3 – 1
13.
9x2 – 16
14.
64x3 – 125y3
15.
2x3 + 16
16.
x2 + 100
Graph the cubic functions.
17)
 
3
f (x)  x 1  5
Initial Point: (
,
)
18)


3
f (x)  2 x  3  4
 y
 y




Initial Point: (
x

19)
















3
Initial Point: (
,
)
20)
f (x) 
1 3
x 4
2

Initial Point: (
 y




x




x











21. Fill out the table on end behavior!
,
)

 y

)
x

f (x)  3 x  4
,


Function
Degree
LC
Rough Sketch
(with maximum turns)
End Behavior
A
f (x ) 4x 3  2x 1
x  , f(x)  _____
x  , f(x)  _____
B
f (x ) 10x 3
x  , f(x)  _____
x  , f(x)  _____
C
f (x ) 2(x 5)(x  2)
x  , f(x)  _____
x  , f(x)  _____
D
f (x ) 2x 7  x 4  x 1
x  , f(x)  _____
x  , f(x)  _____
E
f (x ) x(x 3)(x 5)
x  , f(x)  _____
x  , f(x)  _____
F
f (x ) 5x(x 3)2(x 1)3
x  , f(x)  _____
x  , f(x)  _____
G
f (x ) 5(x  2)(x 3)3
x  , f(x)  _____
x  , f(x)  _____
H
f (x )1x 4  2x 3 1x 6
x  , f(x)  _____
x  , f(x)  _____
I
f (x ) 5x 2
x  , f(x)  _____
x  , f(x)  _____
J
f (x ) x(x  2)6
x  , f(x)  _____
x  , f(x)  _____
22.
Given the factors, find the zeros.
a.
Factor: (x + 3)
Zero: ________
d.
Factor: (x – 5)
Zero: ________
b.
Factor: (x)
Zero: ________
e.
Factor: (x – 100)
Zero: ________
c.
Factor: (2x + 6)
Zero: ________
f.
Factor: (5x – 1)
Zero: ________
23.
Given the zeros, find the factors.
a.
Zero: 9
Factor: ___________
d.
Zero: –3
Factor: _____________
b.
Zero: 0
Factor: ___________
e.
Zero: –3i
Factor: _____________
c.
Zero:
3
5
Factor: ___________
f.
Zero: 7
Factor: _____________
24.
Given a zero, find another zero.
a.
Zero: 5i, ______________
c.
Zero: 4 – 7i, ______________
b.
Zero:  17 , ______________
d.
Zero: 5 2 3 , ______________
Use your calculator to find the following information:
25.
26.
27.
Function:
y = –4x3 – 2x2 + x – 4
Degree: ____________
Lead Coeff: _____________
# of Zeros: __________
# of Real Zeros:__________
Function:
Rough Sketch:
# of Imaginary Zeros: __________
y = 3x4 + 2x2 – 3x – 7
Degree: ____________
Lead Coeff: _____________
# of Zeros: __________
# of Real Zeros:__________
y = – (x – 1)2 (x + 2)2
Rough Sketch:
# of Imaginary Zeros: __________
Graph Sketch
 y

Degree: ________
Leading Coefficient: ________
End behavior:
x  , f(x)  _____
x  , f(x)  _____
Real zeros: _______
_______
_______
Multiplicity: _______
_______
_______
28.
y = –3 (x – 1)2 (x + 4)3
Graph Sketch
 y
Degree: ________

Leading Coefficient: ________
End behavior:
x  , f(x)  _____
x  , f(x)  _____

x




Real zeros: _______
_______
_______

Multiplicity: _______
_______
_______

29.


y = x (x + 4)3 (x – 2)2
Graph Sketch
 y
Degree: ________

Leading Coefficient: ________
End behavior:

x  , f(x)  _____
x  , f(x)  _____

x




Real zeros: _______
_______
_______

Multiplicity: _______
_______
_______




30.
Find the polynomial with a leading coefficient of 2 that has the given zeros: 1, –2i
Write f(x) in factored form: ________________________________________
Change to Standard Form:
Use f(x) = 5x4 – 8x2 + 5x – 2 to answer the questions below.
31.
Use DIRECT substitution given x = 2.
Write as an ordered pair: ____________
Write in function notation: ___________
Is the ordered pair a zero? ____________
32.
Use SYNTHETIC substitution given x = 1.
Write as an ordered pair: ____________
Write in function notation: ___________
Is the ordered pair a zero? ____________
33.
Use SYNTHETIC substitution given x = –2.
Write as an ordered pair: ____________
Write in function notation: ___________
Is the ordered pair a zero? ____________
SOLUTIONS
1.
5d3 – 3d2 – 7d + 1
E.
Deg = 3, LC = 1, End Beh:  , 
2.
–2x2 – 6x + 6
F.
Deg = 6, LC = 5, End Beh:  , 
3.
4x2 + 28x + 49
G.
Deg = 4, LC = 5, End Beh:  , 
4.
2x3 – 24x2 + 96x – 128
H.
Deg = 4, LC = 1, End Beh:  , 
5.
f(g(x)) = 6x – 2
I.
Deg = 2, LC = 5, End Beh:  , 
6.
g(f(x)) = x2 – 4x + 7
J.
Deg = 7, LC = 1, End Beh:  , 
7.
f(g(x)) = 9x4 – 9
22.
a) –3
b) 0
c) –3
8.
g(f(x)) = 8x2 + 44
d) 5
e) 100
f) 1
9.
f(g(–2)) = 50
10.
f(g(–1)) = 128
11.
(x + 3) (x2 – 3x + 9)
12.
(2x – 1) (4x2 + 2x + 1)
13.
(3x – 4) (3x + 4)
14.
(4x – 5y) (16x2 + 20xy + 25y2)
15.
2 (x + 2) (x2 – 2x + 4)
16.
Prime
17.
18.
19.
20.
Initial Point: (–1, –5) Check with calc.
Initial Point: (–3, –4) Check with calc.
Initial Point: (–4, 0) Check with calc.
Initial Point: (0, 4) Check with calc.
21.
A.
23.
24.
Deg = 3, LC = –10, End Beh:  , 
C.
Deg = 2, LC = 2, End Beh:  , 
D.
Deg = 7, LC = 2, End Beh:  , 
5
c) (5x – 3)
f) x  7
25.
Deg = 3, LC = –4
#zeros = 3, #real = 1, #imag = 2
26.
Deg = 4, LC = 3
#zeros = 4, #real = 2, #imag = 2
27.
Deg = 4, LC = –1, End Beh:  , 
Real Zeros: 1, 1,
–2, –2
Multiplicity: 2 (bounce) 2 (bounce)
28.
Deg = 5, LC = –3, End Beh:  , 
Real Zeros: 1, 1,
–4, –4, –4
Multiplicity: 2 (bounce) 3 (cross)
29.
Deg = 6, LC = 1, End Beh:  , 
Real Zeros: 0
–4, –4, –4
2, 2
Multiplicity: 1 (C) 3 (C)
2 (B)
30.
y = 2x3 – 2x2 + 8x – 8
31.
(2, 56), f(2) = 56, No, output is not 0.
32.
(1, 0),
33.
(–2, 36), f(–2) = 36 No, output is not 0.
Deg = 3, LC = –4, End Beh:  , 
B.
a) (x – 9)
b) x
d) (x + 3)
e) (x + 3i)
a) –5i b) 17
c) 4 + 7i
d) 5 2 3
f(1) = 0,
Yes, output is 0.