Download Use synthetic division to find the function value.

Math 7A
CHAPTER 4 TEST
Name:_______________________________________________
*Use synthetic division to find the function value.
1
1) f(x) = 6x 4 + 41x 3 + 37x 2 -x - 4; find f - .
3
-
1 6
3
6
41 37
-2 -13
39 24
-1 -4
-8 3
-9 -1
f-
1
=- 1
3
*Use the rational zero theorem to find all possible rational zeros for the polynomial function.
2) P(x) = 22x 3 + 88x 2 + 3x - 6
p = ± 1, 2, 3, 6
p
1 3 1 2 3 6 1 3
= ± 1, 2, 3, 6, , ,
,
, , , , ,
q
2 2 11 11 11 11 22 22
q = ± 1, 2, 11, 22
*Find all of the real and imaginary roots, stating the multiplicity of each.
3) f(x) = -4x2 (3x - 8)(x + 1)3
3x - 8 = 0
3x = 8
8
x=
3
0 with multiplicity 2, - 1 with multiplicity 3,
8
with a multiplicity of 1
3
*Describe the behavior of the function's graph at its x-intercepts.
2
4) f(x) = x 4 (x - 1)2 (x + 7)
3
touch at 0, 0 and 1, 0
pass at - 7, 0
*Find a polynomial equation with real coefficients that has the given roots.
5) -6, 2 + 4i
x + 6 x - 2 + 4i x - 2 - 4i
x + 6 x - 2 - 4i x - 2 + 4i
x + 6 x - 2 - 4i x - 2 + 4i
(x + 6) x - 2 2 - 4i 2
(x + 6) x 2 - 4x + 4 + 16
(x + 6) x 2 - 4x + 20
x3 - 4x 2 + 6x 2 - 24x + 20x + 120
x3 + 2x 2 - 4x + 120
1
*Use Descartes's rule of signs to discuss the possibilities for the roots of the equation. Do not solve the equation.
6) P(x) = x 5 + 7x4 + 4x 3 - 8x 2 + 2x - 9
P(-x) = x 5 + 7(-x)4 + 4(-x)3 - 8(-x)2 + 2(-x) - 9 = - x 5 + 7x 4 - 4x 3 - 8x 2 - 2x - 9
P
3
3
1
1
N
2
0
2
0
I
0
2
2
4
*Use the theorem on bounds to establish the best integral bounds for the roots of the equation.
7) P(x) = 2x 4 - 3x 2 - 4x + 4
1 2
0
2
2
2
-1 2
2
-3 -4 4
2
-1
0 -3 -4 4
-2 2
-2 -1
2 2
0
4
4
2
-2 2
2
-3 -4 4
8 10 12
5 6 16
0 -3 -4
-4 8 -10
-4 5 -14
4
28
32
-2 < x < 2
*Determine whether y
or y
3
2
8) y = -7x + 5x + 4x + 6
as x
y
-
- as x
as x
- and x
-
.
y
2
Find all of the real and imaginary zeros for the polynomial function.
9) f(x) = 3x 4 - 10x 3 + 20x 2 - 40x + 32
2,
4
, ± 2i
3
Sketch the graph of the function. Find and indicate the type of all asymptotes and intercepts (if there are any).
2x + 1
10) f(x) =
x+3
x+3=0
HA: y =
2
=2
1
x=- 3
2x + 1 = 0
y-intercept 0,
2x = - 1
1
x=2
VA: x = - 3
x-intercept -
3
1
,0
2
1
3
11) f(x) =
x2 + 9
x+2
x+2=0
-2 1
1
x = -2
VA: x = - 2
0 9
-2 4
-2 13
x2 + 9 = 0
x2 = - 9
no x-intercept
OA: y = x - 2
Sketch the graph of the polynomial function.
4
y-intercept 0,
9
2
12) f(x) =
x
-3
-2
-1
1
2
3
4
1
x(x - 5)(x + 4)2
4
y
6
14
13.5
-25
-54
-73.5
-64
5