Rational Zero Theorem Worksheet
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1MATH 7A
CH 4 QUIZ #2
Name______________________________________
Use the rational zero theorem, Descartes's rule of signs, and the theorem on bounds as aids in finding all real and
imaginary roots to the equation.
1) P(x) = 6x 3 + 25x 2 + 12x - 7
P(- x) = 6(- x) 3 + 25(- x) 2 + 12(- x) - 7 = - 6x 3 + 25x 2 - 12x - 7
P
N I
p
1
1
1
7
7
7
p = ± 1, 7
q = ± 1, 2, 3, 6
= ± 1,
,
,
,
,
,
, 7
1
2 0
q
6
3
2
6
3
2
1
0 2
1 6
25 12 -7
- 1 6
25
12 -7
upper bound
6
31
43
7
-6 -19
6
31 43 36
6
19
- 7
0
6x 2 + 19x - 7 = 0
2x + 7 3x - 1 = 0
2x + 7 = 0
3x - 1 = 0
2x = - 7
3x = 1
7
1
1
7
x = -
x =
- 1,
, -
2
3
3
2
Find all of the real and imaginary roots, stating the multiplicity of each. Sketch a graph.
2) f(x) = -5x 2 (x - 6)(x + 3) 3
0 with multiplicity 2, 6 with multiplicity 1, -3 with multiplicity 3
BONUS. For the given function, find all asymptotes (if there are any), and indicated the type . (2 points)
3 1
4
-8
x 2 + 4x - 8
3) f(x) =
vertical asymptote: x = 3
oblique asymptote: y = x + 7
3
21
x - 3
1
7
13
1
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