Math 3
Chapter 3 Quiz
NAME:_____________________________________
Complete, organized, and correct work using the methods taught in class including correct use of symbols is required to
receive full credit. Partial or no credit will be given when work is missing. Each problem is worth 5 points. Leave final
answers in simplest form.
SHOW ALL WORK. LEAVE FINAL ANSWERS IN SIMPLEST FORM. DO NOT ROUND UNLESS INSTRUCTED TO
DO SO.
Use the rational zero theorem to find all possible rational zeros for the polynomial function.
1) P(x) = 3x 3 + 44x 2 + 44x + 27
p = ± 1, 3, 9, 27
q = ± 1, 3
p
1
= ± 1, , 3, 9, 27,
q
3
Find all of the real and imaginary roots, stating the multiplicity of each.
2) f(x) = -5x2 (x - 6)(x + 3)3
-3 with multiplicity 3
0 with multiplicity 2
6 with multiplicity 1
Use Descartes's rule of signs to discuss the possibilities for the roots of the equation. Do not solve the equation.
3 sign changes
3) 7x 5 + 2x 4 + 9x 3 - 6x 2 + 9x - 7 = 0
7(- x)5 + 2(- x)4 + 9(- x)3 - 6(- x)2 + 9(- x) - 7 = - 7x 5 + 2x 4 - 9x 3 - 6x 2 - 9x - 7
P
3
3
1
1
N
2
0
2
0
2 sign changes
I
0
2
2
4
Use the theorem on bounds to establish the best integral bounds for the roots of the equation.
4) 3x 4 - 8x 3 - 8x - 9 = 0
-1 < x < 4
BONUS. 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. (5 POINTS)
5) 3x 4 - 10x3 + 20x 2 - 40x + 32 = 0
4
, 2, -2i, 2i
3
1