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Math 1314 Chapter 3 Review
1. Graph the parabola whose equations is f (x) = −x^{2} − 2x
+ 8 . Label the vertex. Also
label one point on each branch of the parabola. Find the y intercept and any
xintercepts. State the domain and range of the function.
2. Use synthetic division to find the quotient and
remainder if 2x^{4} − 3x^{3} + x^{2} − 5 is
divided by x + 3.
3. Use synthetic division and the reminder theorem to find
f (3) if
f (x) = x^{4} + 2x^{3} −8x^{2} +10x − 5 .
4. Use synthetic division and the factor theorem to decide
if x + 3 is a factor of
f (x) = x^{3} + 4x^{2} + x − 6 .
5. Find a polynomial function of degree three with real
coefficients having zeros 1, and
2 – 3i, and f (2) = 27.
6. If i is a zero of f (x) = x^{4} + 2x^{3} − 7x^{2} + 2x −8 ,
find the remaining zeros by using
synthetic division.
7. State all the zeros and their multiplicities of :
f (x) = x^{3} (2x +1)(x − 4)^{2} (x −1− i)(x −1+ i)
8. Completely factor f (x) = 6x^{3} + 25x^{2} − 24x + 5 if 5 is a zero of f (x) .
9. Use Descartes’ Rule of Signs to determine the possible
number if positive and
negative zeros of f (x) = 3x^{3} −13x^{2} +17x +15 .
10. Determine if 2 is a lower bound and also determine if
1 is an upper bound for the
zeros of f (x) = 2x^{3} + 2x^{2} + 4x + 2 .
11. Find all the zeros of f (x) = 2x^{3} − 5x^{2} − 46x + 24
by using the rational zero theorem
and synthetic division.
12. Find all the zeros of f (x) = 2x^{3} −8x^{2} −14x + 20 by
using the rational zero theorem
and synthetic division and then graph f (x) by labeling the yintercept, the
xintercepts,
and by determining the end behavior.
13. Use the intermediate value theorem to determine if
there is a zero between 0 and 1
for f (x) = x^{3} − 4x^{2} + 2 .
14. Use the intermediate value theorem and synthetic
division to estimate the zero
between x = 3 and x = 4 to the nearest tenth of f (x) = x^{3} −10x − 3 .
For problems 15 and 16, Graph each rational function by
identifying any
yintercept, xintercept(s), and vertical asymptote(s), and horizontal
asymptote, or
any slanted asymptote.
Answer Sheet
1.
Vertex (1, 9)
yint (0, 8)
xint (4, 0) (2, 0)
D = (−∞,∞) R = (−∞,9]
3. f (−3) = −80
4. x + 3 is a factor
5. f (x) = 3(x −1)(x − 2 + 3i)(x − 2 − 3i)
6. Remaining zeros are –i,  4, 2
8. f (x) = 6(x + 5)(x −1 3)(x −1 2)
9. 2 or no positive zeros , 1 negative zero.
10. 2 is a lower bound, 1 is an upper bound
11. zeros: 4, 1/2 , 6
12. zeros: 2, 1, 5 yint (0, 20)
13. There is a zero between 0 and 1.
14. 3.3
15.
yint (0, 3)
xint(3, 0) (1, 0)
Vert. Asym.: x = 1
Horiz. Asym.: y = 1
16.
yint (0, 2)
xint (2, 0) (1, 0)
Vert. Asym.: x = 1
Slant Asym.: y = x
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