# 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

x-intercepts. 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 y-intercept, 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
y-intercept, x-intercept(s), and vertical asymptote(s), and horizontal
asymptote, or
any slanted asymptote.
**

**Answer Sheet**

1.

Vertex (-1, 9)

y-int (0, 8)

x-int (-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 y-int (0, 20)

13. There is a zero between 0 and 1.

14. 3.3

15.

y-int (0, 3)

x-int(3, 0) (1, 0)

Vert. Asym.: x = -1

Horiz. Asym.: y = 1

16.

y-int (0, 2)

x-int (2, 0) (-1, 0)

Vert. Asym.: x = 1

Slant Asym.: y = x

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