MATH 105 Exam 3
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Q1. Find the domain of the rational function.
g(x) =
a. all real numbers
b. {x|x ≠ -7, x ≠ 7, x ≠ -5}
c. {x|x ≠ -7, x ≠ 7}
d. {x|x ≠ 0, x ≠ -49}

Q2. Use the Factor Theorem to determine whether x - c is a factor of f(x).
8x3 + 36x2 - 19x - 5; x + 5
a. Yes
b. No

Q3. Determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then find that value.
f(x) = x2 - 2x - 5
a. maximum; 1
b. minimum; 1
c. maximum; - 6
d. minimum; - 6

Q4. Use the intermediate value theorem to determine whether the polynomial function has a zero in the given interval.
f(x) = -2x4 + 2x2 + 4; [-2, -1]
a. f(-2) = 20 and f(-1) = 5; no
b. f(-2) = -20 and f(-1) = 4; yes
c. f(-2) = 20 and f(-1) = -4; yes
d. f(-2) = -20 and f(-1) = -4; no

Q5. Find the vertex and axis of symmetry of the graph of the function.
f(x) = -3x2 - 6x - 2
a. (-1, 1) ; x = -1
b. (2, -26) ; x = 2
c. (1, -11) ; x = 1
d. (-2, -8) ; x = -2

Q6. Determine whether the rational function has symmetry with respect to the origin, symmetry
with respect to the y-axis, or neither.
f(x) =
a. symmetry with respect to the y-axis
b. symmetry with respect to the origin
c. neither

Q7. Find all of the real zeros of the polynomial function, then use the real zeros to factor f over the real numbers.
f(x) = 3x4 - 6x3 + 4x2 - 2x + 1
a. no real roots; f(x) = (x2 + 1)(3x2 + 1)
b. 1, multiplicity 2; f(x) = (x - 1)2(3x2 + 1)
c. -1, 1; f(x) = (x - 1)(x + 1)(3x2 + 1)
d. -1, multiplicity 2; f(x) = (x + 1)2(3x2 + 1)

Q8. Find all zeros of the function and write the polynomial as a product of linear factors.
f(x) = 3x4 + 4x3 + 13x2 + 16x + 4
a. f(x) = (3x - 1)(x - 1)(x + 2)(x - 2)
b. f(x) = (3x + 1)(x + 1)(x + 2i)(x - 2i)
c. f(x) = (3x - 1)(x - 1)(x + 2i)(x - 2i)
d. f(x) = (3x + 1)(x + 1)(x + 2)(x - 2)

Q9. Solve the inequality.
(x - 5)(x2 + x + 1) > 0
a. (-∞, -1) or (1, ∞)
b. (-1, 1)
c. (-∞, 5)
d. (5, ∞)

Q10. Use the graph to find the vertical asymptotes, if any, of the function.
a. y = 0
b. x = 0, y = 0
c. x = 0
d. none

Q11. State whether the function is a polynomial function or not. If it is, give its degree. If it is not,
tell why not.
f(x) =
a. Yes; degree 3
b. No; x is a negative term
c. No; it is a ratio
d. Yes; degree 1

Q12. Find the domain of the rational function.
f(x) = .
a. {x|x ≠ -3, x ≠ 5}
b. {x|x ≠ 3, x ≠ -5}
c. all real numbers
d. {x|x ≠ 3, x ≠ -3, x ≠ -5}

Q13. Determine whether the rational function has symmetry with respect to the origin, symmetry
with respect to the y-axis, or neither.
f(x) =
a. symmetry with respect to the origin
b. symmetry with respect to the y-axis
c. neither

Q14. Give the equation of the oblique asymptote, if any, of the function.
h(x) =
a. y = 4x
b. y = 4
c. y = x + 4

d. no oblique asymptote

Q15. Find the power function that the graph of f resembles for large values of |x|.
f(x) = (x + 5)2
a. y = x10
b. y = x25
c. y = x2
d. y = x5

Q16. Find k such that f(x) = x4 + kx3 + 2 has the factor x + 1.
a. -3
b. -2
c. 3
d. 2

Q17. A developer wants to enclose a rectangular grassy lot that borders a city street for parking. If the developer has 320 feet of fencing and does not fence the side along the street, what is the largest area that can be enclosed?
a. 25,600 ft2
b. 19,200 ft2
c. 12,800 ft2
d. 6400 ft2

Q18. Use the Theorem for bounds on zeros to find a bound on the real zeros of the polynomial
function.
f(x) = x4 + 2x2 - 3
a. -4 and 4
b. -3 and 3
c. -6 and 6
d. -5 and 5

Q19. Find the indicated intercept(s) of the graph of the function.
y-intercept of f(x) =
a. (0, 3)
b. (0, 4)
c.
d.

Q20. Determine, without graphing, whether the given quadratic function has a maximum value or a
minimum value and then find that value.
f(x) = -x2 - 2x + 2
a. minimum; - 1
b. maximum; 3
c. minimum; 3
d. maximum; - 1