# Algebra multiple choice questions

(See attached file for full problem description for PROPER EQUATIONS AND SYMBOLS)

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If f(x) = x(x + 3)(x ¨C 1), use interval notation to give all values of x

where f(x) > 0.

a. (¨C3, 1)

b . (¨C3, 0) ¡È (1, ¡Þ)

c . (1, 3)

d . (0, 1) ¡È (3, ¡Þ)

If f(x) = x(x ¨C 1)(x ¨C 4)2, use interval notation to give all values of x

where f(x) > 0.

a. (¨C¡Þ, 0) ¡È (4, ¡Þ)

b . (¨C¡Þ, 1) ¡È (4, ¡Þ)

c . (0, 1) ¡È (4, ¡Þ)

d . (¨C¡Þ, 0) ¡È (1, 4) ¡È (4, ¡Þ)

The table shows several value of the function f(x) = ¨Cx3 + x2 ¨C x + 2.

Complete the missing values in this table, and then use these

values and the intermediate value theorem to determine (an)

interval(s) where the function must have a zero.

x -2 -1 0 1 2

F(x) 16 -4

Find the quotient and remainder of f(x) = x3 ¨C 4x2 + 5x + 5 divided

by p(x) = x ¨C 1.

a. x2 + 2x + 2; 7

b . x2 ¨C 3x + 3; ¨C5

c . x2 ¨C 3x + 2; 7

d . x2 ¨C 2x + 3; ¨C5

Find the quotient and remainder of f(x) = x4 ¨C 2 divided by p(x) = x

¨C 1.

a. x3 + x2 + 1; ¨C1

b . x3 + x2 + x + 1; ¨C1

c . x3 + x + 1; ¨C1

d . x3 ¨C x2 ¨C x ¨C 1; ¨C1

Find a polynomial with leading coefficient 1 and degree 3 that has ¨C

1, 1, and 3 as roots.

a. x3 ¨C 3x2 ¨C x + 3

b . x3 ¨C 3x2 + x ¨C 3

c . x3 + 3x2 ¨C x ¨C 3

d . x3 + 3x2 + x + 3

The polynomial f(x) divided x ¨C 3 results in a quotient of x2 + 3x ¨C 5

with a remainder of 2. Find f(3).

a. ¨C5

b . ¨C2

c . 2

d . 3

Let f(x) = x3 ¨C 8x2 + 17x ¨C 9. Use the factor theorem to find other

solutions to f(x) ¨C f(1) = 0, besides x = 1.

a. ¨C2, 5

b . 2, ¨C3

c . 2, 5

d . 2, 10

Find the polynomial f(x) of degree three that has zeroes at 1, 2, and

4 such that f(0) = ¨C16.

a. f (x) = x3 − 7x2 +14x −16

b . f (x) = 2x3 −14x 2 + 28x −16

c . f (x) = 2x3 −14x 2 +14x −16

d . f (x) = 2x3 + 7x2 +14x +16

The degree three polynomial f(x) with real coefficients and leading

coefficient 1, has 4 and 3 + i among its roots. Express f(x) as a

product of linear and quadratic polynomials with real coefficients.

a. f (x) = (x + 4)(x 2 + 6x +10)

b . f (x) = (x − 4)( x 2 − 6x − 9)

c . f (x) = (x − 4)( x 2 − 6x +10)

d . f (x) = (x − 4)( x 2 − 6x + 9)

Given that (3x ¨C a)(x ¨C 2)(x ¨C 7) = 3x3 ¨C 32x2 + 81x ¨C 70, determine

the value of a.

a. 1

b . 3

c . 5

d . 7

Find all roots of the polynomial x3 ¨C x2 + 16x ¨C 16.

a. 1, 4, ¨C4

b . ¨C1, 4, ¨C4

c . ¨C1, 4i, ¨C4i

d . 1, 4i, ¨C4i

Find the vertical asymptote of the rational function f (x)= 3x-12

4x-2

a. x = 1/2

b . x = 3/4

c . x = 2

d . x = 4

Find the horizontal asymptote of the rational function f(x)=8x-12

4x-2

a. y = 1/2

b . y = 3/2

c . y = 2

d . y = 4

For f(x)+ ___1___, find the interval(s) where f(x) < 0.

x 2-2x-8

a. (¨C4, 2)

b . (¨C2, 4)

c . (2, 4)

d . (2, 8)

Express the following statement as a formula with the value of the

constant of proportionality determined with the given conditions: w

varies directly as x and inversely as the square of y. If x = 15 and

y = 5, then w = 36.

a. w=3__ x _

y2

b w=12__ x _

y2

c w=36__ x _

y2

d w=60__ x _

y2

The electrical resistance R of a wire varies directly as its length L

and inversely as the square of its diameter. A wire 20 meters long

and 0.6 centimeters in diameter made from a certain alloy has a

resistance of 36 ohms. What is the resistance of a piece of wire 60

meters long and 1.2 centimeters in diameter made from the same

material?

a. 24 ohms

b . 27 ohms

c . 30 ohms

d . 48 ohms

The period of a simple pendulum is directly proportional to the

square root of its length. If a pendulum has a length of 6 feet and a

period of 2 seconds, to what length should it be shortened to achieve

a 1 second period?

a 1 foot

b 1.5 feet

c 2 feet

d 3 feet

For the function f(x) shown in problem 1, find the domain and range

of f ¨C1(x).

a. Domain = [0, 6], Range [ 2, 5]

b . Domain = [0, 5], Range [ 2, 6]

c. Domain = [2, 5], Range [ 0, 6]

d. Domain = [2, 6], Range [ 0, 5]

For the function defined by f(x) =5x ¨C 4, find a formula for f ¨C1(x).

a. f- -1(x)=-5x+4

b. f- -1(x)= ___1___

5x-4

c. f- -1(x)= x+4

5

d. f- -1(x)=x+4

5

For the function defined by f(x) = 2 − x 2 , 0 ¡Ü x, use a sketch to help

find a formula for f ¨C1(x).

a. f −1(x) = x2 − 2, x ¡Ü 2

b. f −1(x) = 1 , 0 ¡Ü x

2-x2

c. f −1(x) =.-¡Ì2+¡Ìx, 0 ¡Ü x

d. f −1(x) = ¡Ì2-x, x¡Ü2

From the information in the table providing values of f(x) and g(x),

evaluate ( f o g)−1(3).

X 1 2 3 4 5

F(x) 5 3 5 1 2

g(x) 4 5 1 3 2

a. 1

b. 2

c. 4

d. 5

Solve the equation 42x+1=23x+6

a.-5

b. 2

c. 4

d. 5

Find an exponential function of the form f(x)=bax+c with y- intercept 2, horizontal asymptote y=-2, that passes through the point P(1,4).

a. f(x)=-2(2x)

b. f(x)=2(2x)-2

c. f(x)=2(1.5x)-2

d. f(x)=4(1.5x)-2

A bacteria culture started with a count of 480 at 8:00 A.M. and after

t hours is expected to grow to f(t)=480 (3/2)t. Estimate the number of bacteria in the culture at noon the same day.

a. 810

b. 1920

c. 2430

d. 4800

If a piece of real estate purchased for $75,000 in 1998 appreciates at

the rate of 6% per year, then its value t years after the purchase will

be f (t) = 75,000(1.06t ) . According to this model, by how much will the

value of this piece of property increase between the years 2005 and

2008?

a. $14,300

b. $21,500

c. $37,800

d. $59,300

The amount A in an account after t years of an initial principle P

invested at an annual rate r compounded continuously is given by A

= Pert where r is expressed as a decimal. What is the amount in the

account if $500 is invested for 10 years at the annual rate of 5%

compounded continuously?

a. $750.00

b. $800.00

c. $814.45

d. $824.36

The amount of a radioactive tracer remaining after t days is given

by A = A0 e¨C0.18t, where A0 is the starting amount at the beginning of

the time period. How much should be acquired now to have 40

grams remaining after 3 days?

a. 47.9 gm

b. 48.8 gm

c. 61.6 gm

d. 68.6 gm

Find the number log 5(1/5).

a. ¨C5

b. ¨C1

c. 0.2

d. 1

Solve loga (8x + 5) = loga (4x + 29)

a. 4

b. 5

c. 6

d. 8

The amount A in an account after t years from an initial principle P

invested at an annual rate r compounded continuously is given by A

= Pert where r is expressed as a decimal. Solve this formula for t in

terms of A, P, and r.

a. t=1n (AP)

r

b. t=1n(A)

rp

c. t=r1n (a)

p

d. t=1 1n (a)

r p

The decibel level of sound is given by D=10 log (I), where I is the sound intensity

10-12

measured in watts per square meter. Find the decibel level of a whisper at an intensity of 5.4*10-10 watts per square meter.

a. 2.73 decibels

b. 3.73 decibels

c. 27.3 decibels

d. 37.3 decibels

Given that loga (x) = 3.58 and loga (y) = 4.79, find loga (3)

a. 1.21 x

b. 1.34

c. 8.37

d. 17.1

Write the expression loga ( y + 5) + 2 loga (x +1) as one logarithm.

a. loga ( y + 2x + 7)

b. loga ( y + x 2 + 7)

c. loga [2( y + 5)(x +1)]

d. loga [( y + 5)(x +1)2 ]

Solve the equation ln(x + 5) ¨C ln(3) = ln(x ¨C 3).

a. 2.5

b. 4.5

c. 5

d. 7

The population P of a certain culture is expected to be given by a

model P = 100 ert where r is a constant to be determined and t is a

number of days since the original population of 100 was established.

Find the value of r if the population is expected to reach 200 in 3

days.

a. 0.231

b. 0.549

c. 1.098

d. 1.50

Find the exact solution to the equation 3x+5 = 9 x.

a. 5/3

b. 5/2

c. 5

d. 6

The amount A in an account after t years from an initial principle P

invested at an annual rate r compounded continuously is given by A

= Pert where r is expressed as a decimal. How many years will it

take an initial investment of $1,000 to grow to $1,700 at the rate of

4.42% compounded continuously?

a. 10 years

b. 11 years

c. 12 years

d 13 years

The amount of a radioactive tracer remaining after t days is given

by A = A0 e¨C0.058t, where A0 is the starting amount at the beginning of

the time period. How many days will it take for one half of the

original amount to decay?

a. 10 days

b. 11 days

c. 12 days

d. 13 days