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Algebra II – Wilsen
Unit 6: Higher Order Polynomials
Review for Midterm
BLOCK 5
I. Evaluate
4 + 2i
3− i
1)
(4 − i 2 )
5)
3 3 –1
3 2 –1
8)
Find the reciprocal of 5 + 2i in standard form.
9)
If m and na are real (not complex) numbers, under which of the following conditions
m + ni
will
be a non-zero real number? Show your work.
1+ i
2
2)
i 57
3)
6)
i 33 ( 2 + i ) ( 3 – i ) ( 2 + 4i )
(a)
m = 0 and n ≠ 0
b)
m ≠ 0 and n = 0
(d)
m=n≠0
e)
m = 2n ≠ 0
4)
7)
i –19
c)
10)
Factor –24x 3 + 81 over the integers.
11)
Factor 100x 2 + 329x – 408 over the integers.
12)
Factor 4x 2 – 16 over the complex numbers.
13)
Factor into linear factors: x 3 – 5x 2 – 2x + 24
14)
Which of the following is a factor of f (x) = x 4 + 2x 3 – 13x 2 – 14x + 24 ?
(
(a)
15)
–45
m = –n ≠ 0
)
x+3
(b)
x+4
(c)
x–2
(d)
x+1
x 6 + y 6 can be factored into
(a)
(x
(d)
cannot be factored
3
+ y3 )
2
(b)
(x
2
+ y2 )( x 4 – x 2 y2 + y 4 )
(c)
(x
2
+ y2 )
3
16)
Factor each over the complex numbers:
(a)
17)
18)
(b)
2x 3 – 72x
3x 2 + 9x – 84
(c)
9x 2 + 25
Given f (x) = – ( 2x – 8 ) – 1
2
a)
find and simplify f (x – 1) – f (x + 1)
b)
graph f(x), labeling coordinates of vertex and all intercepts (x and y).
Given the polynomial f (x) = 3x 3 – 4x 2 – 17x + 6
list all possible zeros
a)
19)
Solve over the complex numbers:
20)
6
x =
3
12 – 2
x
21)
x 2 – 4y 2
( x – 2y )2
22)
21
3
+
x – 49 x + 7
12 +
2
b)
factor the polynomial
x 3 729
–
=0
125 64
23)
Simplify each:
24)
Simplify
a)
25)
1
x+2
– 2
2x – 3 4x – 9
1
1–
2x + 3
b)
x 3 + 8 x 2 – 2x + 4
x +1
÷
• 2
2
x –4
6x – 12
3x + 6x + 3
Given f (x) = – ( 2x – 5 ) , find and simplify f (x + 1)
2
II.
Evaluate
1)
If f (x) = –x 2 and g(x) = 2x , find f ⎡⎣ g ( a + 2 ) ⎤⎦
2)
x 2 + 3x – 4
The expression
is undefined for what values of x?
2x 2 + 10x + 8
3)
Given f (x) = x – 4 and g(x) = x 2 – 5x + 4 , find
a)
g ⎡⎣ f ( 2i ) ⎤⎦
d)
f (x)
… simplify, state the domain and range, and graph
g(x)
b)
f ⎡⎣ g ( x ) ⎤⎦
c)
g ⎡⎣ f ( x ) ⎤⎦
1
.
2
4)
Find the value of the remainder obtained when 6x 4 + 5x 3 – 2x + 8 is divided by x –
5)
Given the function f (x) = x 2 – 4x + 39 , evaluate f (2 + 5i) .
6)
A ball is thrown upward from a height of 1 meter with an initial velocity of 14
meters/second. (The acceleration due to gravity is either 9.8 m / sec 2 or 32 ft / sec 2 .
(a) Write the equation for the height (h) of the ball in meters, in terms of time in
seconds.
(b) How long will the ball be in the air?
(c) When does the ball reach its maximum height?
(d) What is that maximum height?
7)
Given the function f (x) = 2x 2 + 4x – 9 , evaluate f (i) and f (3 – 2i) .
III.
Graph
1)
What are the coordinates of the vertex of y = 3x 2 – 6x + 2 ?
2)
The function f (x) = x 4 + 5x 2 + 4 is
a)
d)
odd
b)
even
neither odd nor even
c)
both odd and even
3)
Graph f (x) = 3x 2 – 12x + 20 , including coordinates of vertex and all intercepts.
4)
Find the domain of f (x) =
5)
Graph each of the following, and give their domain and range:
6)
7)
2x 2 + 8x + 6
.
x 4 + 7x 3 + 17x 2 + 17x + 6
a)
f (x) = 2 – x – 3
b)
f (x) = 2 [ x ] + 1 , –2 < x < 2
c)
g(x) = –
d)
f –1 ( x ) where f (x) = 4 – x 2 + 1
x–2
Given the graph of f (x) shown below, graph each
of the following:
a)
f (2x + 3) – 1
b)
⎛1 ⎞
f ⎜ x⎟
⎝2 ⎠
c)
f ( –x – 1)
d)
– f (–x)
e)
f ( x) – 1
f)
f 2x
g)
1
f (x)
h)
2 f (x – 1) + 1
Give all of the following: standard form, intercept form, vertex (max or min?),
y-intercept, axis of symmetry, symmetric point, real x-ints (if any), sketch.
(a)
(c)
f (x) = – ( x – 1) + 3
2
one of the x-ints is 2 + 3
(b)
f (x) =
1 2
x –x+3
2
8)
How many roots, and what kind? 2x 3 + 10x – 3 = 0
9)
10)
11)
A telephone company charges a basic charge for any call and then a fixed charge per
minute for every additional minute or portion of a minute. Sketch a reasonable graph
relating the cost of a call and the length of time a person talks.
12)
⎧2
⎪ 3 x + 4, – 7 ≤ x ≤ –2
⎪
– 2 < x ≤1
Sketch the graph of f (x) = ⎨ x 2 ,
⎪1,
x >1
⎪
⎩
13)
Given p(x) = x and r(x) = [ x ] , sketch p(x) + r(x) .
IV
Writing Equations
1)
What is the equation of the perpendicular bisector of the line segment whose end
points are (2, 6) and (–4, 3)?
2)
Line L passes through the origin and point (a, b). If ab ≠ 0, and line L has a slope
greater than 1, then which of the following must be true?
3)
Find all three forms of the equation of the parabola with vertex at (–1, 16) and an xintercept at 3.
4)
You are driving on the highway, having started with a full tank of gas. As you drive,
the number of minutes since you had the tank filled and the number of gallons
remaining in the tank are related by a linear function. After 30 minutes, you have 15
gallons left, and after 45 minutes, you have 11 gallons in the tank.
a)
b)
c)
d)
e)
f)
Write an equation in slope-intercept form for this function.
Sketch the graph, labeling the axes and all intercepts.
What the y-intercept of the line represent?
After how many minutes will you run out of gas?
What are the domain and range of the line?
What are the units of the slope of this function, and what does the slope
represent?
5)
A quadratic function has a root of 3 – 2i 3 and a y-intercept of –7. Write the
equation in standard form.
6)
Find a quadratic function that has a root of 3 + i 5 and a vertex at (3, –7.5).
7)
Find the domain of a parabola passing through (1, 3), (–2, 3) and (0, 1).
8)
2
Find a polynomial in standard form with integer coefficients that has zeros at – ,
3
4 + 5 and 4 – 5 .
9)
Given a linear function f (x) , with f (1800) = 1625 and f (2200) = 875 ,
a)
b)
find f (600)
find the value of x when f (x) = 3500
10)
Which of the following could be an equation of the
graph below?
11)
Which of the following could be an equation of the graph
below?
12)
Find equations for each of the following:
a)
c)
(b)
V
Inverses
1)
If f (x) = 5x 3 , then f –1 (x) is
2)
For which of the following functions is the inverse also a function?
3)
Given f (x) = 16 – x 2 , what is f –1 ⎡⎣ f –1 ( x ) ⎤⎦ ?
4)
What is the domain of the inverse of f (x) = 4 – x 2 + 1 ?
5)
What is the inverse of w(x) =
x–4
?
2x + 5
VI
Solve
1)
3– i
= x + yi
i + i18
2)
If f (x) = 2x and f ⎡⎣ g ( x ) ⎤⎦ = –x , then find g(x) .
3)
Solve for x and graph your solution on a number line:
37
(a)
2 ≤ 3x + 4 ≤ 7
(b)
(c)
3x + 7
x + 3 − a = 1– x
x + 15
x – 3 3– x
x−
x+3
x + 1+
4)
Solve for a:
5)
Given that 3x – 2 and 3x – 4 are two factors of the polynomial P(x) = 18x 3 – 9x 2 + ax + b
a)
b)
solve for a and b
find the third factor
2x – 1
> 4 is
x+3
6)
The solution set for
7)
Find c so that the line through (3, 2c + 5) and (5, 4) is perpendicular to the line
through (6, c + 1) and (4, 4).
2 – 2i ( 3 + i ) – 2yi
=
3– i
x
2
8)
5x
4
54x + 5
+
= 2
x – 5 x + 6 x + x – 30
10)
Find a if the y-intercept is 16 for y – 4 = –a ( x + 2 )
11)
Using the discriminant, it can be determined that the solutions to 9x 2 + 6x + 1 = 0 will
be:
9)
2
12)
Solve each of these for all complex solutions:
(a)
x 2 + 4x + 5 = 0
(b)
x 3 − x 2 – 13x – 13 = 0
(c)
x 2 – 4x + 9 = 0
(d)
x +1
x+2
7x + 1
+ 2
= 2
x – 2x – 3 x – 3x x + x
(e)
2x + 5 = 4
(f)
2x + 5 > –3
(g)
–2 2x + 5 = 4
(h)
–2 2x – 1 + 2 = –4x
(i)
–2 2x – 1 + 2 > –4x
(j)
(k)
(m)
2x + 25 – 2 x + 4 = 1
3
<1
x–5
2
7 – 6x + 3x = 2
(l)
x 2 – 3x + 4 ≤ 0
(m)
( 4x – 9 )2 ≤ 0
13)
Determine values of k so that ks 2 + 6x + k = 0 will have at least one real root.
14)
Solve 2 x – 4x – 3 =
1
4x – 3
VII
Systems of Equations
1)
(a)
3)
Determine j and k so that ( 2x – 1) and ( x + 3) are factors of 6x 3 + 17x 2 + jx + k . Then
find the third factor.
4)
Two roots of the equation 4x 3 – px 2 + qx – 2 p = 0 are 4 and 7. What is the third root?
5)
If a particular parabola has the points (0, 4), (1, 1) and (2, 0), what are the values of a, b
and c?
(b)
VIII
Applications
1)
A theater has a certain number of rows of seats with the same number of seats in each
row. If the theater could put 5 more seats to a row, it would need 20 rows less to seat
the same number. But if each row had 3 fewer seats, it would 20 more rows to seat the
same number. How many people will the theater seat?
2)
Bobbi’s goals per game average during the regular season was 1.25. During the
playoffs, she scored 1 goal in 2 games and her season goals per game average became
1.2. How many goals in how many games had she scored during the regular season?
3)
How much of an 85% salt solution should be added to 4 ounces of water to produce a
15% salt solution?
4)
A woman walks the first part of her journey at 3 mph. She continues her journey
jogging at 7 mph. Her journey is a total of 8 miles, in 1.5 hours. How much time does
she spend walking?
5)
Jane and Joe work together on a job for 3 hours, and then Jane stops. Joe takes over
the job alone and finishes it in another 3 hours. If Jane and Joe had worked together
the whole time, it would have taken them 4 hours total to do the job. How long would
it have taken each of them to do the job entirely alone?
6)
A rectangular parking lot bordered on one side by a highway is to be enclosed on the
other 3 sides by a total of 420 feet of fencing. Write a quadratic function representing
the area of the parking lot as of function of either the width or the length of the lot.
What should be the dimensions of the lot to get the maximum amount of area
possible?
7)
A handful of change is made up of nickels, dimes and quarters. The nickels and
quarters together are worth $1.95, and the dimes and quarters together are worth
$2.25. The nickels and dimes together account of 9 of the coins. How many of each
type of coin are there?
8)
Determine the dimensions of a rectangle having a perimeter of 53 feet and an area of
175 square feet.
9)
There are available 10 tons of coal containing 2.5% sulfur, and also supplies of coal
containing 0.80% and 1.10% sulfur respectively. How many tons of each of the latter
should be mixed with the original 10 tons to give 20 tons containing 1.7% sulfur?
10)
A tank can be filled by two pipes separately in 10 minutes and 15 minutes respectively.
When a third pipe is used simulataneously with the first two pipes, th tank can be
filled in 4 minutes. How long would it take the third pipe alone ot fill the tank?
11)
By increasing her average speed by 10 mph, a motorist could save 36 minutes in
traveling a distance of 120 miles. Find her actual average speed.
12)
The tens digit of a certain two-digit number is twice the units digit. If the number is
multiplied by the sum of its digits, the product is 63. Find the number.
13)
Team A can count the ballots in Palm Beach County in 5 days. If Team B helps, the
job will be done in 3 days. Find how long it would take Team B to count the ballots
alone.
14)
You make a road trip that takes 4.5 hours. When you are on the highway, your
average speed is 55 mph. On secondary roads, your average speed is 36 mph. If the
length of the trip is 200 miles, how many miles of the trip are on the highway?
15)
How much of an 80% sugar solution must be added to 3 gallons of a 10% solution to
make a 50% solution?
16)
A ball is thron up into the air, reaches its maximum height, and then comes back
down. One second after it is thrown, it is at a height of 30.1 meters; 2 seconds after
being thrown, it is at a height of 40.4 meters; and 3 seconds after it’s thrown, it is at a
height of 40.9 meters. What’s the maximum height the ball reaches, and when does it
reach that height? When does the ball hit the ground? When does the ball reach a
height of 35 meters? From what height was the ball thrown?
IX
Other
1)
Given g(x) = – ( x – 2 ) – 1 ,
3
a)
Find the equation for g –1 (x) .
b)
Evaluate g(4), g –1 (–2) and g ⎡⎣ g –1 ( 98 ) ⎤⎦
Is g –1 (x) a function?
Graph g(x) and g –1 (x) .
c)
d)
2)
For each below, state the domain and range:
(a)
(b)
f (x) = 4x + 8
(d)
k(x) = 3x 2 − 1
(f)
f (x) =
just domain now:
(c)
(e)
f (x) =
g(x) =
1
6 − 2 1− x
x 2 – 7x + 12
x 2 – 16
2x − 1
−1
x+5
3)
4)
Which one of the following could possible by a root of 4x 3 – px 2 + qx – 6 = 0 ?
5)
Give the first eight terms of the sequence of numbers given by
⎧ fn–1 – 4 when fn–1 ≥ 5
f1 = 7 and fn = ⎨
⎩2 • fn–1 + 3 when fn–1 < 5
6)
Divide each of the following:
a)
7)
3x 3 + 17x 2 + 13x + 2
3x + 2
(b)
4x 3 – 5x – 6
2x 2 + 3x + 2
c)
x 3 + 216
x+6
Complete the square and use your equation to solve for the x-intercepts:
2mx 2 – 3nx + 4 p = 0
8)
Tell whether each of the following is even, odd, or neither:
a)
9)
10)
f (x) = x – 4
b)
f (x) = – x – 4
c)
y = x 5 – 2x 3
a)
Given the two equations 3x – 4y = 12 and y = cx + 10 , find the value for c so that
these two lines are parallel.
b)
Given the two equations y = –3x + 6 and dx + y = 8 , find the value for d so that
these two lines are perpendicular.
Questions about types of numbers. True or false? If false, give a counterexample.
When you subtract a whole number from a whole number, the answer is always
a whole number.
The sum of an irrational number and a rational number is always irrational.
The product of two rational numbers is always a rational number.
The quotient of two irrational numbers is always irrational.
11)
Graph the solution of the system:
y > 2|x – 3|
y<4–x
12.
Simplify:
(a) 4 98 pq 5 r 6
( 3x y )( 5xy )
(x ) y
–2
(d)
13.
(b) – 3 27m 3n 6
3
–8
–3 4
e)
–2
4 – 3 2 • 3–2
8 –19 –1
Graph and show the solution of this set of inequalities: (Remember: The solution is
where all the graphs overlap.)
1
y ≥ − x +1
2
y < −3x + 5
(
y >2 x +4
14.
(c) 2 3 –54x 5 y 3
)
2
+1
For the graph, determine its equation in factored
form. The y-intercept is (0, -9).
15.
Solve for k so that x – 4 will be one of the factors of x 3 + kx 2 – 17x – 60 .
16.
Simplify without leaving any parentheses or negative exponents in your answer.
( 4x y )( 5xy )
( 2x ) y
–3 3
–7
–3 5
⎛ 3–5 ⎞
⎜⎝ 2 –5 ⎟⎠
–4
–2
( 4x ) ( –2x
3 2
⎛ –4x 2n ⎞
⎜⎝ x –3n z 2 ⎟⎠
–3
– 9x 2 – 6xy + y 2
3x
–1 3
x
17.
⎛ 6⎞
⎜⎝ ⎟⎠
7
1
( –4 )–2
+x y
5 3
2 3
x 7 y+1
x 7 y–5
)
–3 –1 3
y
32 y−2 x ⋅ 3y− x
3y−3x
( –n )4 ( 2xy 3n )
–1
(
m –3 4m 2 + m –5
3
(
+ 4xy 3n 2
)
) ( –3xn )
2
3
⎛ n 2 1 ⎞
n
⎜⎝ 25 ⋅ 5 ⋅ − n ⎟⎠ − 125
5
3n
5 ⋅ 25
Factor completely:
(a)
3x 2 + 2x – 1
(c)
axz + 3z – 5ax – 15
(e)
x 2 z 2 – 25x 2 – 16z 2 + 400
(b)
125x 3 y 3 – c 3d 3
(d)
40cx 2 – 57cx + 20c
18.
Sketch y = a(x – 1)3 (x + 5)2 (x + 3) if a is negative.
19.
The domain of a function f (x) is x > 5 . What is the domain of the function that has
1
been transformed by f (x – 8) + 2 ?
2
The range of that same function is –8 ≤ y ≤ 7 . What is the range of the function that
1
has been transformed also by f (x – 8) + 2 ?
2
21.
Three points, A (–2, 2) and B (1, –1) and C (5, 4) undergo the transformation
⎛1
⎞
–2 f ⎜ x – 3⎟ – 4 . Give the new coordinates of A, B and C. You do not need to graph
⎝2
⎠
them.
22.
(a)
Solve for x:
(b)
Solve for y:
(c)
Solve for a:
1
2
a ( r + 2 p ) – a ( 2r + p ) = x + 2
3
5
(d)
Solve for z:
1 1 1
+ + =1
x y z
23.
2
1
y + xy =
5
4
7
1
xy = x + y
2
3
Write a single absolute value inequality that represents each of the graphs below:
(a)
(c)
(b)