Algebra II/IIH Semester 1 Exam: Units 3,4,5 ___ Class: _ Date: ___ ID: A 1 Algebra II/IIH Semester 1 Exam: Units 3,4,5 Multiple Choice Identify the choice that best completes

Name: ________________________ Class: ___________________ Date: __________ ID: A

1

Algebra II/IIH Semester 1 Exam: Units 3,4,5

Multiple Choice

Identify the choice that best completes the statement or answers the question.

23 The parent function f(x) = x2 is reflected across

the x-axis, vertically stretched by a factor of 10,

and translated right 10 units to create g. Use the

description to write the quadratic function in

vertex form.

A g(x) = 10(x + 10)2

B g(x) = −10(x − 10)2

C g(x) = 10(x − 10)2

D g(x) = −10(x + 10)2

24 Identify the axis of symmetry for the graph of

f(x) = x2+ 2x − 3.

A x = −1

B y = −4

C y = −1

D x = −4

25 Find the minimum or maximum value of

f(x) = x2− 2x − 6. Then state the domain and

range of the function.

A The maximum value is 1. D: {all real

numbers}; R: {y | y ≥ –7}

B The minimum value is –7. D: {x | x ≥ –7 };

R: {all real numbers}

C The maximum value is 1. D: {x | x ≥ –7 }; R:

{all real numbers}

D The minimum value is –7. D: {all real

numbers}; R: {y | y ≥ –7}

26 What quadratic function does the graph represent?

A f(x) = x2+ 8x − 14 C f(x) = −x

2+ 8x + 14

B f(x) = −x2+ 8x − 14 D f(x) = −x

2− 8x − 14

Name: ________________________ ID: A

2

27 Find the zeros of the function h x( ) = x2+ 23x + 60 by factoring.

A x = 4 or x = 15 C x = −4 or x = −15

B x = −20 or x = −3 D x = 20 or x = 3

28 Write a quadratic function in standard form with zeros 6 and –8.

A 0 = x2+ 2x − 48 C f(x) = x

2− 2x − 48

B f(x) = x2+ 2x − 48 D f(x) = x

2− 4x + 4

29 Complete the square for the expression

x2− 16x + ____. Write the resulting expression

as a binomial squared.

A x − 8( )2

B x + 8( )2

C x + 16( )2

D x − 16( )2

30 Solve the equation x2− 10x + 25 = 54.

A x = 5±3 6

B x = 5+3 6

C x = 5−3 6

D x = 5± 6 3

31 Express 8 −84 in terms of i.

A −16i 21

B −5376

C −16 21

D 16i 21

32 Find the zeros of the function f(x) = x2+ 6x + 18.

A x = 3i or –3i

B x = –3 + 3i or –3 – 3i

C x = –3 + 3i

D x = –6 + 3i or –6 – 3i

33 Find the complex conjugate of 3i + 4.

A −4− 3i

B −4+ 3i

C 4 + 3i

D 4 − 3i

34 Solve the inequality x2− 14x + 45 ≤ −3 by using

algebra.

A x ≤ 5 or x ≥ 9

B x ≤ 6 or x ≥ 8

C 6 ≤ x ≤ 8

D 5 ≤ x ≤ 9

35 Determine whether the data set could represent a

quadratic function. Explain.

x –4 –2 0 2 4

y 15 5 –1 –3 –1

A The x-values are not evenly spaced, so this

could not be a quadratic function.

B The first differences between y-values are

constant for equally spaced x-values, so it

could represent a quadratic function.

C The 2nd differences between y-values are

constant for equally spaced x-values, so it

could represent a quadratic function.

D The 2nd differences between y-values are not

constant, so this could not be a quadratic

function.

36 Subtract. Write the result in the form a + bi.

(5 – 2i) – (6 + 8i)

A –1 – 10i

B 7 – 2i

C –3 – 8i

D 11 + 6i

37 What expression is equivalent to (3 − 2i)2?

A 13− 12i

B 13

C 5 − 12i

D 9 + 4i

Name: ________________________ ID: A

3

38 Graph the complex number 4 + 2i.

A

B

C

D

39 Identify the degree of the monomial −5r3s5.

A 5

B 8

C 3

D –5

40 Rewrite the polynomial 12x2 + 6 – 7x5 + 3x3 + 7x4

– 5x in standard form. Then, identify the leading

coefficient, degree, and number of terms. Name

the polynomial.

A −7x5+ 7x

4+ 3x

3+ 12x

2− 5x + 6

leading coefficient: –7; degree: 5; number of

terms: 6; name: quintic polynomial

B 6− 5x + 12x2+ 7x

3+ 3x

4− 7x

5

leading coefficient: 6; degree: 0; number of


C −7x5+ 7x

4+ 12x

3+ 3x

2− 5x + 6

leading coefficient: –7; degree: 5; number of


D 6− 5x + 12x2+ 3x

3+ 7x

4− 7x

5

leading coefficient: 6; degree: 0; number of


Name: ________________________ ID: A

4

41 Add. Write your answer in standard form.

(5a5− a

4) + (a

5+ 7a

4− 2)

A 6a5+ 6a

4C 6a

5+ 6a

4− 2

B 6a10+ 6a

8− 2 D 5a

5+ 7a

4− 2

42 Find the product 5x − 3( )(x3− 5x + 2).

A 5x4− 3x

3+ 25x

2− 5x − 6

B 5x3− 28x

2+ 25x − 6

C 5x4− 3x

3− 25x

2+ 25x − 6

D 5x3+ 22x

2− 5x − 6

43 Find the product (x − 2y)3.

A x3− 6x

2y + 12xy

2− 8y

3

B x3+ 8y

3

C x3+ 6x

2y + 12xy

2+ 8y

3

D x3− 8y

3

44 Divide by using long division:

(5x + 6x3− 8) ÷ (x − 2).

A 6x2− 12x + 29 −

64

(x − 2)

B 6x2+ 12x + 29

C 6x2+ 12x + 29 +

50

(x − 2)

D 6x2+ 5−

8

(x − 2)

45 Divide by using synthetic division.

(x2− 9x + 10) ÷ x − 2( )

A x − 11+32

x − 2C 2x − 18+

10

x − 2

B x − 9 +6

x − 2D x − 7 +

−4

x − 2

46 Factor the expression 81x6+ 24x

3y3.

A 3x3(3x + 2y)(9x

2− 6xy + 4y

2) C 3x

3(3x + 2y)

3

B 3x3(3x + 2y)(9x

2+ 6xy + 4y

2) D 3x

3(27x

3+ 8y

3)

47 Identify the roots of −3x3− 21x

2+ 72x + 540 = 0. State the multiplicity of each root.

A x + 5 is a factor once, and x − 6 is a factor twice.

The root 5 has a multiplicity of 1, and the root −6 has a multiplicity of 2.

B x − 5 is a factor once, and x + 6 is a factor twice.

The root 5 has a multiplicity of 1, and the root −6 has a multiplicity of 2.

C x + 5 is a factor once, and x − 6 is a factor twice.

The root −5 has a multiplicity of 1, and the root 6 has a multiplicity of 2.

D x − 5 is a factor once, and x + 6 is a factor twice.

The root −5 has a multiplicity of 1, and the root 6 has a multiplicity of 2.

Name: ________________________ ID: A

5

48 Write the simplest polynomial function with the zeros 2− i, 5 , and −2.

A P x( ) = x6− 4x

5− 4x

4+ 36x

3− 25x

2− 80x + 100 = 0

B P x( ) = x5− 2x

4− 8x

3− 20x

2− 65x − 50 = 0

C P x( ) = x5− 2x

4− 8x

3+ 20x

2+ 15x − 50 = 0

D P x( ) = x5− 2x

4− 10x

3+ 16x

2+ 25x − 30 = 0

49 Identify whether the function graphed has an odd

or even degree and a positive or negative leading

coefficient.

A The degree is even, and the leading

coefficient is negative.

B The degree is odd, and the leading coefficient

is negative.

C The degree is odd, and the leading coefficient

is positive.

D The degree is even, and the leading

coefficient is positive.

50 Write a function that transforms f(x) = 2x3+ 4 in

the following way:

stretch vertically by a factor of 6 and shift 5 units

left.

A g(x) = 12(x + 5)3+ 24

B g(x) = 12(x + 5)3+ 4

C g(x) = 12x3+ 9

D g(x) = 12(x − 5)3+ 4

51 The daily profit of a bicycle store can be modeled

by f(x) = x3− 5x

2+ 2x + 2 where x is the number

of bicycles sold. Let g(x) = f(x + 4). Find the rule

for g, and explain the meaning of the

transformation in terms of daily profit.

A g(x) = x3− 5x

2+ 2x + 6

The shop makes the same profit after selling

4 fewer bicycles.

B g(x) = x3+ 7x

2+ 10x − 6


4 more bicycles.

C g(x) = x3− 5x

2+ 2x + 6


4 fewer bicycles.

D g(x) = x3+ 7x

2+ 10x − 6


4 fewer bicycles.

Name: ________________________ ID: A

6

52 What quartic function does the graph represent?

A f(x) = x + 2( ) 2x + 1( ) 2x − 1( ) 2x − 3( )

B f(x) = x + 2( )(1

2x + 1)(

1

2x − 1)(

1

2x − 3)

C f(x) = x − 2( ) 2x − 1( ) 2x + 1( ) 2x + 3( )

D f(x) = x − 2( )(1

2x − 1)(

1

2x + 1)(

1

2x + 3)

53 Graph the function f(x) = 4 x + 53

, and identify

its domain and range.

A

The domain is the set of all real numbers.

The range is also the set of real numbers.

B



C



D



Name: ________________________ ID: A

7

54 Simplify the expression 256z164

. Assume that

all variables are positive.

A 2564

z4

B 4z4

C 4z11

D 2564

z11

55 Write the expression 8

5

3

in radical form, and simplify. Round to the nearest whole number if necessary.

A ( 85

)3; 3 C 8

5

3

; 32

B ( 83

)5; 32 D

85

83; 64

56 Write the expression 10811

by using rational

exponents.

A 10−3

B 10

8

11

C 10

11

8

D 103

57 The parent function f(x) = x is stretched

horizontally by a factor of 4, reflected across the

y-axis, and translated left 2 units. Write the

square-root function g.

A g(x) = −1

4x − 2( )

B g(x) = −1

4x + 2( )

C g(x) = −4 x + 2( )

D g(x) = −1

4x + 2( )

58 Solve 11x = 3 x + 2 .

A x = 14

B x = 11

C x = 9

D x = 18

59 Solve x + 31 = x + 1.

A No solution.

x + 31 is not defined for x < −31.

B x = 5 or x = −6

C x = 5

D x = –6

Name: ________________________ ID: A

8

Short Answer

60 Consider the function f(x) = −4x2− 8x + 10. Determine whether the graph opens upward or downward. Find the

axis of symmetry, the vertex and the y-intercept. Graph the function.

ID: A

1

Algebra II/IIH Semester 1 Exam: Units 3,4,5

Answer Section

MULTIPLE CHOICE

23 ANS: B

Identify how each transformation affects the coefficients in vertex form.

reflection across x-axis a is negative.

vertical stretch by a factor of 10 a| | = 10

translated right 10 units h = 10

Write the transformed function, using the vertex form g(x) = a(x – h)2 + k.

g(x) = −10(x − 10)2+ 0 Substitute –10 for a, 10 for h, and 0 for k.

g(x) = −10(x − 10)2 Simplify.

Feedback

A The vertex form is g(x) = a(x-h)^2 + k.

B Correct!

C When reflecting across the x-axis, a becomes negative.

D The variable h is positive when translating to the right.

PTS: 1 DIF: Average REF: 156454f6-4683-11df-9c7d-001185f0d2ea

OBJ: 2-1.4 Writing Transformed Quadratic Functions NAT: NT.CCSS.MTH.10.9-12.F.BF.3

LOC: MTH.C.10.07.06.04.003 TOP: 2-1 Using Transformations to Graph Quadratic Functions

KEY: quadratic | graph | parent function MSC: DOK 3

ID: A

2

24 ANS: A

If a function has one zero, use the x-coordinate of the vertex to find the axis of symmetry.

If a function has two zeros, use the average of the two zeros to find the axis of symmetry.

Feedback

A Correct!

B The axis of symmetry of a parabola is a vertical line. All the points it contains have the

same x-value, so the variable in the equation should be x and not y.

C The axis of symmetry of a parabola is a vertical line. All the points it contains have the

same x-value, so the variable in the equation should be x and not y.

D Look at the graph. Does the line you found divide the parabola into two symmetrical

halves?

PTS: 1 DIF: Average REF: 1566b752-4683-11df-9c7d-001185f0d2ea

OBJ: 2-2.1 Identifying the Axis of Symmetry NAT: NT.CCSS.MTH.10.9-12.F.IF.7

LOC: MTH.C.10.07.06.01.007 TOP: 2-2 Properties of Quadratic Functions in Standard Form

MSC: DOK 3

ID: A

3

25 ANS: D

Step 1 Determine whether the function has a minimum or maximum value.

Because a is positive, the graph opens upward and has a minimum value.

Step 2 Find the x-value of the vertex.

x = −

b

2a= −

−2

2(1)= 1

Step 3 Find the y-value of the vertex.

f(1) = (1)2 – 2(1) – 6 = –7

The minimum value is –7. The domain is {all real numbers}. The range is {y | y ≥ –7}.

Feedback

A The maximum or minimum value is the y-value of the vertex.

B Check the domain and range.

C If a parabola opens upward, then there is a minimum value. If a parabola opens

downward, then there is a maximum value.

D Correct!

PTS: 1 DIF: Average REF: 156940be-4683-11df-9c7d-001185f0d2ea

OBJ: 2-2.3 Finding Minimum or Maximum Values NAT: NT.CCSS.MTH.10.9-12.F.IF.7

LOC: MTH.C.10.07.06.015 | MTH.C.10.07.06.017

TOP: 2-2 Properties of Quadratic Functions in Standard Form KEY: maximum | minimum

MSC: DOK 3

26 ANS: B

For f(x) = ax2+ bx + c , where a, b, and c are real numbers and a ≠ 0, the parabola has these properties:

1. The parabola opens upward if a > 0 and downward if a < 0.

2. The axis of symmetry is x = −

b

2a.

3. The vertex is the point −b

2a,f −

b

2a

Ê

Ë

ÁÁÁ

ˆ

¯

˜̃˜

Ê

Ë

ÁÁÁÁ

ˆ

¯

˜̃˜̃ .

4. The y-intercept is c.

Feedback

A The parabola opens downward so the coefficient for x^2 must be negative.

B Correct!

C The vertex of the graph must be a solution of the equation.

D The vertex of the graph must be a solution of the equation.

PTS: 1 DIF: Advanced REF: 156dde66-4683-11df-9c7d-001185f0d2ea

LOC: MTH.C.10.07.06.013 TOP: 2-2 Properties of Quadratic Functions in Standard Form

MSC: DOK 3

ID: A

4

27 ANS: B

h x( ) = x2+ 23x + 60

x2+ 23x + 60 = 0 Set the function equal to 0.

(x + 20)(x + 3) = 0 Factor: Find factors of 60 that add to 23.

x + 20 = 0 or x + 3 = 0 Apply the Zero-Product Property.

x = −20 or x = −3 Solve each equation.

Feedback

A To factor h(x), find factors of the constant term whose sum is the coefficient of the x

term. Set each factor equal to 0 and solve for x.

B Correct!

C To factor h(x), find factors of the constant term whose sum is the coefficient of the x


D To factor h(x), find factors of the constant term whose sum is the coefficient of the x


PTS: 1 DIF: Basic REF: 157040c2-4683-11df-9c7d-001185f0d2ea

OBJ: 2-3.2 Finding Zeros by Factoring

NAT: NT.CCSS.MTH.10.9-12.A.REI.4 | NT.CCSS.MTH.10.9-12.F.IF.8

LOC: MTH.C.10.07.06.018 TOP: 2-3 Solving Quadratic Equations by Graphing and Factoring

KEY: solve quadratic equations MSC: DOK 3

28 ANS: B

x = 6 or x = −8 Write the zeros as solutions for two equations.

x − 6 = 0 or x + 8 = 0 Rewrite each equation so that it is equal to 0.

0 = (x − 6)(x + 8)Apply the converse of the Zero-Product Property to write a

product that is equal to 0.

0 = x2+ 2x − 48 Multiply the binomials.

f(x) = x2+ 2x − 48 Replace 0 with f(x)

Feedback

A Replace 0 with f(x) for a function in standard form.

B Correct!

C Write two initial equations where the zeros given are set equal to x.

D Set each of the zeros given equal to x; do not combine them.

PTS: 1 DIF: Average REF: 1575057a-4683-11df-9c7d-001185f0d2ea

OBJ: 2-3.5 Using Zeros to Write Function Rules NAT: NT.CCSS.MTH.10.9-12.A.APR.2

LOC: MTH.C.10.07.06.04.001 | MTH.C.10.07.06.04.005

TOP: 2-3 Solving Quadratic Equations by Graphing and Factoring

MSC: DOK 3

ID: A

5

29 ANS: A

−16

2

Ê

ËÁÁÁ

ˆ

¯˜̃˜2

= −8( )2

= 64 Find b

2

Ê

Ë

ÁÁÁ

ˆ

¯

˜̃˜

2

.

x2− 16x + 64 Add.

x − 8( )2 Factor.

Feedback

A Correct!

B Is b positive or negative?

C Add (b/2)^2 to the given expression, then factor.

D Add (b/2)^2 to the given expression, then factor.

PTS: 1 DIF: Average REF: 157767d6-4683-11df-9c7d-001185f0d2ea

OBJ: 2-4.2 Completing the Square NAT: NT.CCSS.MTH.10.9-12.A.REI.4

STA: NV.NVAS.MTH.06.9-12.2.12.6.1 LOC: MTH.C.10.06.04.01.007

TOP: 2-4 Completing the Square KEY: complete the square | solve quadratic equations

MSC: DOK 3

30 ANS: A

x2− 10x + 25 = 54

x − 5( )2

= 54 Factor the perfect square trinomial.

x − 5 = ± 54 Take the square root of both sides.

x = 5± 54 Add 5 to each side.

x = 5±3 6 Simplify.

Feedback

A Correct!

B Does the right side of the equation have only a positive square root?

C Does the right side of the equation have only a negative square root?

D You switched the number to the left of the radical sign with the number under the

radical sign.

PTS: 1 DIF: Average REF: 15752c8a-4683-11df-9c7d-001185f0d2ea

OBJ: 2-4.1 Solving Equations by Using the Square Root Property

NAT: NT.CCSS.MTH.10.9-12.A.REI.4 STA: NV.NVAS.MTH.06.9-12.2.12.6.1

LOC: MTH.C.10.06.04.01.002 TOP: 2-4 Completing the Square

MSC: DOK 3

ID: A

6

31 ANS: D

8 −84

= 8 (−1)(84) Factor out –1.

= 8 −1 84 Product Property

= 16 21 −1 Simplify.

= 16i 21 Express in terms of i.

Feedback

A The imaginary unit is the same as the square root of –1.

B Simplify the square root.

C The imaginary unit is in the solution.

D Correct!

PTS: 1 DIF: Average REF: 157c2c8e-4683-11df-9c7d-001185f0d2ea

OBJ: 2-5.1 Simplifying Square Roots of Negative Numbers NAT: NT.CCSS.MTH.10.9-12.N.CN.1

LOC: MTH.C.10.03.003 | MTH.C.10.03.01.006 TOP: 2-5 Complex Numbers and Roots

KEY: complex numbers MSC: DOK 2

32 ANS: B

x2 + 6x + 18 = 0 Set f(x) = 0.

x2 + 6x = –18 Rewrite.

x2 + 6x + 9 = −18+ 9Add

b

2

Ê

Ë

ÁÁÁ

ˆ

¯

˜̃˜

2

to both sides of the equation.

(x + 3)2 = –9 Factor.

x + 3 =± −9

Take square roots.

x = −3± 3i Simplify.

Feedback

A Solve for x by completing the square.

B Correct!

C There are two complex roots.

D Add –b/2 to both sides of the equation.

PTS: 1 DIF: Average REF: 1580f146-4683-11df-9c7d-001185f0d2ea

OBJ: 2-5.4 Finding Complex Zeros of Quadratic Functions

NAT: NT.CCSS.MTH.10.9-12.N.CN.7 | NT.CCSS.MTH.10.9-12.A.REI.4

LOC: MTH.C.10.07.06.018 | MTH.C.10.07.06.019 TOP: 2-5 Complex Numbers and Roots


ID: A

7

33 ANS: D

3i + 4 = 4 + (3)i Rewrite as a + bi.

= 4 − (3)i Find a − bi.

= 4 − 3i Simplify.

Feedback

A You changed the sign of both the real and imaginary parts. Only change the sign of the

real part.

B You changed the sign of the real part. Only change the sign of the imaginary part.

C This is the number in a + bi form. Now find the complex conjugate.

D Correct!

PTS: 1 DIF: Basic REF: 158353a2-4683-11df-9c7d-001185f0d2ea

OBJ: 2-5.5 Finding Complex Conjugates NAT: NT.CCSS.MTH.10.9-12.N.CN.3

LOC: MTH.C.10.03.009 TOP: 2-5 Complex Numbers and Roots


ID: A

8

34 ANS: C

Step 1

x2− 14x + 45 = −3 Write the related equation.

x2− 14x + 48 = 0 Write the equation in standard form.

(x − 6)(x − 8) = 0 Factor.

Step 2x − 6 = 0 or x − 8 = 0 Find the critical values.x = 6 or x = 8 The critical values are 6 and 8.

Step 3

The critical values divide the number line into three intervals: x < 6, 6<x < 8, or x > 8.

Test an x-value in each interval in the original inequality to determine which intervals make the inequality true.

The solution is 6 ≤ x ≤ 8.

Feedback

A Subtract values from both sides of the equal sign when solving an equation.

B The critical values are correct. Test values of x to find the intervals satisfy the

inequality.

C Correct!

D Subtract values from both sides of the equal sign when solving an equation.


OBJ: 2-7.3 Solving Quadratic Inequalities Using Algebra NAT: NT.CCSS.MTH.10.9-12.A.REI.4

LOC: MTH.C.10.08.04.01.01.002 TOP: 2-7 Solving Quadratic Inequalities

KEY: quadratic inequality MSC: DOK 3

ID: A

9

35 ANS: C

x –4 –2 0 2 4

y 15 5 –1 –3 –1

1st differences −10 −6 −2 2

2nd differences 4 4 4

As the 2nd differences are constant for equally spaced x-values, the data set could represent a quadratic

function.

Feedback

A The x-values are evenly spaced.

B The first differences of y-values do not need to be evenly spaced for this to be a

quadratic function.

C Correct!

D The second differences of y-values are evenly spaced.

PTS: 1 DIF: Average REF: 1591a1ca-4683-11df-9c7d-001185f0d2ea

OBJ: 2-8.1 Identifying Quadratic Data LOC: MTH.C.10.07.06.001 | MTH.C.10.07.06.002

TOP: 2-8 Curve Fitting with Quadratic Models KEY: quadratic inequality

MSC: DOK 3

36 ANS: A

To add complex numbers, add the real parts and the imaginary parts. To subtract complex numbers, subtract the

real parts and the imaginary parts.

(5 – 2i) – (6 + 8i) = (5 – (6)) + (–5 – (5))i = –1 – 10i

Feedback

A Correct!

B Add or subtract real parts and imaginary parts.

C Add or subtract real parts and imaginary parts.

D Check whether you should add or subtract the two complex numbers.

PTS: 1 DIF: Average REF: 159db4a6-4683-11df-9c7d-001185f0d2ea

OBJ: 2-9.3 Adding and Subtracting Complex Numbers NAT: NT.CCSS.MTH.10.9-12.N.CN.2

LOC: MTH.C.10.03.01.002 TOP: 2-9 Operations with Complex Numbers

MSC: DOK 3

ID: A

10

37 ANS: C

(3 − 2i)2

= (3 − 2i)(3− 2i)

= 9− 6i − 6i + 4i2 Multiply.

= 9− 12i − 4 Combine like terms. i2= −1.

= 5 − 12i Simplify.

Feedback

A i squared is equal to –1.

B First, expand the square and multiply. Then, combine like terms and simplify.

C Correct!

D First, expand the square and multiply. Then, combine like terms and simplify.

PTS: 1 DIF: Advanced REF: 15a71706-4683-11df-9c7d-001185f0d2ea

NAT: NT.CCSS.MTH.10.9-12.N.CN.2 LOC: MTH.C.10.03.01.003

TOP: 2-9 Operations with Complex Numbers MSC: DOK 3

38 ANS: B

The real axis is the x-axis, and the imaginary axis is the y-axis. Think of a + bi as x + yi. Thus the complex

number 4 + 2i is at (4, 2).

Feedback

A The real axis is the x-axis, and the imaginary axis is the y-axis.

B Correct!

C If the imaginary part is positive, the point lies above the real (or x) axis.

D The real axis is the x-axis, and the imaginary axis is the y-axis.

PTS: 1 DIF: Basic REF: 1598c8de-4683-11df-9c7d-001185f0d2ea

OBJ: 2-9.1 Graphing Complex Numbers NAT: NT.CCSS.MTH.10.9-12.N.CN.4

LOC: MTH.C.10.03.012 TOP: 2-9 Operations with Complex Numbers

KEY: graph complex number MSC: DOK 2

39 ANS: B

Add the exponents of the variables. 3 + 5 = 8

The degree is 8.

Feedback

A Add the exponents of the variables.

B Correct!

C Add the exponents of the variables.

D The degree of the monomial is the sum of the exponents of the variables.

PTS: 1 DIF: Basic REF: 15d6ed46-4683-11df-9c7d-001185f0d2ea

OBJ: 3-1.1 Identifying the Degree of a Monomial LOC: MTH.C.10.05.08.006

TOP: 3-1 Polynomials MSC: DOK 2

ID: A

11

40 ANS: A

The standard form is written with the terms in order from highest to lowest degree.

In standard form, the degree of the first term is the degree of the polynomial.

The polynomial has 6 terms. It is a quintic polynomial.

Feedback

A Correct!

B The standard form is written with the terms in order from highest to lowest degree.

C Find the correct coefficient of the x-cubed term.

D The standard form is written with the terms in order from highest to lowest degree.


OBJ: 3-1.2 Classifying Polynomials

LOC: MTH.C.10.05.08.004 | MTH.C.10.05.08.006 | MTH.C.10.05.08.007


41 ANS: C

(5a5− a

4) + (a

5+ 7a

4− 2)

= (5a5+ 7a

4) + (−a

4+ a

5) + −2( )

Identify like terms. Rearrange terms to get like terms

together.

= 6a5+ 6a

4− 2 Combine like terms.

Feedback

A Check that you have included all the terms.

B When adding polynomials, keep the same exponents.

C Correct!

D First, identify the like terms and rearrange these terms so they are together. Then,

combine the like terms.

PTS: 1 DIF: Basic REF: 15db8aee-4683-11df-9c7d-001185f0d2ea

OBJ: 3-1.3 Adding and Subtracting Polynomials NAT: NT.CCSS.MTH.10.9-12.A.APR.1

STA: NV.NVAS.MTH.06.9-12.2.12.3.1 LOC: MTH.C.10.05.08.03.001


ID: A

12

42 ANS: C

5x − 3( )(x3− 5x + 2)

= 5x(x3− 5x + 2) − 3(x

3− 5x + 2) Distribute 5x and −3.

= 5x(x3) + 5x(−5x) + 5x 2( ) − 3(x

3) − 3 −5x( ) − 3 2( ) Distribute 5x and −3 again.

= 5x4− 25x

2+ 10x − 3x

3+ 15x − 6 Multiply.

= 5x4− 3x

3− 25x

2+ 25x − 6 Combine like terms.

Feedback

A Check the signs.

B Combine only like terms.

C Correct!

D Combine only like terms.

PTS: 1 DIF: Average REF: 15e2b202-4683-11df-9c7d-001185f0d2ea

OBJ: 3-2.2 Multiplying Polynomials NAT: NT.CCSS.MTH.10.9-12.A.APR.1

LOC: MTH.C.10.05.08.03.02.002 TOP: 3-2 Multiplying Polynomials

MSC: DOK 3

43 ANS: A

Write in expanded form.

(x − 2y)(x − 2y)(x − 2y)

Multiply the last two binomial factors.

(x − 2y)(x2− 4xy + 4y

2)

Distribute the first term, distribute the second term, and combine like terms.

x3− 6x

2y + 12xy

2− 8y

3

Feedback

A Correct!

B To find the product, write out the three binomial factors and multiply in two steps.

C Remember that the second term is negative.

D To find the product, write out the three binomial factors and multiply in two steps.

PTS: 1 DIF: Average REF: 15e53b6e-4683-11df-9c7d-001185f0d2ea

OBJ: 3-2.4 Expanding a Power of a Binomial NAT: NT.CCSS.MTH.10.9-12.A.APR.1

TOP: 3-2 Multiplying Polynomials MSC: DOK 3

ID: A

13

44 ANS: C

To divide, first write the dividend in standard form. Include missing terms with a coefficient of 0.

6x3+ 0x

2+ 5x − 8

Then write out in long division form, and divide.

6x2+ 12x + 29

x − 2 6x3+ 0x

2+ 5x − 8

−(6x3− 12x

2)

12x2+ 5x

−(12x2− 24x)

29x − 8

−(29x − 58)

50

Write out the answer with the remainder to get 6x2+ 12x + 29 +

50

(x − 2).

Feedback

A Be careful when subtracting the terms.

B Remember to include the remainder in the answer.

C Correct!

D Remember to divide by the "–2".

PTS: 1 DIF: Average REF: 15ea0026-4683-11df-9c7d-001185f0d2ea

OBJ: 3-3.1 Using Long Division to Divide Polynomials NAT: NT.CCSS.MTH.10.9-12.A.APR.6

LOC: MTH.C.10.05.08.03.03.002 TOP: 3-3 Dividing Polynomials

MSC: DOK 3

ID: A

14

45 ANS: D

For x − 2( ) , a = 2.

2 1 –9 10 Write the coefficients of the expression.

Bring down the first coefficient. Multiply and add each

column.2 –14

1 –7 –4

Write the remainder as a fraction to get x − 7 +−4

x − 2.

Feedback

A The value 'a' occurs in the divisor as 'x – a'.

B Multiply each column by the value 'a'.

C Begin synthetic division at the second coefficient.

D Correct!

PTS: 1 DIF: Average REF: 15ec3b72-4683-11df-9c7d-001185f0d2ea

OBJ: 3-3.2 Using Synthetic Division to Divide by a Linear Binomial

NAT: NT.CCSS.MTH.10.9-12.A.APR.6 LOC: MTH.C.10.05.08.03.03.003

TOP: 3-3 Dividing Polynomials MSC: DOK 3

46 ANS: A

Factor out the GCF.

3x3(27x

3+ 8y

3)

Write as a sum of cubes.

3x3((3x)

3+ (2y)

3)

Factor.

3x3(3x + 2y)((3x)

2− 6xy + (2y)

2) = 3x

3(3x + 2y)(9x

2− 6xy + 4y

2)

Feedback

A Correct!

B In a sum of cubes, the plus and minus signs alternate.

C Check the formula for the sum of cubes.

D After factoring out the GCF, see if the result can be factored further.

PTS: 1 DIF: Basic REF: 15f36286-4683-11df-9c7d-001185f0d2ea

OBJ: 3-4.3 Factoring the Sum or Difference of Two Cubes NAT: NT.CCSS.MTH.10.9-12.A.SSE.2

LOC: MTH.C.10.05.08.03.04.005 TOP: 3-4 Factoring Polynomials

MSC: DOK 3

ID: A

15

47 ANS: B

−3x3− 21x

2+ 72x + 540 = 0

−3x3− 21x

2+ 72x + 540 = −3 x − 5( ) x + 6( ) x + 6( )

x − 5 is a factor once, and x + 6 is a factor twice.

The root 5 has a multiplicity of 1.

The root −6 has a multiplicity of 2.

Feedback

A You reversed the operation signs of the factors. Also, if x – a is a factor of the equation,

a is a root of the equation.

B Correct!

C You reversed the operation signs of the factors.

D If x – a is a factor of the equation, then a is a root of the equation.

PTS: 1 DIF: Average REF: 15fa899a-4683-11df-9c7d-001185f0d2ea

OBJ: 3-5.2 Identifying Multiplicity NAT: NT.CCSS.MTH.10.9-12.A.APR.3

TOP: 3-5 Finding Real Roots of Polynomial Equations MSC: DOK 3

48 ANS: C

There are five roots: 2− i, 2+ i, 5 , − 5 , and −2. (By the Irrational Root Theorem and Complex Conjugate

Root Theorem, irrational and complex roots come in conjugate pairs.) Since it has 5 roots, the polynomial must

have degree 5.

Write the equation in factored form, and then multiply to get standard form.

P x( ) = 0

(x − 2− i( ))(x − 2 + i( ))(x − 5)(x − (− 5))(x − −2( )) = 0

(x2− 4x + 5)(x

2− 5) x + 2( ) = 0

(x4− 4x

3+ 20x − 25) x + 2( ) = 0

P x( ) = x5− 2x

4− 8x

3+ 20x

2+ 15x − 50 = 0

Feedback

A Only the irrational roots and the complex roots come in conjugate pairs. There are five

roots in total.

B –4x(–5) = 20x

C Correct!

D i squared is equal to –1, so the opposite is equal to 1.

PTS: 1 DIF: Average REF: 1601b0ae-4683-11df-9c7d-001185f0d2ea

OBJ: 3-6.3 Writing a Polynomial Function with Complex Zeros

NAT: NT.CCSS.MTH.10.9-12.N.CN.9 | NT.CCSS.MTH.10.9-12.A.APR.2

LOC: MTH.C.10.06.05.008 TOP: 3-6 Fundamental Theorem of Algebra

MSC: DOK 3

ID: A

16

49 ANS: C

As x → −∞ , P(x) → −∞ and as x → ∞ , P(x) → ∞ .

P(x) is of odd degree with a positive leading coefficient.

Feedback

A The degree is even if the curve approaches the same y-direction as x approaches

positive or negative infinity, and is odd if the curve increases and decreases in opposite

directions. The leading coefficient is positive if the graph increases as x increases and

negative if the graph decreases as x increases.

B The leading coefficient is positive if the graph increases as x increases and negative if

the graph decreases as x increases.

C Correct!

D The degree is even if the curve approaches the same y-direction as x approaches

positive or negative infinity, and is odd if the curve increases and decreases in opposite

directions.

PTS: 1 DIF: Basic REF: 1608d7c2-4683-11df-9c7d-001185f0d2ea

OBJ: 3-7.2 Using Graphs to Analyze Polynomial Functions NAT: NT.CCSS.MTH.10.9-12.F.IF.7.c

LOC: MTH.C.10.07.07.003 | MTH.C.10.07.07.005

TOP: 3-7 Investigating Graphs of Polynomial Functions MSC: DOK 2

50 ANS: A

g(x) = 6f(x + 5)

g(x) = 6(2(x + 5)3+ 4)

g(x) = 12(x + 5)3+ 24

Feedback

A Correct!

B The vertical stretch factor will effect the y-intercept.

C The left shift value is added to the x value before it is cubed.

D A shift to the left involves adding, not subtracting.

PTS: 1 DIF: Average REF: 16126132-4683-11df-9c7d-001185f0d2ea

OBJ: 3-8.4 Combining Transformations

NAT: NT.CCSS.MTH.10.9-12.A.CED.2 | NT.CCSS.MTH.10.9-12.F.BF.3

TOP: 3-8 Transforming Polynomial Functions MSC: DOK 3

ID: A

17

51 ANS: D

g(x) = f(x + 4)

g(x) = (x + 4)3− 5(x + 4)

2+ 2(x + 4) + 2

g(x) = x3+ 12x

2+ 48x + 64 − 5x

2− 40x − 80+ 2x + 8+ 2

g(x) = x3+ 7x

2+ 10x − 6

The transformation represents a horizontal shift left of 4 units, which corresponds to making the same profit for

selling 4 fewer bicycles.

Feedback

A The transformation is f(x + 4), not f(x) + 4.

B The transformation is a horizontal shift left.

C The transformation is f(x + 4), not f(x) + 4.

D Correct!

PTS: 1 DIF: Average REF: 1614c38e-4683-11df-9c7d-001185f0d2ea

OBJ: 3-8.5 Application

NAT: NT.CCSS.MTH.10.9-12.A.CED.2 | NT.CCSS.MTH.10.9-12.F.BF.3

TOP: 3-8 Transforming Polynomial Functions MSC: DOK 4

52 ANS: A

f(x) = (x − a1)(x − a

2)(x − a

3)(x − a

4)

If r is a root of P(x), then x − r is

a factor of P(x).

f(x) = x − −2( )ÈÎÍÍÍ

˘˚˙̇˙ x − (−

1

2)

È

Î

ÍÍÍÍÍ

˘

˚

˙̇˙̇˙ x − (

1

2)

È

Î

ÍÍÍÍÍ

˘

˚

˙̇˙̇˙ x − (

3

2)

È

Î

ÍÍÍÍÍ

˘

˚

˙̇˙̇˙

Substitute the roots from the

graph.

f(x) = x + 2( )(x +1

2)(x −

1

2)(x −

3

2) Simplify.

f(x) = x + 2( ) 2x + 1( ) 2x − 1( ) 2x − 3( ) Multiply by 8 and simplify.

Feedback

A Correct!

B Find the roots of the graph and subtract these values from x. Multiply these factors

together to create the polynomial.

C Each factor of the polynomial subtracts a root from x.

D Find the zeros of the graph and subtract these values from x. Multiply these factors

together to create the polynomial.

PTS: 1 DIF: Advanced REF: 161beaa2-4683-11df-9c7d-001185f0d2ea

NAT: NT.CCSS.MTH.10.9-12.A.CED.2 TOP: 3-9 Curve Fitting with Polynomial Models

MSC: DOK 3

ID: A

18

53 ANS: B

Make a table of values. Plot enough ordered pairs to see the shape of the curve. Choose both negative and

positive values for x.

x 4 x + 53 (x, f(x))

–13 4 −13+ 53

= 4 −83

= −8 (–13, –8)

–6 4 −6 + 53

= 4 −13

= −4 (–6, –4)

–5 4 −5 + 53

= 4 03

= 0 (–5, 0)

–4 4 −4 + 53

= 4 13

= 4 (–4, 4)

3 4 3 + 53

= 4 83

= 8 (3, 8)

The domain is the set of all real numbers. The range is also the set of all real numbers.

Feedback

A The function whose graph you need to draw is a product of a number and a cube root.

B Correct!

C The function whose graph you need to draw is a product of a number and a cube root.

D You reversed the values of the numbers under the radical sign and outside of it when

graphing the function.

PTS: 1 DIF: Average REF: 16cebb96-4683-11df-9c7d-001185f0d2ea

OBJ: 5-7.1 Graphing Radical Functions NAT: NT.CCSS.MTH.10.9-12.F.IF.7.b

LOC: MTH.C.10.07.10.002 | MTH.C.10.07.10.003 | MTH.C.10.07.10.005

TOP: 5-7 Radical Functions MSC: DOK 2

54 ANS: B

256z164

= 256 · z4· z

4· z

4· z

44 Factor into perfect powers of four.

= 4 · z · z · z · z Use the Product Property of Roots.

= 4z4 Simplify.

Feedback

A Use the Product Property of Roots to simplify the coefficient.

B Correct!

C To find the nth root of a product, factor the product into powers of n and use the

Product Property of Roots.

D To find the nth root of a product, factor the product into powers of n and use the

Product Property of Roots.

PTS: 1 DIF: Basic REF: 16c53226-4683-11df-9c7d-001185f0d2ea

OBJ: 5-6.2 Simplifying Radical Expressions NAT: NT.CCSS.MTH.10.9-12.N.RN.2

STA: NV.NVAS.MTH.06.9-12.2.12.3.2 LOC: MTH.C.10.05.10.02.003 | MTH.C.10.05.10.05.01.002

TOP: 5-6 Radical Expressions and Rational Exponents MSC: DOK 1

ID: A

19

55 ANS: B

8

5

3

= ( 83

)5 Write with a radical.

= (2)5 Evaluate the root.

= 32 Evaluate the power.

Feedback

A The exponent m/n indicates the nth root raised to the m-th power.

B Correct!

C The exponent m/n indicates the nth root raised to the m-th power.

D The exponent m/n indicates the nth root raised to the m-th power.

PTS: 1 DIF: Average REF: 16c55936-4683-11df-9c7d-001185f0d2ea

OBJ: 5-6.3 Writing Expressions in Radical Form NAT: NT.CCSS.MTH.10.9-12.N.RN.2

LOC: MTH.C.10.05.03.04.002 TOP: 5-6 Radical Expressions and Rational Exponents

MSC: DOK 1

56 ANS: B

10mn

= 10

m

n

10811

= 10

8

11

Feedback

A The exponent is a fraction; the numerator is the power of the radicand, and the

denominator is the index of the radical.

B Correct!

C The exponent is a fraction; the numerator is the power of the radicand, and the


D The exponent is a fraction; the numerator is the power of the radicand, and the


PTS: 1 DIF: Average REF: 16c79482-4683-11df-9c7d-001185f0d2ea

OBJ: 5-6.4 Writing Expressions by Using Rational Exponents NAT: NT.CCSS.MTH.10.9-12.N.RN.2

LOC: MTH.C.10.05.03.04.004 TOP: 5-6 Radical Expressions and Rational Exponents

MSC: DOK 1

ID: A

20

57 ANS: D

Step 1 Identify how each transformation affects the function.

Horizontal stretch by a factor of 4: b| | = 4

Reflection across the y-axis: b is negative

Translation left 2 units: h = −3

Step 2 Write the transformed function.

g(x) =1

bx − h( )

g(x) =1

−4x − −2( )ÈÎÍÍÍ

˘˚˙̇˙ Substitute –4 for b and –2 for h.

g(x) = −1

4x + 2( ) Simplify.

Feedback

A This is a translation of f(x) to the right.

B This is a reflection of f(x) across the x-axis.

C This is a horizontal compression of f(x) by 1/4.

D Correct!

PTS: 1 DIF: Average REF: 16d3804e-4683-11df-9c7d-001185f0d2ea

OBJ: 5-7.4 Writing Transformed Square-Root Functions NAT: NT.CCSS.MTH.10.9-12.F.BF.3

TOP: 5-7 Radical Functions MSC: DOK 2

58 ANS: C

( 11x )2= (3 x + 2 )

2 Square both sides.

11x = 9 x + 2( ) Simplify.

11x = 9x + 18 Distribute 9.

2x = 18 Solve for x.

x = 9

Feedback

A Square both terms when squaring a product, and distribute correctly.

B Square both terms when squaring a product.

C Correct!

D Distribute correctly.

PTS: 1 DIF: Average REF: 16daa762-4683-11df-9c7d-001185f0d2ea

OBJ: 5-8.2 Solving Equations Containing Two Radicals NAT: NT.CCSS.MTH.10.9-12.A.REI.2

LOC: MTH.C.10.06.07.003 TOP: 5-8 Solving Radical Equations and Inequalities

MSC: DOK 1

ID: A

21

59 ANS: C

Step 1 Solve for x.

x + 31Ê

ËÁÁÁ

ˆ

¯˜̃˜2

= x + 1( )2

Square both sides.

x + 31 = x2+ 2x + 1 Simplify.

0 = x2+ x − 30 Write in standard form.

0 = x − 5( ) x + 6( ) Factor.

x − 5 = 0 or x + 6 = 0 Solve for x.

x = 5 or x = −6

Step 2 Use substitution to check for extraneous solutions.

x + 31 = x + 1

5+ 31 5 + 1

36 6

6 6

x + 31 = x + 1

−6+ 31 −6+ 1

25 −5

5 −5

Because x = −6 does not satisfy the original equation, it is extraneous.

The only solution is x = 5.

Feedback

A Square both sides and solve for x. Check whether each possible solution satisfies the

original equation.

B Check whether each possible solution satisfies the original equation.

C Correct!

D Square both sides and solve for x. Check whether each possible solution satisfies the

original equation.

PTS: 1 DIF: Average REF: 16dace72-4683-11df-9c7d-001185f0d2ea

OBJ: 5-8.3 Solving Equations with Extraneous Solutions NAT: NT.CCSS.MTH.10.9-12.A.REI.2

LOC: MTH.C.10.06.07.003 | MTH.C.10.06.07.005

TOP: 5-8 Solving Radical Equations and Inequalities MSC: DOK 2

ID: A

22

SHORT ANSWER

60 ANS:

The parabola opens downward.

The axis of symmetry is the line x = −1.

The vertex is the point (−1,14).

The y-intercept is 10.

Because a is −2, the graph opens downward.

The axis of symmetry is given by x =−(−8)

2(−4)=

8

−8= −1.

x = −1 is the axis of symmetry.

The vertex lies on the axis of symmetry, so x = −1.

The y-value is the value of the function at this x-value.

f(−1) = −4(−1)2− 8(−1) + 10 = −4+ 8+ 10 = 14

The vertex is (−1,14).

Because the last term is 10, the y-intercept is 10.

PTS: 1 DIF: Average REF: 156919ae-4683-11df-9c7d-001185f0d2ea

OBJ: 2-2.2 Graphing Quadratic Functions in Standard Form NAT: NT.CCSS.MTH.10.9-12.F.IF.7

LOC: MTH.C.10.07.06.01.001 TOP: 2-2 Properties of Quadratic Functions in Standard Form

MSC: DOK 3

Algebra II/IIH Semester 1 Exam: Units 3,4,5 _____ Class: _____ Date: _____ ID: A 1 Algebra II/IIH Semester 1 Exam: Units 3,4,5 Multiple Choice Identify the choice that best completes

Documents

Algebra II/IIH Semester 1 Exam: Units 3,4,5 ___ Class: _ Date: ___ ID: A 1 Algebra II/IIH Semester 1 Exam: Units 3,4,5 Multiple Choice Identify the choice that best completes