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MATH ACCUPLACER TEST REVIEW BOOKLET

PROPERTY OF TUTORING CENTER

DO NOT WRITE IN BOOKLET

2 MSU Denver SASC Tutoring

TABLE OF CONTENTS

Basic Rule Overview………………………………………………………………… 3

Real Numbers…………………………………………………………………………6

Linear Equations, Inequalities, System of Linear Equations………………………… 14

Algebraic Expressions and Equations………………………………………………... 23

Word Problems and Applications……………………………………………………. 27

Quadratic Expressions, Quadratic Equations………………………………………… 33

Practice Exams……………………………………………………………………….. 37

Solutions……………………………………………………………………………... 50


BASIC RULE OVERVIEW

BASIC RULE OVERVIEW


BASIC RULE OVERVIEW:

Distributive Property:

3(5𝑥𝑥) → 15𝑥𝑥 𝑜𝑜𝑜𝑜 3(5𝑥𝑥 + 2𝑦𝑦) → 15𝑥𝑥 + 6𝑦𝑦

The three “distributes” to each piece inside the parenthesis.

Associative Property:

3 + (2 + 1) = 6 → (3 + 2) + 1 = 6

When adding, we can add in any order.

Commutative Property:

3(2 ∗ 4) = 24 → (3 ∗ 2)4 = 24

When multiplying, we can multiply in any order.

Exponentials:

23 → 2 × 2 × 2 = 8 (−2)2 → (−2)(−2) = 4 −22 → −(2 × 2) = −4

Absolute Values:

|−3| = 3 |3| = 3

Absolute value means you take the positive of whatever is inside. It is a measure of the distance from the number to zero on the number line. Radicals:

√𝑎𝑎 = √𝑎𝑎2 A square root with no “magnitude” is assumed to be “magnitude 2.”

√𝑎𝑎 ∗ 𝑏𝑏𝑛𝑛 = √𝑎𝑎𝑛𝑛 ∗ √𝑏𝑏𝑛𝑛 When multiplying with roots, numbers can be split into their own roots.

�𝑎𝑎𝑏𝑏

𝑛𝑛 = √𝑎𝑎𝑛𝑛

√𝑏𝑏𝑛𝑛 When dividing with roots, numbers can be split into their own roots.

𝑎𝑎𝑚𝑚𝑛𝑛 = √𝑎𝑎𝑚𝑚𝑛𝑛 Roots can be written as fractional powers.

(𝑎𝑎𝑚𝑚)1𝑛𝑛 = �𝑎𝑎

1𝑛𝑛�

𝑚𝑚 Roots follow the same rules as powers raised to another power.

𝑎𝑎𝑛𝑛𝑛𝑛 = 𝑎𝑎1 = 𝑎𝑎 Any number to the power of 1 is equal to the original number.

BASIC RULE OVERVIEW


Addition or subtraction of Fractions:

𝑎𝑎𝑏𝑏

+𝑐𝑐𝑑𝑑

= �𝑑𝑑𝑑𝑑�𝑎𝑎𝑏𝑏

+𝑐𝑐𝑑𝑑�𝑏𝑏𝑏𝑏� =

𝑎𝑎 ∗ 𝑑𝑑 + 𝑏𝑏 ∗ 𝑐𝑐 𝑏𝑏 ∗ 𝑑𝑑

To add or subtract fractions, we find a common denominator. We can do this by multiplying the top and bottom of both fractions by the denominator of the other. Once the 2 fractions have the same denominator, you can add or subtract the numerators normally. Multiplication of Fractions:

𝑎𝑎𝑏𝑏∗𝑐𝑐𝑑𝑑

=𝑎𝑎 ∗ 𝑐𝑐𝑏𝑏 ∗ 𝑑𝑑

To multiply fractions, we can multiply the numerators together on the top and all of the denominators together on the bottom. Division of Fractions:

𝑎𝑎𝑏𝑏

÷𝑐𝑐𝑑𝑑

=𝑎𝑎𝑏𝑏∗𝑑𝑑𝑐𝑐

=𝑎𝑎 ∗ 𝑑𝑑𝑏𝑏 ∗ 𝑐𝑐

To divide fractions, we can “flip” the second fraction upside down, and then multiply the resulting fractions together. This is called, “multiplying by the reciprocal.” EXPONENTS (also called “powers”):

𝑎𝑎𝑚𝑚 ∗ 𝑎𝑎𝑛𝑛 = 𝑎𝑎𝑚𝑚+𝑛𝑛 When multiplying like bases, we add the powers.

(𝑎𝑎 ∗ 𝑏𝑏)𝑚𝑚 = 𝑎𝑎𝑚𝑚 ∗ 𝑏𝑏𝑚𝑚 When raising parenthesis to a power, all elements are raised to that power.

𝑎𝑎𝑚𝑚

𝑎𝑎𝑛𝑛= 𝑎𝑎𝑚𝑚−𝑛𝑛 When we divide like bases with powers, subtract the bottom power from the top.

(𝑎𝑎𝑚𝑚)𝑛𝑛 = 𝑎𝑎𝑚𝑚∗𝑛𝑛 When raising a power to another power, we multiply the powers.

�𝑎𝑎𝑏𝑏�𝑛𝑛

= 𝑎𝑎𝑛𝑛

𝑏𝑏𝑛𝑛 When raising a fraction to a power, both top and bottom are raised to that power.

�𝑎𝑎𝑏𝑏�−𝑛𝑛

= 𝑏𝑏𝑛𝑛

𝑎𝑎𝑛𝑛 A fraction raised to a negative power is flipped.

𝑎𝑎−𝑛𝑛 = 1𝑎𝑎𝑛𝑛

A number to a negative power is equal to one divided by that number and power.

𝑎𝑎0 = 1 Any number to the power of 0 is equal to 1.


REAL NUMBERS

REAL NUMBERS


MULTIPLICATION MEMORIZATION:

Before beginning anything else, these times tables should be MEMORIZED. Below is a table which you can use as a cheat sheet, but you should refrain from using is and try to just know it as you won’t be allowed a calculator in the test.

To multiply large numbers, follow the process below in the example:

4025 × 15

REAL NUMBERS


SQUARE ROOT AND SQUARES MEMORIZATION:

These are the roots and squares that should be memorized for the test.

12 = 1

22 = 4

32 = 9

42 = 16

52 = 25

62 = 36

72 = 49

82 = 64

92 = 81

102 = 100

112 = 121

122 = 144

√1 = 1

√4 = 2

√9 = 3

√16 = 4

√25 = 5

√36 = 6

√49 = 7

√64 = 8

√81 = 9

√100 = 10

√121 = 11

√144 = 12

REAL NUMBERS


ORDER OF OPERATIONS (PEMDAS)

P –Parenthesis ( ), [ ], { }

E – Exponents 𝑥𝑥2 the number that x is raised to.

M – Multiplication x

D – Division /

A – Addition +

S – Subtraction -

PRACTICE PROBLEMS:

Solve the following problems.

1) 2 × (8 + 5) − 5

2) 5 + (3 ∗ 6 + 2)2

3) −22 + 32

4) (−2)2∗223

5) (5∗2)(6−3)6÷2

6) �3+(3−7)�−252−42÷2

7) (−3)2

(4−2)3∗ |4 − 6|

8) ((90 ÷ 10) ∗ (2 ÷ 3))2(5 + 32)1

9) ((5− 3)2)2

10) (5 − 6 + 4)(100÷50)

11) 1446∗2

12) � 102

−100−50�

13) 8(5+√16)22

− 6(2−42)(1−22)

14) (4 + 2)2 + (−2)3 − 32

15) (6 + [2 − 23]) + 20

PEMDAS is the mnemonic to remember the order in which to perform mathematic operations. Start with parentheses and end with addition and subtraction.

Note: Multiplication/Division and Addition/Subtraction are performed from left to right. Multiplication/Division is still performed prior to Addition/Subtraction.

REAL NUMBERS


FRACTIONS:

- ADDITION AND SUBTRACTION 12

+ 23

= 1×32×3

+ 2×23×2

= 36

+ 46

= 76

- MULTIPLICATION

12

× 23

= 26

= 13

- DIVISION

12

÷ 23

= 12

× 32

= 34

PRACTICE PROBLEMS:


1) 37

+ 23

2) �−35� �−15

9�

3) 8 �15�

4) 13

÷ 19

5) 2757

6) 25 (6)34

7) �12�2

+ 17− 3

4

8) 56�−910� �−4

5�

9) 14 + 23

11

10) −53

[35

+ 12�34

÷ 12�]

11) 37

÷ 87

+ �−35� �1

3�

NOTE: CD: 3 * 2 = 6

CD= Common Denominator

Multiply bottom number to reflect CD. Remember to multiply top number by the same number used to obtain the CD on bottom.

1. Flip Right side 2. Division sign => Multiplication sign

REAL NUMBERS


RADICALS:

√4 = 2 √83 = 2 √2 × √15 = √30

√108 = √𝟑𝟑 × 𝟑𝟑 × 3 × 𝟐𝟐 × 𝟐𝟐 = (𝟑𝟑 × 𝟐𝟐)√3 = 6√3

�49

= √4√9

= 23

823 = �823 = √643 = 4

PRACTICE PROBLEMS:


1) √2 × √15

2) �√2�2

3) −�2536

4) 3 − √5 + �√3�2

√83 ÷ 2

5) 1628

6) �√83 �2

7) �√−56 �6

8) 83√64

9) √7 + 5√7 − 2√7

10) √−1253 − √814

Note: 108 = 2 x 54 = 2 x 2 x 27

= 2 x 2 x 3 x 9 = 2 x 2 x 3 x 3 x 3

We are just reducing 108 to what it is composed of. Try the final answer in a calculator to make sure it’s 108.

This section also has a 10% - 20% chance of showing up on the test.

REAL NUMBERS


EXPONENTS:

23 → 2 × 2 × 2 = 8 (−2)2 → (−2)(−2) = 4 −22 → −(2 × 2) = −4

(32)2 = 92 = 81

2 ∗ 2−3 = 2 ∗ �1

23� =

223

=28

=14

PRACTICE PROBLEMS:


1) (32 ∗ 23) ∗ 3−3 + (3 ÷ 33)

2) 25

−22+ 2

23

3) (23)−2(22 ∗ 13)2

4) 17−32+7

42−32

5) 2(−5+3)−22

− 3(−32+2)

3−( − 12 )

REAL NUMBERS


SUBSTITUTION:

2𝑥𝑥2 + 3𝑦𝑦3 − 5𝑦𝑦 = 2(3)2 + 3(2)3 − 5(2) = 2(6) + 3(8) − 5(2) = 12 + 24 − 10 = 26

When: x = 3 and y = 2

PRACTICE PROBLEMS:

Solve for the following problems with the given values.

1. −𝑥𝑥2𝑦𝑦 When x = 2 and y = 3

2. √𝑦𝑦 − 𝑧𝑧3

𝑥𝑥 When x = 3, y = 4, and z = 2

3. 𝑥𝑥4

𝑥𝑥5+ 𝑧𝑧 When x = 4 and z = −5

4

4. 𝑥𝑥𝑛𝑛−1

𝑛𝑛−1− 𝑦𝑦𝑛𝑛−1

𝑛𝑛−1 When x = 2, y = 1, n = 3

5. 𝑥𝑥 − √𝑦𝑦 + √𝑥𝑥2

√𝑧𝑧4 ÷ 𝑧𝑧 When x = 3, y = 25, z = 2

6. 5(𝑥𝑥+ℎ)−5(𝑥𝑥)ℎ

When x = 3 and h = 4

7. −𝑥𝑥3 When x = - 2

8. 𝑥𝑥2

4+ 𝑦𝑦2

9 When x = 2 and y = 3

9. 𝑥𝑥𝑦𝑦 − 3

√𝑦𝑦 When x = 1 and y = 4

10. −𝑏𝑏+√𝑏𝑏2−4∗𝑎𝑎∗𝑐𝑐2∗𝑎𝑎

When a = 3, b = 4, c = 1

Note: Place parenthesis around all variables (letters) first then place numbers in afterwards.


LINEAR EQUATIONS, INEQUALITIES, SYSTEM OF EQUATIONS

LINEAR EQUATIONS, INEQUALITIES, SYSTEM OF LINEAR EQUATIONS


SOLVING LINEAR EQUATIONS:

Solve for x, given 3𝑥𝑥 + 7 = 1

3𝑥𝑥 + 7 = 1

-7 -7

3𝑥𝑥 = −6 → 3𝑥𝑥3

=−63

x = -2

PRACTICE PROBLEMS:

Solve the following problems for the variable.

1) 7x + 8 = 1

2) 5r − 4 = 21

3) 5x + 2 = 3x− 6

4) 2(G + 3) = −4(G + 1)

5) −4t + 5t − 8 + 4 = 6t − 4

6) 3(2w + 1) − 2(w− 2) = 5

7) 7[2 − (3 + 4x)]− 2x = −9 + 2(1 − 10x)

8) −[3x − (2x + 5)] = −4 − [3(2x− 4)]

9) −59

k = 2

10) m2

+ m3

= 5

11) G − 103

+ 25

= −G3

12) 4t + 13

= t + 56

+ t − 36

13) y3(y + 4)

= 1(2 + 4)∗3

14) 2a+12a

= 15+a

+ 1

15) 2x + 57

− x + 45

= 2

Note: The overall goal for these problems is to isolate x on one side of the equation and send all the numbers to the other side.

Rules:

• We are solving for x = some number. • What you do to one side you must do to the other

side. • Generally, start with numbers that do not contain

x’s. • To undo addition, subtract from both sides and

vice versa. • To undo multiplication, divide from both sides

and vice versa.



SOLVING MULTI-VARIABLE EQUATIONS:

Given the equation 𝐴𝐴 = 𝑃𝑃1−𝑃𝑃𝑃𝑃

; Solve for P

• Step 1: Multiply both sides by (1 − 𝑃𝑃𝑃𝑃) to get rid of the division sign.

(1 − 𝑃𝑃𝑃𝑃) × (𝐴𝐴) = � 𝑃𝑃1−𝑃𝑃𝑃𝑃

� × (1 − 𝑃𝑃𝑃𝑃) =≫ (𝐴𝐴)(1− 𝑃𝑃𝑃𝑃) = 𝑃𝑃

• Step 2: Distribute the (A) into (1 − 𝑃𝑃𝑃𝑃) on the left side and add APQ to both sides.

𝐴𝐴 − 𝐴𝐴𝑃𝑃𝑃𝑃 = 𝑃𝑃

+𝐴𝐴𝑃𝑃𝑃𝑃 + 𝐴𝐴𝑃𝑃𝑃𝑃

𝐴𝐴 = 𝑃𝑃 + 𝐴𝐴𝑃𝑃𝑃𝑃

• Step 3: Factor out P from the left side and divide everything else over.

𝐴𝐴 = 𝑃𝑃(1 + 𝐴𝐴𝑃𝑃) =≫ 𝐴𝐴

(1 + 𝐴𝐴𝑃𝑃) =𝑃𝑃(1 + 𝐴𝐴𝑃𝑃)(1 + 𝐴𝐴𝑃𝑃)

• Solution: 𝑃𝑃 = 𝐴𝐴(1+𝐴𝐴𝑃𝑃)

PRACTICE PROBLEMS:

Solve the following equations for the stated variable.

1. 𝑆𝑆 = P(1 + rt); Solve for r

2. 𝐴𝐴 = 𝜋𝜋𝑅𝑅2; Solve for R

3. 4𝐴𝐴 + 3𝐵𝐵 = 30; Solve for B

4. 3𝐺𝐺𝐺𝐺 + 12𝐺𝐺 = 𝑌𝑌; Solve for G

5. 𝐷𝐷 = 𝐺𝐺+𝐽𝐽+𝐹𝐹+𝐾𝐾60

; Solve for J

6. 𝑌𝑌𝐻𝐻

+ 6 = 𝑍𝑍; Solve for H



INEQUALITIES:

Solve for x, given −3𝑥𝑥 + 7 ≤ 1

−3𝑥𝑥 + 7 ≤ 1

-7 -7

−3𝑥𝑥 ≤ −6 → −3𝑥𝑥−3

≤−6−3

x ≥ 2

PRACTICE PROBLEMS:


1. x − 3 ≥ 12

2. 4x < −16

3. 5w + 2 ≤ −48

4. −3(z − 6) > 2z − 2

5. −2n + 8 ≤ 2n

6. 2s −5−4

> 5

7. 23

(3k − 1) ≥ 32

(2k − 3)

8. x − 43

+ 3 ≤ x8

9. 5y + 1 < 2(3y − 4) + 3

10. 6 − 2(x − 1) < 3x

Note: Treat the Inequality sign as if it were an “=” sign and solve for x.

Rules:

• Solve the problem for x, where x = some number • Flip sign if you Multiply by a negative number • Flip sign if you Divide by a negative number



GRAPHING:

6x + 2y = 12

Method 1

If y = 0 then solve for x:

6𝑥𝑥 + 2(0) = 12

6𝑥𝑥 = 12

𝑥𝑥 = 2

If x = 0 then solve for y:

6(0) + 2𝑦𝑦 = 12

2𝑦𝑦 = 12

𝑦𝑦 = 6



Method 2

Solving for “y” results in an equation in slope intercept form (y = m x + b)

- “m” and “b” are numbers - “x” and “y” stay as variables (letters) - “m” number represents the slope of this equation or the rise over run and should be

written as a fraction - “b” number represents the same thing as y = # and y-intercept (where the line crosses the

y-axis.

In our case, y = -3x + 6 when solved for y.

PRACTICE PROBLEMS:

Graph the following equations.

1. 𝑦𝑦 = 𝑥𝑥

2. 𝑦𝑦 = 4𝑥𝑥 − 1

3. 3𝑦𝑦 − 12 = −6𝑥𝑥

4. 𝑦𝑦 = 12𝑥𝑥 + 3

5. 𝑥𝑥 = 5

6. 𝑦𝑦 = −2.5

7. 𝑥𝑥 + 6𝑦𝑦 = 7



GRAPHING INEQUALITIES:

Example: Graph the following inequality.

𝑦𝑦 ≥ −3 𝑥𝑥 + 6

Step 1: Solve for y = m x + b form, then draw the graph.

Step 2: Shade the region above or below the line:

- If “y is greater than,” shade above since above y values are greater than those on the line - If “y is less than” shade below since all the y values below will be less than on the line

Step 3: Check a test point to make sure it works (for example the point x = 0 and y = 0)

6(0) + 2(0) ≥ 12

0 ≱ 12 Decision: The point (0,0) should not fall in the shaded region

Notice:

Since the equation given at the top of the page states that “y is greater than or equal to - 3 x + 6” the region above the graph is shaded.

The line is SOLID since it could also be equal to. For any inequality that is strictly greater than or less than and NOT EQUAL TO, the line should be dotted.

Practice problems on the next page



GRAPHING INEQUALITIES:

PRACTICE PROBLEMS:

Graph the following inequalities.

1. 𝑦𝑦 ≤ 𝑥𝑥

2. 𝑦𝑦 < 4𝑥𝑥 − 1

3. 3𝑦𝑦 − 12 ≤ −6𝑥𝑥

4. 𝑦𝑦 ≥ 12𝑥𝑥 + 3

5. 𝑥𝑥 ≥ 5

6. 𝑦𝑦 ≥ −2.5

7. 𝑥𝑥 + 6𝑦𝑦 ≤ 7



SYSTEMS OF LINEAR EQUATIONS:

• Given 2 equations solve for x and y.

3𝑥𝑥 + 3𝑦𝑦 = 9

2𝑥𝑥 + 𝑦𝑦 = 5 → 𝑦𝑦 = 5 − 2𝑥𝑥 (Remember this)

• Plug (y = 5 – 2x) back into first equation, simplify, and solve for x.

3𝑥𝑥 + 3(5 − 2𝑥𝑥) = 9 → 3𝑥𝑥 + 15 − 6𝑥𝑥 = 9 → −3𝑥𝑥 + 15 = 9

−3𝑥𝑥 = −6 → −3𝑥𝑥−3

=−6−3

→ 𝑥𝑥 = 2

• Knowing that x = 2, Plug that x-value back into an early equation and solve for y.

Remember: 𝑦𝑦 = 5 − 2𝑥𝑥

If 𝑥𝑥 = 2, then 𝑦𝑦 = 5 − 2(2) → 𝑦𝑦 = 1

PRACTICE PROBLEMS:

Solve the following systems of linear equations for their variables.

1. 3x + y = 12 and y = −2x + 10

2. 5y − 2x = 10 and 3x + y = 138

3. 2x + y = 5 and x − 2y = 15

4. 16y − 12x = 20 and 8y − 6 2x

= 23

5. 12

x − 32

y = 5 and x + 7y2y

= 154

6. x + 29 = 6y − 33 and 2x − 3y−5

= 5

7. 3x − y = 9 and 2x + 3y = 28

8. 2 �9−5x7� = 3y and 3 �10−3y

2x� = 1

9. yx

= 43

and x + 1 = y

10. 4 �x+y7� = 2x − 4y and x−2y

2 =6

I choose to use the second equation because the “y” did not have a front number (coefficient). An equation having a variable without a coefficient is the best one to start off with.


ALGEBRAIC EXPRESSIONS AND EQUATIONS



DISTRIBUTION/FACTORING REVIEW:

3(2𝑥𝑥) = 6𝑥𝑥 3(2𝑥𝑥 + 𝑦𝑦) = 6𝑥𝑥 + 3𝑦𝑦 𝑥𝑥(2𝑥𝑥) = 2𝑥𝑥2 2𝑥𝑥(3𝑥𝑥 + 𝑦𝑦) = 6𝑥𝑥2 + 2𝑥𝑥𝑦𝑦

Given: 3x + 6 --There are two sections, described by the underline, separated by a + or a – sign.

3𝑥𝑥3

+ 63

= 3(𝑥𝑥 + 2)

PRACTICE PROBLEMS:

Use distribution and factoring for the following problems.

1. 3𝑘𝑘5 − 6𝑘𝑘3 + 3𝑘𝑘2

2. 2𝑥𝑥(10𝑦𝑦 + 7𝑥𝑥)

3. 2𝐺𝐺6(3 + 𝑦𝑦7)

4. 24𝑥𝑥 + 32𝑦𝑦

5. 2𝑥𝑥𝑦𝑦2 + 4𝑥𝑥2𝑦𝑦3

Note: The two underlined sections can both be divisible by 3. After division, place the 3 in front of the parenthesis to notate that division was performed. This process is called factoring out.

Rules:

• All sections must be divisible by the same number or variable. • Always notate what was divided out by placing a marker in front

of the parenthesis for the final answer.



FACTORING OUT COMMON FACTORS:

Reference the Distribution/Factoring Review section.

Factor the given equation: 3𝑥𝑥5 + 21𝑥𝑥3 − 3𝑥𝑥2

3𝑥𝑥5 + 21𝑥𝑥3 − 3𝑥𝑥2

3𝑥𝑥5

3+ 21𝑥𝑥3

3− 3𝑥𝑥2

3= 3(𝑥𝑥5 + 7𝑥𝑥3 − 𝑥𝑥2)

3(𝑥𝑥5 + 7𝑥𝑥3 − 𝑥𝑥2) → 3�𝑥𝑥5

𝑥𝑥2+

7𝑥𝑥3

𝑥𝑥2−𝑥𝑥2

𝑥𝑥2� → 3𝑥𝑥2(𝑥𝑥3 + 7𝑥𝑥 − 1)

PRACTICE PROBLEMS:

Factor the following problems.

1. 6𝑥𝑥 + 2𝑦𝑦

2. 4𝑜𝑜2 + 20𝑜𝑜

3. 12𝑧𝑧2𝑦𝑦 + 144𝑦𝑦2

4. 6𝑘𝑘3 − 36𝑘𝑘4 − 48𝑘𝑘5

5. −25𝑥𝑥7𝑦𝑦2 + 10𝑥𝑥𝑦𝑦5 − 50𝑥𝑥12𝑦𝑦7

Note: Imagine that this equation has 3 sections that are divided by a plus or minus sign.

All sections have a common factor (Divisible by) of 3. So by dividing all sections by 3, we have to notate what was factored out with a three outside of the parenthesis.



SIMPLIFYING ALGEBRAIC EXPRESSIONS:

2𝑦𝑦3 ∗ 4𝑦𝑦2 = (2 ∗ 4)𝑦𝑦(3+2) = 8𝑦𝑦5

𝐻𝐻4 ∗ 𝑦𝑦12 ∗ 𝐻𝐻3 ∗ 𝑦𝑦5 = 𝐻𝐻(4+3) ∗ 𝑦𝑦(12+5) = 𝐻𝐻7𝑦𝑦17

(3𝑥𝑥2𝑦𝑦4)2 = 32𝑥𝑥2∗2𝑦𝑦4∗2 = 9𝑥𝑥4𝑦𝑦8

3𝑥𝑥2𝑦𝑦5

6𝑥𝑥4𝑦𝑦3=

3 ∗ 𝑥𝑥 ∗ 𝑥𝑥 ∗ 𝑦𝑦 ∗ 𝑦𝑦 ∗ 𝑦𝑦 ∗ 𝑦𝑦 ∗ 𝑦𝑦6 ∗ 𝑥𝑥 ∗ 𝑥𝑥 ∗ 𝑥𝑥 ∗ 𝑥𝑥 ∗ 𝑦𝑦 ∗ 𝑦𝑦 ∗ 𝑦𝑦

=1 ∗ 𝑦𝑦2

2 ∗ 𝑥𝑥2

2 ∗ 𝑥𝑥−3 = 2 ∗ �1𝑥𝑥3� =

2𝑥𝑥3

𝑥𝑥2+4𝑥𝑥+4𝑥𝑥2−4

= (𝑥𝑥+2)(𝑥𝑥+2)(𝑥𝑥+2)(𝑥𝑥−2)

= 𝑥𝑥+2𝑥𝑥−2

PRACTICE PROBLEMS:

Simplify the following problems.

1) (𝑎𝑎2 ∗ 𝑎𝑎5) ∗ 𝑎𝑎−12 + (𝑎𝑎 ÷ 𝑎𝑎3)

2) 𝑎𝑎5

𝑎𝑎2+ 𝑎𝑎6

𝑎𝑎3

3) (𝑥𝑥2𝑦𝑦3𝑧𝑧0)−3(𝑥𝑥2𝑦𝑦1𝑧𝑧3)3

4) �𝑥𝑥3𝑦𝑦5𝑧𝑧7

𝑥𝑥2𝑦𝑦3𝑧𝑧4�−1

5) �𝑞𝑞3

𝑝𝑝�4

+ 𝑟𝑟𝑞𝑞�𝑝𝑝𝑞𝑞

2𝑟𝑟3

𝑟𝑟�2

6) �𝑥𝑥𝑦𝑦𝑧𝑧�3

+ �𝑧𝑧𝑥𝑥𝑦𝑦�4

7) 3𝑥𝑥𝑦𝑦

+ 12𝑥𝑥

+ 2𝑥𝑥𝑦𝑦2

8) (4𝑟𝑟𝑟𝑟)3𝑟𝑟2

4𝑟𝑟2𝑟𝑟6

9) 𝑥𝑥2+8𝑥𝑥+12𝑥𝑥2−36

10) 𝑥𝑥2+5𝑥𝑥+6𝑥𝑥2+9𝑥𝑥+20

÷ 2𝑥𝑥2+8𝑥𝑥+62𝑥𝑥2+12𝑥𝑥+10


WORD PROBLEMS AND APPLICATIONS



WORD PROBLEMS:

Key Words to Focus on:

Addition:

- Sum - More than - Plus - Added to - Increased by

Subtraction:

- Less - Less than - Minus - Decreased by - Subtracted from - Difference between

Multiplication:

- Times - Multiplied by - (fraction) of a number - (fraction) as much as - Twice a number - Product of

Division:

- Quotient of - Divided by - Ratio of

PRACTICE PROBLEMS:

1. Ralph the farmer wants to fence off a rectangular area of 100 square feet on his farm and wants the side closest to his house to be 25 feet long. How many feet of fencing does the farmer need to purchase to fence off his farm?

A. 29 feet

B. 58 feet

C. 100 feet

D. 75 feet

2. Alison makes 𝑥𝑥𝑦𝑦 dollars selling lemonade while Lidia makes 𝑥𝑥 − 𝑦𝑦

𝑦𝑦 dollars as a cook, and

Brandi only makes 23 as much as what Lidia makes. Which of the following equation represents

the total combined income of all three people?

A. 2𝑥𝑥 − 2𝑦𝑦3𝑦𝑦

B. 1 + 2𝑥𝑥 − 2𝑦𝑦3𝑦𝑦

C. 1 + 6𝑥𝑥 − 3𝑦𝑦3𝑦𝑦

D. 8𝑥𝑥 − 5𝑦𝑦3𝑦𝑦



3. If a price of an item is represented by K, how much is an item worth if it was discounted by 23%?

A. 0.23𝐺𝐺

B. 0.77𝐺𝐺

C. 𝐺𝐺 − 0.23

D. 𝐺𝐺 + 0.77𝐺𝐺

4. If Matt can walk 2 miles in one hour, how many miles could he walk in 4 ½ hours?

A. 8 ¼ Miles

B. 8 ½ Miles

C. 2 ¼ Miles

D. 9 Miles

5. Cody drops a ball from a cliff that is 200 meters high. If the distance of the ball down the cliff is represented by 𝐷𝐷 = 1

2𝑔𝑔𝑡𝑡2, how long will it take the ball to travel 122.5 meters down the

cliff assuming 𝑔𝑔 = 9.8 𝑚𝑚 𝑠𝑠2� ?

A. 5 Seconds

B. 10 Seconds

C. 12 Seconds

D. 15 Seconds

6. If Joey is half the age of Tyler and Tyler is three times the age of Eugene, what is Joey’s age in 1995 if Eugene was born in 1983?

A. 12 Years Old

B. 18 Years Old

C. 16 Years Old

D. 36 Years Old

7. The area of a right triangle is given by the equation 𝐴𝐴 = 12𝐵𝐵ℎ, where B represents the

base and h represents the height. If a triangle has an area 10 and a height of 5, what is the length of its base?

A. 5

B. 2

C. 4

D. 25



8. In the beginning of the month, Emery owes Mike, Lizzie, and Jack $3.50 each. If he receives a paycheck of $250 and repays Jack, Mike, and Lizzie, how much will he have left?

A. $239.50

B. $241.50

C. $31.50

D. $10.50

9. Nick has $90 in his checking account. He makes purchases of $22 and $54. He overdrafts

his account with a purchase of $37 resulting in a $35 fee. The next day he deposits a check

for $185. What is the balance of his bank account?

A. -$58

B. $58

C. -$127

D. $127

10. The temperatures on planet Zeeborg can be as high as 102°C during the day and as low as

-35°C at night. How many degrees dose the temperature drop from day to night?

A. 67°C

B. 102°C

C. 137°C

D. 35°C

11. To calculate the Body Mass Index of an individual their weight (in kg) is divided by the

square of their height (in m). A BMI between 18.5 and 25 is considered healthy weight and

above is overweight. If Mike weighs 108 kg and stands 2m tall what is his BMI

(Wikipedia.org)

A. 27

B. 24

C. 22

D. 18



12. Kyle needs to be at work in 10 minutes. He is on the highway where the speed limit is

55mph. If he is 10 miles away how much does he need to speed in order to get there on

time?

A. 0mph

B. 5mph

C. 10mph

D. 15mph

13. Jasmine buys a car at a major dealership. She pays a single payment of $5,775 for it. If

she pays 5% in taxes and fees what was the actual price of the car to the nearest dollar?

A. $6,064

B. $5,500

C. $3,850

D. $5,486

14. The perimeter of a rectangle is 3 times the length. The length is 6in longer than the width.

What are the dimensions of this rectangle?

A. 13in x 25in

B. 15in x 9in

C. 18in x 10in

D. 10in x 4in

15. Brenna is going on a 7 day rafting trip. Out of Breath Rafting charges $500 in gear rental

and $150 a day, and Xtr3m3 Rafting charges $210 a day (including all gear). How much

would each cost for her 7 day trip?

A. $1,270, $1,400

B. $1,850, $1,540

C. $1,550, $1,470

D. $1,380, $1,420



16. Daniela is moving into a new house. The boxes he is using have a volume of 2,000in3 the

base has an area of 100in2. What is the height?

A. 20in

B. 200in

C. 10in

D. 100in

17. The area of a rectangle is 160in2. The length is 12in longer than the width. What are the

dimensions (L xW) of the rectangle?

A. 8in x 20in

B. 20in x 8in

C. 16in x 10in

D. 10in x 16in

18. Kristen and Ryan want to know how tall a building is. They measure the shadow the

building cast as 10ft long. They notices a tree next to the building which cast a shadow 2ft

long. If the tree is 7ft tall how tall is the building?(HINT: the shadow length is directly

proportional to the height of the object that cast the shadow)

A. 75ft

B. 35ft

C. 50ft

D. 70ft


QUADRATIC EXPRESSIONS, QUADRATIC EQUATIONS



DISTRIBUTION AND FOIL METHOD FOR A POLYNOMIAL:

Recall: 3(2𝑥𝑥) = 6𝑥𝑥 𝑎𝑎𝑎𝑎𝑑𝑑 3(2𝑥𝑥 + 𝑦𝑦) = 6𝑥𝑥 + 3𝑦𝑦

So then if we had: (x + 2)(2x + y)

Same concepts apply: 2𝑥𝑥2 + 4𝑥𝑥 + 𝑥𝑥𝑦𝑦 + 2𝑦𝑦

OR

Foil Method: (First Outside Inside Last)

(𝑥𝑥 + 2)(2𝑥𝑥 + 𝑦𝑦) = 2𝑥𝑥2 + 𝑥𝑥𝑦𝑦 + 4𝑥𝑥 + 2𝑦𝑦

EXAMPLE PROBLEMS:

Distribute the following problems.

1. (𝑥𝑥 + 3)(𝑥𝑥 + 2)

2. (𝑥𝑥 + 5)(𝑥𝑥 − 1)

3. (𝑡𝑡 − 8)(𝑡𝑡 − 4)

4. (3𝑥𝑥 + 2)(𝑥𝑥 + 4)

5. (𝑤𝑤 − 5)(3𝑤𝑤 + 9)

6. (2𝑦𝑦 + 1)(3𝑦𝑦 + 8)

7. (4𝑦𝑦 − 2)(5𝑦𝑦 + 6)

8. (𝑜𝑜 + 5)2

9. (𝑡𝑡 − 3)2

10. (2𝑥𝑥 + 7)2

This section now represents where the 3 used to be.

F

O



FACTORING POLYNOMIALS:

• Given: 𝑥𝑥2 + 5𝑥𝑥 + 6

Step 1: Make 2 sets of parentheses

𝑥𝑥2 + 5𝑥𝑥 + 6 = ( ) ( )

Step 2: Place x’s after both beginning parenthesis.

𝑥𝑥2 + 5𝑥𝑥 + 6 = (𝑥𝑥 ) (𝑥𝑥 )

Step 3: Make a multiplication table that solves for the last term number.

# x # = 6 3 x 2 = 6 1 x 6 = 6

Step 4: Use the numbers from the previous table and change the multiplication sign into addition sign. Do the math and look for the middle term number as the answer.

# + # = 5 3 + 2 = 5 1 + 6 = 7

Step 5: Place the numbers, with the correct sum, from the previous table into the answer.

𝑥𝑥2 + 5𝑥𝑥 + 6 = (𝑥𝑥 + 3) (𝑥𝑥 + 2)

EXAMPLE PROBLEMS:

Factor the following expressions.

1. 𝑜𝑜2 + 8𝑜𝑜 + 12 2. 𝑦𝑦2 + 2𝑦𝑦 − 35 3. 𝑘𝑘2 − 11𝑘𝑘 + 30

4. 2𝑘𝑘2 − 𝑘𝑘 − 15 5. 3𝑥𝑥2 − 20𝑥𝑥 + 12

Note: We are dissembling the equation into 2 separate

t

Last

Middle Term

First Term

Take note of the signs.



SOLVING POLYNOMIAL EQUATIONS:

Recall from Factoring Trinomials: 𝑥𝑥2 + 5𝑥𝑥 + 6 = (𝑥𝑥 + 3) (𝑥𝑥 + 2)

• Solve for x when: 𝑥𝑥2 + 5𝑥𝑥 + 6 = 0

(𝑥𝑥 + 3) (𝑥𝑥 + 2) = 0

Answer: 𝑥𝑥 = −3,−2

EXAMPLE PROBLEMS:


1. 𝑥𝑥2 − 6𝑥𝑥 + 8 = 0

2. 9𝑡𝑡2 − 3𝑡𝑡 = 0

3. 3𝑤𝑤2 + 8𝑤𝑤 + 4 = 0

4. −𝑥𝑥2 − 5𝑥𝑥 + 36 = 0

5. −2𝑥𝑥2 + 13𝑥𝑥 + 24 = 0

6. (𝑥𝑥 − 5)2 = 36

7. (2𝑥𝑥 − 3)2 = 81

8. (3𝑥𝑥 + 2)(𝑥𝑥 − 3) = 7𝑥𝑥 − 1

Note: In other words, the question is asking what values of x will make the = 0 statement true. Remember to factor.


PRACTICE EXAMS

PRACTICE EXAM #1


PRACTICE EXAM #1:

1. Solve (−2)2∗223

A. 12

B. 2

C. 1

D. 14

2. Solve 1319

A. 13

B. 3

C. 127

D. 27

3. Solve √2 × √15

A. √30

B. √17

C. 17

D. 30

4. Solve for −𝑥𝑥2𝑦𝑦, when x = 2 and y = 3

A. -12

B. 12

C. -6

D. 6

5. Solve for the variable w given 3(2w + 1) − 2(w − 2) = 5

PRACTICE EXAM #1


A. 2

B. 3

C. 32

D. −12

6. Solve for z given −3(𝑧𝑧 − 6) > 2𝑧𝑧 − 2

A. z<4

B. z>4

C. z>3

D. z>2

7. Which of the following graphs represents −2𝑦𝑦 + 4 = 12𝑥𝑥

A.

B.

C.

D.

8. Solve 3𝑥𝑥 + 𝑦𝑦 = 12 𝑎𝑎𝑎𝑎𝑑𝑑 𝑦𝑦 = −2𝑥𝑥 + 10

PRACTICE EXAM #1


A. x=1 y=3

B. x=2 y=6

C. x=6 y=4

D. x=0 y=1

9. Which of the following is equivalent to (𝑜𝑜 + 5)2

A. r2+5r+10

B. r2+25

C. r2+10r+25

D. 2r+10

10. Given 𝑎𝑎 = 𝑎𝑎𝑎𝑎𝑎𝑎 𝑜𝑜𝑟𝑟𝑎𝑎𝑎𝑎 𝑎𝑎𝑛𝑛𝑚𝑚𝑏𝑏𝑟𝑟𝑜𝑜𝑠𝑠, then (𝑎𝑎2 ∗ 𝑎𝑎5) ∗ 𝑎𝑎−12 + (𝑎𝑎 ÷ 𝑎𝑎3)

A. 1+𝑎𝑎3

𝑎𝑎5

B. 𝑎𝑎2

C. 1−𝑎𝑎5

𝑎𝑎3

D. 1

11. Ralph the farmer wants to fence off a rectangular area of 100 square feet on his farm and wants the side closest to his house to be 25 feet long. How many feet of fencing does the farmer need to purchase to fence off his farm?

A. 29 feet B. 58 feet C. 100 feet D. 75 feet

12. If Matt can walk 2 miles in one hour, how many miles could he walk in 4 ½ hours?

PRACTICE EXAM #1


A. 8 ¼ Miles B. 8 ½ Miles C. 2 ¼ Miles D. 9 Miles

PRACTICE EXAM #2

42

MSU Denver SASC Tutoring

PRACTICE EXAM #2:

1. Solve (5∗2)(6−3)6÷2

A. 5

B. 10

C. 4

D. 1

2. Solve 56�−910� �−4

5�

A. - 35

B. 35

C. 3

D. 5

3. Solve �√2�2

A. -2

B. 4

C. 2

D. 1

4. Which of the following is equivalent to �𝑝𝑝3𝑞𝑞2𝑟𝑟3�

2

[𝑝𝑝2𝑟𝑟0𝑞𝑞1]2

A. 𝑝𝑝2𝑞𝑞2𝑜𝑜6

B. 𝑝𝑝𝑞𝑞𝑜𝑜3

C. pq

D. 1

PRACTICE EXAM #2


5. Solve for G given 2(G + 3) = −4(G + 1)

A. 3

B. 13

C. −53

D. -5

6. Given 5𝑦𝑦 + 1 < 2(3𝑦𝑦 − 4) + 3, solve for y

A. y > 6

B. y < 6

C. y < 3

D. y > 3

7. How many unique solution sets does 5𝑦𝑦 − 2𝑥𝑥 = 10 𝑎𝑎𝑎𝑎𝑑𝑑 3𝑥𝑥 + 𝑦𝑦 = 138 have?

A. One, x = 40, y = 18

B. One, x = 18, y = 18

C. Infinite sets of solutions

D. None

8. Find a solution for 9𝑡𝑡2 − 3𝑡𝑡 = 0

A. t = 1

B. t = 2

C. t = 0

D. t = 3

PRACTICE EXAM #2


9. Which of the following is not equivalent to 2𝑥𝑥𝑦𝑦2 + 4𝑥𝑥2𝑦𝑦3

A. 2𝑥𝑥(𝑦𝑦2 + 2𝑥𝑥𝑦𝑦3)

B. 2𝑥𝑥𝑦𝑦2(1 + 2𝑥𝑥𝑦𝑦)

C. 2𝑦𝑦−3(𝑥𝑥𝑦𝑦 + 2𝑥𝑥𝑦𝑦6)

D. 2 𝑦𝑦𝑥𝑥

( 𝑥𝑥2

𝑦𝑦−1+ 2𝑥𝑥3𝑦𝑦)

10. Sarah makes 𝑥𝑥𝑦𝑦 dollars selling lemonade while James makes 𝑥𝑥 − 𝑦𝑦

𝑦𝑦 dollars as a cook, and Lisa only

makes 23 as much as what James makes. Which of the following equation represents the total

combined income of all three people?

A. 2𝑥𝑥 − 2𝑦𝑦3𝑦𝑦

B. 1 + 2𝑥𝑥 − 2𝑦𝑦3𝑦𝑦

C. 1 + 6𝑥𝑥 − 3𝑦𝑦3𝑦𝑦

D. 8𝑥𝑥 − 5𝑦𝑦3𝑦𝑦

11. The area of a right triangle is given by the equation 𝐴𝐴 = 12𝐵𝐵ℎ, where B represents the base and h

represents the height. If a triangle has an area 5 and a height of 5, what is the length of its base?

A. 5 B. 2 C. 3.5 D. 25

PRACTICE EXAM #2


12. Which of the following graph represents < 4𝑥𝑥 − 1 ?

A.

C.

B.

D.

PRACTICE EXAM #3


PRACTICE EXAM #3:

1. Solve (4+2)2

24+ (−2)3

12− (3

2)2

A. 1712

B. −1712

C. 1217

D. −1217

2. Solve 37

87 + �−3

5� �1

3�

A. 24245

B. 407

C. 740

D. 24524

3. Which of the following is equivalent to �𝑥𝑥𝑦𝑦𝑧𝑧�3

+ �𝑧𝑧𝑥𝑥𝑦𝑦�4

A. 𝑥𝑥7𝑦𝑦−1𝑧𝑧

B. 𝑥𝑥7𝑦𝑦7𝑧𝑧7

C. 𝑥𝑥3𝑦𝑦7+𝑥𝑥4𝑧𝑧7

𝑦𝑦4𝑧𝑧3

D. 𝑥𝑥2𝑦𝑦𝑧𝑧3

+ 𝑧𝑧𝑦𝑦

PRACTICE EXAM #3


4. Solve 𝑥𝑥2

4+ 𝑦𝑦2

9 when x = 2 and y = 3

A. 2

B. 4

C. 14

D. -2

5. Solve for x when −[3x − (2x + 5)] = −4 − [3(2x − 4)]

A. 15

B. −15

C. −35

D. 35

6. Solve for H when 𝑦𝑦𝐻𝐻

+ 6 = 𝑧𝑧

A. yz-6y

B. 𝑦𝑦𝑧𝑧−6

C. 𝑦𝑦𝑧𝑧− 6

D. 𝑦𝑦𝑧𝑧 − 6

7. Solve for “x” and “y” given x + 29 = 6y − 33 and 2x − 3y−5

= 5

A. x=2 y=5

B. x=-5 y=2

C. x=6 y=12

D. x=4 y=11

PRACTICE EXAM #3


8. Which of the following is equivalent to (2𝑥𝑥 + 7)2 ?

A. 4𝑥𝑥2 + 14𝑥𝑥 + 49

B. 2𝑥𝑥2 + 49

C. 4𝑥𝑥2 + 49

D. 4x+14

9. Solve for x when 𝑥𝑥2 − 6𝑥𝑥 + 8 = 0

A. x=-4,-2

B. x=4,6

C. x=2,4

D. x=-2,4

10. Which of the following is equivalent to 3𝑘𝑘5 − 6𝑘𝑘3 + 3𝑘𝑘2

A. 3𝑘𝑘(3𝑘𝑘4 − 6𝑘𝑘2 + 3𝑘𝑘)

B. 6𝑘𝑘7

C. 0

D. 3𝑘𝑘2(𝑘𝑘3 − 2𝑘𝑘 + 1)

11. If Sally is half the age of Ben and Ben is three times the age of Ron, what is Sally’s age in 1995 if Ron was born in 1983?

A. 12 Years Old

B. 18 Years Old

C. 16 Years Old

D. 36 Years Old

PRACTICE EXAM #3


12. Joe has $90 in his checking account. He makes purchases of $22 and $54. He overdrafts his account with a purchase of $37 resulting in a $35 fee. The next day he deposits a check for $185. What is the balance of his bank account?

A. $245 B. $320 C. $146 D. $12


SOLUTIONS

SOLUTIONS


REAL NUMBERS

Order of operations:

1. 21 2. 405 3. 5 4. 1 5. 10 6. -3/17 7. 9/4 8. 504

9. 16 10. 9 11. 12 12. 2/3 13. -10 14. 19 15. 1

Fractions:

1. 23/21 2. 1 3. 8/5 4. 3 5. 2/5 6. 16/5

7. -5/14 8. 3/5 9. 1/12 10. -9/4 11. 7/40

Exponents:

1. 25/9 2. -31/4 3. ¼

4. 15/7 5. 7

Radicals:

1. √30 𝑜𝑜𝑜𝑜 5√3 2. 2 3. -5/6 4. 6 − √5 5. 2

6. 4 7. -5 8. 1/3 9. 4√7 10. -8

Substitution:

1. -12 2. -2 3. -1 4. 3/2 5. 1/2

6. 5 7. 8 8. 2 9. 5/2 10. -1/3

SOLUTIONS


LINEAR EQUATIONS

Solving linear equations:

1. -1 2. 5 3. -4 4. -5/3 5. 0 6. -1/2 7. 0 8. 3/5

9. -18/5 10. 6 11. 22/5 12. 0 13. 4/5 14. 5 15. 73/3

Solving multi-variable equations:

1. r = 𝑆𝑆−𝑃𝑃𝑃𝑃𝑃𝑃

𝑜𝑜𝑜𝑜 𝑜𝑜 =𝑆𝑆𝑃𝑃 − 1

𝑃𝑃

2. 𝑅𝑅 = ±�𝐴𝐴𝜋𝜋

3. 𝐵𝐵 = 30−4𝐴𝐴3

4. 𝐺𝐺 = 𝑌𝑌3𝐾𝐾 + 12

5. 𝐽𝐽 = 60𝐷𝐷 − 𝐺𝐺 − 𝐹𝐹 − 𝐺𝐺 6. 𝐻𝐻 = 𝑌𝑌

𝑍𝑍−6

Inequalities:

1. 𝑥𝑥 ≥ 15 [15, ∞) 2. 𝑥𝑥 < −4 (-∞ , 4) 3. 𝑤𝑤 ≤ −10 (-∞, -10] 4. 𝑧𝑧 < 4 (-∞, 4) 5. 𝑎𝑎 ≥ 2 [2, ∞) 6. 𝑠𝑠 < −15

2 (-∞, −𝟏𝟏𝟏𝟏

𝟐𝟐 )

7. 𝑘𝑘 ≤ 236

(-∞, 𝟐𝟐𝟑𝟑𝟔𝟔

) 8. 𝑥𝑥 ≤ −8 (-∞, -8] 9. 𝑦𝑦 > 6 (6, ∞) 10. 𝑥𝑥 > 8

5 (𝟖𝟖

𝟏𝟏 , ∞)

SOLUTIONS


Graphing:

1.

3.

2.

4.

5.

SOLUTIONS


6.

7.

SOLUTIONS


Graphing inequalities: 1.

3.

2.

4.

SOLUTIONS


5.

6.

7.

SOLUTIONS


System of linear equations:

1. x = 2 y = 6

2. x = 40 y = 18

3. x = 5 y = - 5

4. x = - 6/7 y = 17/28

5. x = - 2 y = - 4

6. x = 4 y = 11

7. x = 5 y = 6

8. x = - 39/4 y = 11/2

9. x = 3 y = 4

10. x = 32 y = 10

SOLUTION



Distribution/Factoring review:

1. 3 𝑘𝑘2( 𝑘𝑘3 − 2 𝑘𝑘 + 1) 2. 14 𝑥𝑥2 + 20 𝑥𝑥 𝑦𝑦 3. 6 𝐺𝐺6 + 2 𝐺𝐺6 𝑦𝑦7

4. 8 (3 x + 4 y) 5. 2 𝑥𝑥 𝑦𝑦2(1 + 2 𝑥𝑥 𝑦𝑦)

Factoring out common factors:

1. 2 ( 3 x + y ) 2. 4 r ( r + 5 ) 3. 12 𝑦𝑦 ( 𝑧𝑧2 + 12 𝑦𝑦)

4. 6 𝑘𝑘3(1 − 6 𝑘𝑘 − 8 𝑘𝑘2) 5. 5 𝑥𝑥 𝑦𝑦2(−5 𝑥𝑥6 + 2 𝑦𝑦3 − 10 𝑥𝑥11 𝑦𝑦5

Simplifying algebraic expressions:

1. 1+𝑎𝑎3

𝑎𝑎5

2. 2𝑎𝑎3

3. 𝑧𝑧9

𝑦𝑦6

4. 1𝑥𝑥𝑦𝑦2𝑧𝑧3

5. 𝑞𝑞3(𝑞𝑞9+𝑝𝑝6𝑟𝑟5)

𝑝𝑝4

6. 𝑥𝑥3𝑦𝑦7+𝑥𝑥4𝑧𝑧7

𝑦𝑦4𝑧𝑧3

7. 6𝑥𝑥2𝑦𝑦+𝑦𝑦2+42𝑥𝑥𝑦𝑦2

8. 16𝑟𝑟𝑟𝑟

9. 𝑥𝑥+2𝑥𝑥+6

10. 𝑥𝑥+2𝑥𝑥+4

SOLUTIONS



1. B 2. D 3. B 4. D 5. A 6. B 7. C 8. A 9. D

10. C 11. A 12. B 13. A 14. A 15. C 16. A 17. B 18. B

QUADRATIC EXPRESSION, QUADRATIC EQUATIONS

Distribution and foil method for a polynomial:

1. x2 + 5 x + 6 2. x2 + 4 x - 5 3. t2 – 12 t + 32 4. 3 x2 + 14 x + 8 5. 3 w2 – 6 w - 45

6. 6 y2 + 19 y + 8 7. 20 y2 + 14 y - 12 8. r2 + 10 r + 25 9. t2 – 6 t + 9 10. 4 x2 + 28 x + 49

Factoring Polynomials:

1. ( r + 6 ) ( r + 2 ) 2. ( y – 5 ) ( y + 7 ) 3. ( k – 5 ) ( k – 6 )

4. ( 2 k + 5 ) ( k – 3 ) 5. ( 3 x – 2 ) ( x – 6 )

Solving trinomial equations:

1. x = 2, 4 2. x = 0, 1/3, - 1/3 3. w = - 2, - 2/3 4. x = - 9, 4

5. x = - 3/2, 8 6. x = - 1, 11 7. x = - 3, 6 8. x = - 1/3, 5

SOLUTIONS


PRACTICE EXAMS:

Practice exam #1:

1. C 2. B 3. A 4. A 5. D 6. A

7. A 8. B 9. C 10. A 11. B 12. D

Practice exam #2:

1. B 2. B 3. C 4. A 5. C 6. A

7. A 8. C 9. C 10. D 11. B 12. C

Practice exam #3:

1. B 2. C 3. C 4. A 5. C 6. B

7. D 8. A 9. C 10. D 11. B 12. D

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