# 6.E: Review Exercises and Sample Exam - Mathematics

## Review Exercises

Exercise (PageIndex{1}) Introduction to Factoring

Determine the missing factor.

1. (12x^{3}−24x^{2}+4x=4x( ? ))
2. (10y^{4}−35y^{3}−5y^{2}=5y^{2}( ? ))
3. (−18a^{5}+9a^{4}−27a^{3}=−9a^{3}( ? ))
4. (−21x^{2}y+7xy^{2}−49xy=−7xy( ? ))

1. ((3x^{2}−6x+1))

3. ((2a^{2}−a+3))

Exercise (PageIndex{2}) Introduction to Factoring

Factor out the GCF.

1. (22x^{2}+11x)
2. (15y^{4}−5y^{3})
3. (18a^{3}−12a^{2}+30a)
4. (12a^{5}+20a^{3}−4a)
5. (9x^{3}y^{2}−18x^{2}y^{2}+27xy^{2})
6. (16a^{5}b^{5}c−8a^{3}b^{6}+24a^{3}b^{2}c)

1. (11x(2x+1))

3. (6a(3a^{2}−2a+5))

5. (9xy2(x^{2}−2x+3))

Exercise (PageIndex{3}) Introduction to Factoring

Factor by grouping.

1. (x^{2}+2x−5x−10 )
2. (2x^{2}−2x−3x+3 )
3. (x^{3}+5x^{2}−3x−15 )
4. (x^{3}−6x^{2}+x−6 )
5. (x^{3}−x^{2}y−2x+2y )
6. (a^{2}b^{2}−2a^{3}+6ab−3b^{3})

1. ((x+2)(x−5))

3. ((x+5)(x^{2}−3))

5. ((x−y)(x^{2}−2))

Exercise (PageIndex{4}) Factoring Trinomials of the Form (x^{2}+bx+c)

Are the following factored correctly? Check by multiplying.

1. (x^{2}+5x+6=(x+6)(x−1) )
2. (x^{2}+3x−10=(x+5)(x−2) )
3. (x^{2}+6x+9=(x+3)^{2} )
4. (x^{2}−6x−9=(x−3)(x+3))

1. No

3. Yes

Exercise (PageIndex{5}) Factoring Trinomials of the Form (x^{2}+bx+c)

Factor.

1. (x^{2}−13x−14 )
2. (x^{2}+13x+12 )
3. (y^{2}+10y+25 )
4. (y^{2}−20y+100 )
5. (a^{2}−8a−48 )
6. (b^{2}−18b+45 )
7. (x^{2}+2x+24 )
8. (x^{2}−10x−16 )
9. (a^{2}+ab−2b^{2} )
10. (a^{2}b^{2}+5ab−50)

1. ((x−14)(x+1))

3. ((y+5)^{2})

5. ((a−12)(a+4))

7. Prime

9. ((a−b)(a+2b))

Exercise (PageIndex{6}) Factoring Trinomials of the Form (ax^{2}+bx+c)

Factor.

1. (5x^{2}−27x−18 )
2. (3x^{2}−14x+8 )
3. (4x^{2}−28x+49 )
4. (9x^{2}+48x+64 )
5. (6x^{2}−29x−9 )
6. (8x^{2}+6x+9 )
7. (60x^{2}−65x+15 )
8. (16x^{2}−40x+16 )
9. (6x^{3}−10x^{2}y+4xy^{2})
10. (10x^{3}y−82x^{2}y^{2}+16xy^{3})
11. (−y^{2}+9y+36 )
12. (−a^{2}−7a+98 )
13. (16+142x−18x^{2} )
14. (45−132x−60x^{2})

1. ((5x+3)(x−6) )

3. ((2x−7)^{2})

5. Prime

7. (5(3x−1)(4x−3) )

9. (2x(3x−2y)(x−y) )

11. (−1(y−12)(y+3) )

13. (−2(9x+1)(x−8))

Exercise (PageIndex{7}) Factoring Special Binomials

Factor completely.

1. (x^{2}−81)
2. (25x^{2}−36)
3. (4x^{2}−49)
4. (81x^{2}−1)
5. (x^{2}−64y^{2})
6. (100x^{2}y^{2}−1)
7. (16x^{4}−y^{4})
8. (x^{4}−81y^{4})
9. (8x^{3}−125)
10. (27+y^{3})
11. (54x^{4}y−2xy^{4})
12. (3x^{4}y^{2}+24xy^{5})
13. (64x^{6}−y^{6})
14. (x^{6}+1)

1. ((x+9)(x−9))

3. ((2x+7)(2x−7))

5. ((x+8y)(x−8y))

7. ((4x^{2}+y^{2})(2x+y)(2x−y))

9. ((2x−5)(4x^{2}+10x+25))

11. (2xy(3x−y)(9x^{2}+3xy+y^{2}))

13. ((2x+y)(4x^{2}−2xy+y^{2})(2x−y)(4x^{2}+2xy+y^{2}))

Exercise (PageIndex{8}) General Guidelines for Factoring Polynomials

Factor completely.

1. (8x^{3}−4x^{2}+20x)
2. (50a^{4}b^{4}c+5a^{3}b^{5}c^{2})
3. (x^{3}−12x^{2}−x+12)
4. (a^{3}−2a^{2}−3ab+6b)
5. (−y^{2}−15y+16)
6. (x^{2}−18x+72)
7. (144x^{2}−25)
8. (3x^{4}−48)
9. (20x^{2}−41x−9)
10. (24x^{2}+14x−20)
11. (a^{4}b−343ab^{4})
12. (32x^{7}y^{2}+4xy^{8})

1. (4x(2x^{2}−x+5))

3. ((x−12)(x+1)(x−1))

5. (−1(y+16)(y−1))

7. ((12x+5)(12x−5))

9. ((4x−9)(5x+1))

11. (ab(a−7b)(a^{2}+7ab+49b^{2}))

Exercise (PageIndex{9}) Solving Equations by Factoring

Solve.

1. ((x−9)(x+10)=0 )
2. (−3x(x+8)=0 )
3. (6(x+1)(x−1)=0 )
4. ((x−12)(x+4)(2x−1)=0 )
5. (x^{2}+5x−50=0 )
6. (3x^{2}−13x+4=0 )
7. (3x^{2}−12=0 )
8. (16x^{2}−9=0 )
9. ((x−2)(x+6)=20 )
10. (2(x−2)(x+3)=7x−9 )
11. (52x^{2}−203x=0 )
12. (23x^{2}−512x+124=0)

1. (9, −10)

3. (−1, 1)

5. (−10, 5)

7. (±2)

9. (−8, 4)

11. (0, frac{8}{3})

Exercise (PageIndex{10}) Solving Equations by Factoring

Find a quadratic equation with integer coefficients, given the following solutions.

1. (−7, 6)
2. (0, −10)
3. (−frac{1}{9}, frac{1}{2})
4. (± frac{3}{2})

1. (x^{2}+x−42=0)

3. (18x^{2}−7x−1=0)

Exercise (PageIndex{11}) Applications Involving Quadratic Equations

Set up an algebraic equation and then solve the following.

1. An integer is (4) less than twice another. If the product of the two integers is (96), then find the integers.
2. The sum of the squares of two consecutive positive even integers is (52). Find the integers.
3. A (20)-foot ladder leaning against a wall reaches a height that is (4) feet more than the distance from the wall to the base of the ladder. How high does the ladder reach?
4. The height of an object dropped from the top of a (196)-foot building is given by (h(t)=−16t^{2}+196), where (t) represents the number of seconds after the object has been released. How long will it take the object to hit the ground?
5. The length of a rectangle is (1) centimeter less than three times the width. If the area is (70) square centimeters, then find the dimensions of the rectangle.
6. The base of a triangle is (4) centimeters more than twice the height. If the area of the triangle is (80) square centimeters, then find the measure of the base.

1. {(8, 12)} or {(−6, −16)}

3. (16) feet

5. Length: (14) centimeters; width: (5) centimeters

## Sample Exam

Exercise (PageIndex{12})

1. Determine the GCF of the terms (25a^{2}b^{2}c, 50ab^{4}), and (35a^{3}b^{3}c^{2}).
2. Determine the missing factor: (24x^{2}y^{3}−16x^{3}y^{2}+8x^{2}y=8x^{2}y( ? )).

1. (5ab^{2})

Exercise (PageIndex{13})

Factor.

1. (12x^{5}−15x^{4}+3x^{2})
2. (x^{3}−4x^{2}−2x+8)
3. (x^{2}−7x+12)
4. (9x^{2}−12x+4)
5. (x^{2}−81)
6. (x^{3}+27y^{3})

1. (3x^{2}(4x^{3}−5x^{2}+1))

3. ((x−4)(x−3) )

5. ((x+9)(x−9))

Exercise (PageIndex{14})

Factor completely.

1. (x^{3}+2x^{2}−4x−8)
2. (x^{4}−1)
3. (−6x^{3}+20x^{2}−6x)
4. (x^{6}−1)

1. ((x+2)^{2}(x−2))

3. (−2x(3x−1)(x−3))

Exercise (PageIndex{15})

Solve.

1. ((2x+1)(x−7)=0 )
2. (3x(4x−3)(x+1)=0 )
3. (x^{2}−64=0 )
4. (x^{2}+4x−12=0 )
5. (23x^{2}+89x−16=0 )
6. ((x−5)(x−3)=−1 )
7. (3x(x+3)=14x+2 )
8. ((3x+1)(3x+2)=9x+3)

1. (−frac{1}{2}, 7 )

3. (±8 )

5. (−frac{3}{2}, frac{1}{6})

7. (−frac{1}{3}, 2)

Exercise (PageIndex{16})

For each problem, set up an algebraic equation and then solve.

1. An integer is (4) less than twice another. If the product of the two integers is (70), then find the integers.
2. The sum of the squares of two consecutive positive odd integers is (130). Find the integers.
3. The length of a rectangle is (4) feet more than twice its width. If the area is (160) square feet, then find the dimensions of the rectangle.
4. The height of a triangle is (6) centimeters less than four times the length of its base. If the area measures (27) square centimeters, then what is the height of the triangle?
5. The height of a projectile launched upward at a speed of (64) feet/second from a height of (36) feet is given by the function (h(t)=−16t^{2}+64t+36). How long will it take the projectile to hit the ground?

1. {(7, 10)} or {(−14, −5)}

3. Width: (8) feet; length: (20) feet

5. (4frac{1}{2}) sec

## SAT Math Practice Test

The free SAT Math practice test is specifically designed to ensure that the test-taker is knowledgeable about the SAT and is able to know what to expect when it is time to take the Math portion of the SAT. The Math portion will consist of two sections, a 44 question multiple-choice section, and a 10 question student-response section. These will address six main topics:

• Numbers and Operations
• Algebra and Functions
• Geometry and Measurements
• Data Analysis
• Statistics
• Probability

Due to our foreknowledge of the domains of this test, we have ensured that the SAT Math practice test will cover these topics thoroughly.

When taking this portion of the SAT, the test-taker will have a total of 70 minutes to finish.

The test will be broken down into two 25-minute sections and one 20-minute section.

Test-takers will not be allowed to use a calculator in one of the two sections.

The ACCUPLACER tests are administered by College Board. If you would like to schedule an appointment, you should start by visiting with your college advisor or counselor. If your institution is located in another state, there may be remote testing options available to take your ACCUPLACER test. Your advisor or counselor will be able to guide you further on the necessary steps to receive permission for remote testing and schedule your appointment.

You will be presented with 20 questions on the ACCUPLACER arithmetic subtest. The questions will assess your comprehension of fundamental arithmetic concepts, including the following knowledge and skills categories:

• Decimal Operations (3-5 questions / 15-25%)
• Addition, Division, Multiplication and Subtraction of Decimal Numbers
• Applying Operations to Real-Life Contexts
• Estimation and Rounding
• Order of Operations
• Fraction Operations (3-5 questions / 15-25%)
• Addition, Division, Multiplication and Subtraction of Fractions and Mixed Numbers
• Applying Operations to Real-Life Contexts
• Estimation and Rounding
• Order of Operations
• Comparisons of Differently Formatted Values by Ordering
• Evaluation of Equivalent Number Statements
• Using Equality/Inequality Symbol Notation
• Using the Number Line
• Applying Percent to Real-Life Contexts
• Calculation with Percent (with or without a context)
• Determining the Percent of a Number
• Percent Decrease
• Percent Increase
• Addition, Division, Multiplication and Subtraction of Whole Numbers
• Applying Operations to Real-Life Contexts
• Estimation and Rounding
• Order of Operations

This is one of over 2,400 courses on OCW. Explore materials for this course in the pages linked along the left.

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## Free Math Practice Tests

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## Free TABE Practice Tests

#### TABE Language

The easiest (and fastest) way to prepare for your TABE exam is to use Test-Guide.com's sample TABE questions. Our sample tests require no registration and are completely free. We have organized our questions based on the official TABE test outline.

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Publication Date:September 12, 2018
math-2022/5697/image_dhDgxYKCK2w2wB.png

Getting Started
The College Placement Math Plan 9
Making a Study Schedule 10

Basic Math
Fraction Tips, Tricks and Shortcuts 16
Converting Fractions to Decimals 18
Converting Fractions to Percent 20
Decimal Tips, Tricks and Shortcuts 22
Converting Decimals to Fractions 22
Converting Decimals to Percent 22
Percent Tips, Tricks and Shortcuts 23
Converting Percents to Decimals 24
Converting Percents to Fractions 25
Scientific Notation 25
How to Convert to Scientific Notation 26
Exponents Tips, Shortcuts & Tricks 28
Basic Math Practice 32

Word Problems
How to Solve Word Problems 52
Types of Word Problems 55
Word Practice 70

Basic Geometry
Cartesian Plane and Coordinate Grid 86
Pythagorean Geometry 92
Geometry Practice Questions 105

Basic Algebra
Solving One-Variable Linear Equations 141
Solving Two-Variable Linear Equations 142
Simplifying Polynomials 144
Factoring Polynomials 144
Algebra Practice Questions 151

Trigonometry 190
Sequences 193
Logarithms 194

Basic Math Multiple Choice
Multiple Choice Strategy and Shortcuts 228

How to Study for a Math Test
How to Prepare for a Test

The Strategy of Studying 236
How to Take a Test
How to Take a Test - The Basics 241
In the Test Room – What you MUST do! 245
Avoid Anxiety Before a Test 251
Common Test-Taking Mistakes 253 Over 200 College Placement Math practice questions, plus test tips, how to study math, multiple choice strategies and more!

Written by, Brian Stocker MA., Complete Test Preparation Inc.

Date Published: Friday, September 14th, 2018
Date Modified: Friday, August 7th, 2020

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