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5.1 Number Properties, Exponents, Radicals, and Polynomials

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CAT-ASVAB Computer Adaptive ASVAB

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1 Chapter 1: CAT-ASVAB Overview, Adaptive Mechanics & Test Strategy

1.1 CAT-ASVAB Architecture, MEPS Administration & Subtest Structure 1.2 Computer Adaptive Testing (CAT), IRT Engine & Pacing Rules 1.3 AFQT Scoring Formula, VE Weighting & Service Branch Minimums 1.4 Military Composite Line Scores & MOS/AFSC/Rating Classification

2 Chapter 2: Word Knowledge (WK)

2.1 Greek and Latin Roots, Prefixes, and Suffixes 2.2 Context Clues, Secondary Meanings, and Nuanced Synonyms 2.3 Military, Technical, and Formal Vocabulary in Context 2.4 High-Frequency Word Bank and the 36-Second Word-Attack Protocol

3 Chapter 3: Paragraph Comprehension (PC)

3.1 Identifying Main Ideas, Central Themes, and Author Purpose 3.2 Supporting Details, Factual Extraction, and Structural Analysis 3.3 Inferences, Contextual Meaning, and Tone Evaluation

4 Chapter 4: Arithmetic Reasoning (AR)

4.1 Translating Complex Word Problems into Algebraic Expressions 4.2 Ratios, Proportions, Rates, and Combined Work Problems 4.3 Percentages, Interest, Profit, and Financial Math 4.4 Applied Geometric Word Problems (Area, Perimeter, Volume)

5 Chapter 5: Mathematics Knowledge (MK)

5.1 Number Properties, Exponents, Radicals, and Polynomials 5.2 Linear Equations, Quadratic Equations, and Inequalities 5.3 Geometric Principles, Coordinate Geometry, and Trigonometry Fundamentals 5.4 Basic Probability, Permutations, Combinations, and Statistics 5.5 Fractions, Rational Expressions, and Non-Calculator Computation

6 Chapter 6: General Science (GS)

6.1 Cellular Biology, Human Anatomy, and Biological Systems 6.2 Chemistry Fundamentals, Atomic Structure, and Reactions 6.3 Classical Physics, Energy, Forces, and Motion 6.4 Earth Science, Meteorology, Astronomy, and Environmental Science 6.5 Scientific Method, Measurement Systems, and Laboratory Instruments

7 Chapter 7: Electronics Information (EI)

7.1 Voltage, Current, Resistance, Ohm's Law, and Power 7.2 Circuit Analysis: Series, Parallel, and Combination Circuits with Kirchhoff's Laws 7.3 Electronic Components, Semiconductor Devices, and Schematic Symbols 7.4 Magnetism, Electromagnetism, Motors, and Generators 7.5 Circuit Protection, Wiring, Measurement Instruments, and Electrical Safety

8 Chapter 8: Auto Information (AI)

8.1 Internal Combustion Engines and the Four-Stroke Cycle 8.2 Powertrain, Transmission, Drivetrain, and Fuel Systems 8.3 Braking Systems, Suspension, Steering, and Electrical Diagnostics

9 Chapter 9: Shop Information (SI)

9.1 Hand Tools, Measuring Instruments, and Cutting Implements 9.2 Power Tools, Fasteners, Hardware, and Joining Techniques 9.3 Woodworking, Metalworking, Materials, and Workshop Safety

10 Chapter 10: Mechanical Comprehension (MC)

10.1 Simple Machines: Levers, Classes, and Mechanical Advantage 10.2 Pulleys, Wheel and Axle Systems, and Inclined Planes 10.3 Gears, Belt and Chain Drives, Cams, and Torque Transmission 10.4 Fluid Power, Hydraulics, Pneumatics, and Buoyancy (Pascal & Archimedes) 10.5 Forces, Friction, Work, Power, Energy Conservation, and Wave Mechanics

11 Chapter 11: Assembling Objects (AO)

11.1 Spatial Relationship Fundamentals & 2D Silhouette Synthesis 11.2 Connection Point Problems, Interior Seams, and Labeled Points 11.3 High-Speed Pacing, Elimination Tactics, and Visual Pattern Recognition

5.1 Number Properties, Exponents, Radicals, and Polynomials

Key Takeaways Explain This Topic AI

Mathematics Knowledge (MK) delivers 15 scored adaptive questions in 31 minutes (about 124 seconds per question) and feeds the Armed Forces Qualification Test directly, with no calculator permitted.

The real number system spans rational numbers expressible as a/b (b ≠ 0) and irrational numbers with infinite non-repeating decimals; prime factorization breaks composite integers into prime powers to extract GCF (lowest common powers) and LCM (highest powers).

Strict order of operations (PEMDAS/GEMS) requires resolving equal-precedence operations (multiplication/division and addition/subtraction) from left to right, while negative base exponent rules dictate that (-x)² = +x² but -x² = -(x²).

Exponent and radical laws mandate that negative powers invert bases (x^(-n) = 1/x^n), rational exponents represent roots (x^(m/n) = √n(x^m)), and radical expressions simplify by factoring out the largest perfect square divisors.

Polynomial multiplication relies on the distributive law and the FOIL method, while factoring requires mastering GCF extraction, difference of two squares (a² - b² = (a-b)(a+b)), and perfect square trinomials (a² ± 2ab + b² = (a ± b)²).

Last updated: August 2026

5.1 Number Properties, Exponents, Radicals, and Polynomials

Core Principle: The Mathematics Knowledge (MK) subtest assesses high school-level mathematical principles, theoretical rules, and computational procedures. On the computerized CAT-ASVAB, you have 15 scored adaptive questions in 31 minutes — giving you roughly 124 seconds per question , about twice the allowance of the technical recall subtests. Unlike Arithmetic Reasoning (AR), which evaluates word problems and practical applied scenarios, MK focuses on pure mathematical mechanics: evaluating complex expressions, simplifying radicals, manipulating exponents, factoring polynomials, and solving algebraic systems without a calculator.

1. CAT-ASVAB Mathematics Knowledge (MK) Architecture & Tactical Pacing

Mathematics Knowledge is one of the four academic subtests that determine the Armed Forces Qualification Test (AFQT) percentile score, which dictates overall military enlistment eligibility: AFQT Raw Score = 2 VE + AR + MK \text{AFQT Raw Score} = 2\text{VE} + \text{AR} + \text{MK} AFQT Raw Score = 2 VE + AR + MK

Where VE = WK + PC \text{VE} = \text{WK} + \text{PC} VE = WK + PC (Verbal Expression). While VE carries double weight, MK and AR together comprise 50% of the non-verbal AFQT score. Furthermore, MK is a mandatory composite component for high-tier technical military occupational specialties (MOS/Ratings/AFSCs):

Skilled Technical (ST): GS + VE + MK + MC \text{GS} + \text{VE} + \text{MK} + \text{MC} GS + VE + MK + MC (Required for Army Cyber Operations, Intelligence, and IT)

Electronics (EL): GS + AR + MK + EI \text{GS} + \text{AR} + \text{MK} + \text{EI} GS + AR + MK + EI (Required for Navy Nuclear Field, Air Force Avionics, and Submarine Warfare)

General Maintenance (GM): GS + AS + MK + EI \text{GS} + \text{AS} + \text{MK} + \text{EI} GS + AS + MK + EI (Prerequisite for Aviation Maintenance and Armament)

General Technical (GT): VE + AR \text{VE} + \text{AR} VE + AR (Officer Candidate School and Special Operations prerequisite)

Non-Calculator Execution & Time Management

Calculators are strictly prohibited during the ASVAB. Candidates are issued scratch paper and a pencil at the Military Entrance Processing Station (MEPS). Excelling on the MK subtest demands fluent mental arithmetic, disciplined scratchpad layout, and instant recognition of structural algebraic shortcuts. +-----------------------------------------------------------------------------------------+

  • | MK TIMING BUDGET: 75 SECONDS |
  • +-----------------------+-------------------------+---------------------------------------+
  • | 0 - 15 Seconds | 15 - 55 Seconds | 55 - 75 Seconds |
  • | Classify concept and | Execute algebraic steps | Verify signs, check exponents, and |
  • | identify shortcut | on scratch paper | confirm selected multiple-choice item |
  • +-----------------------+-------------------------+---------------------------------------+
  • 2. Classification of Real Numbers & Foundational Arithmetic Properties

All numbers encountered on the MK subtest belong to the set of Real Numbers ( R \mathbb{R} R ): [ Real Numbers (R) ]

  • ├── [ Rational Numbers (Q) ] (can be expressed as a/b, where a, b ∈ Z, b ≠ 0)
  • │ ├── [ Integers (Z) ] (..., -3, -2, -1, 0, 1, 2, 3, ...)
  • │ │ └── [ Whole Numbers (W) ] (0, 1, 2, 3, ...)
  • │ │ └── [ Natural Numbers (N) ] (1, 2, 3, ...)
  • │ └── [ Fractions & Decimals ] (Terminating e.g. 0.375, Repeating e.g. 0.666...)
  • └── [ Irrational Numbers ] (Cannot be written as a/b; non-terminating, non-repeating: √2, √3, π, e)

Critical Number Definitions:

Prime Numbers: Integers strictly greater than 1 whose only positive integer divisors are 1 and themselves. The first 15 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 . Note that 2 is the only even prime number , and 1 is neither prime nor composite .

Composite Numbers: Positive integers greater than 1 that possess at least one positive factor other than 1 and themselves (e.g., 4, 6, 8, 9, 10, 12, 14, 15...).

Reciprocals (Multiplicative Inverses): Two non-zero numbers whose product is exactly 1. The reciprocal of a b \frac{a}{b} b a ​ is b a \frac{b}{a} a b ​ . The integer n n n has reciprocal 1 n \frac{1}{n} n 1 ​ . Zero has no reciprocal.

Absolute Value ( ∣ x ∣ |x| ∣ x ∣ ): The non-negative geometric distance of a number from zero on the real number line ( ∣ − 9 ∣ = 9 |-9| = 9 ∣ − 9∣ = 9 , ∣ 9 ∣ = 9 |9| = 9 ∣9∣ = 9 ).

Fundamental Arithmetic Properties

  • Property Addition Formulation Multiplication Formulation
  • Commutative a + b = b + a a + b = b + a a + b = b + a a ⋅ b = b ⋅ a a \cdot b = b \cdot a a ⋅ b = b ⋅ a

Associative ( a + b ) + c = a + ( b + c ) (a + b) + c = a + (b + c) ( a + b ) + c = a + ( b + c ) ( a ⋅ b ) ⋅ c = a ⋅ ( b ⋅ c ) (a \cdot b) \cdot c = a \cdot (b \cdot c) ( a ⋅ b ) ⋅ c = a ⋅ ( b ⋅ c )

Distributive a ( b + c ) = a b + a c a(b + c) = ab + ac a ( b + c ) = ab + a c ( a + b ) c = a c + b c (a + b)c = ac + bc ( a + b ) c = a c + b c

Identity a + 0 = a a + 0 = a a + 0 = a (Additive Identity: 0) a ⋅ 1 = a a \cdot 1 = a a ⋅ 1 = a (Multiplicative Identity: 1)

Inverse a + ( − a ) = 0 a + (-a) = 0 a + ( − a ) = 0 (Additive Inverse) a ⋅ 1 a = 1 a \cdot \frac{1}{a} = 1 a ⋅ a 1 ​ = 1 for a ≠ 0 a \neq 0 a  = 0 (Multiplicative Inverse)

  • 3. Rapid Divisibility Tests, Prime Factorization, GCF & LCM
  • Divisibility Diagnostic Table

Mastering divisibility tests allows instantaneous reduction of large fractional coefficients on scratch paper:

  • Divisor Divisibility Condition Applied Example
  • 2 Last digit is even (0, 2, 4, 6, 8) 6 , 438 6,438 6 , 438 ends in 8 → \rightarrow → Divisible

3 Sum of all digits is a multiple of 3 8 , 142 → 8 + 1 + 4 + 2 = 15 8,142 \rightarrow 8+1+4+2 = 15 8 , 142 → 8 + 1 + 4 + 2 = 15 ( 15 ÷ 3 = 5 15 \div 3 = 5 15 ÷ 3 = 5 ) → \rightarrow → Divisible

4 Last two digits form a number divisible by 4 9 , 528 → 28 9,528 \rightarrow 28 9 , 528 → 28 is divisible by 4 → \rightarrow → Divisible

5 Last digit is 0 or 5 4 , 395 4,395 4 , 395 ends in 5 → \rightarrow → Divisible

6 Divisible by both 2 AND 3 7 , 314 → 7,314 \rightarrow 7 , 314 → Even and 7 + 3 + 1 + 4 = 15 7+3+1+4=15 7 + 3 + 1 + 4 = 15 (div by 3) → \rightarrow → Divisible

8 Last three digits form a number divisible by 8 5 , 160 → 160 / 8 = 20 5,160 \rightarrow 160 / 8 = 20 5 , 160 → 160/8 = 20 → \rightarrow → Divisible

9 Sum of all digits is a multiple of 9 6 , 804 → 6 + 8 + 0 + 4 = 18 6,804 \rightarrow 6+8+0+4 = 18 6 , 804 → 6 + 8 + 0 + 4 = 18 ( 18 ÷ 9 = 2 18 \div 9 = 2 18 ÷ 9 = 2 ) → \rightarrow → Divisible

  • 10 Last digit is 0 12 , 750 12,750 12 , 750 ends in 0 → \rightarrow → Divisible
  • Prime Factorization & Finding GCF and LCM

According to the Fundamental Theorem of Arithmetic, every composite number factors uniquely into a product of prime powers. Prime Factorizations for 48, 72, and 120:

  • 48 = 16 × 3 = 2⁴ × 3¹
  • 72 = 8 × 9 = 2³ × 3²
  • 120 = 8 × 15 = 2³ × 3¹ × 5¹

Greatest Common Factor (GCF): Take the lowest exponent of every prime factor common to all sets: GCF ( 48 , 72 , 120 ) = 2 min ⁡ ( 4 , 3 , 3 ) × 3 min ⁡ ( 1 , 2 , 1 ) = 2 3 × 3 1 = 8 × 3 = 24 \text{GCF}(48, 72, 120) = 2^{\min(4,3,3)} \times 3^{\min(1,2,1)} = 2^3 \times 3^1 = 8 \times 3 = 24 GCF ( 48 , 72 , 120 ) = 2 m i n ( 4 , 3 , 3 ) × 3 m i n ( 1 , 2 , 1 ) = 2 3 × 3 1 = 8 × 3 = 24

Least Common Multiple (LCM): Take the highest exponent of every prime factor present in any set: LCM ( 48 , 72 , 120 ) = 2 max ⁡ ( 4 , 3 , 3 ) × 3 max ⁡ ( 1 , 2 , 1 ) × 5 max ⁡ ( 0 , 0 , 1 ) = 2 4 × 3 2 × 5 1 = 16 × 9 × 5 = 720 \text{LCM}(48, 72, 120) = 2^{\max(4,3,3)} \times 3^{\max(1,2,1)} \times 5^{\max(0,0,1)} = 2^4 \times 3^2 \times 5^1 = 16 \times 9 \times 5 = 720 LCM ( 48 , 72 , 120 ) = 2 m a x ( 4 , 3 , 3 ) × 3 m a x ( 1 , 2 , 1 ) × 5 m a x ( 0 , 0 , 1 ) = 2 4 × 3 2 × 5 1 = 16 × 9 × 5 = 720

4. Order of Operations (PEMDAS / GEMS) & Signed Numbers

When evaluating algebraic expressions, operations must strictly follow hierarchical priority: [ P / G ] : Grouping Symbols (Parentheses (), Brackets [], Braces {}, Numerator/Denominator bars)

  • [ E ] : Exponents and Radicals (evaluated left-to-right)
  • [ M / D ] : Multiplication & Division (Equal priority: evaluate Left-to-Right)
  • [ A / S ] : Addition & Subtraction (Equal priority: evaluate Left-to-Right)

ASVAB Trap: Multiplication does not outrank division! They have identical priority and must be executed in left-to-right sequence. For example: 18 ÷ 6 × 3 = 3 × 3 = 9 18 \div 6 \times 3 = 3 \times 3 = 9 18 ÷ 6 × 3 = 3 × 3 = 9 , NOT 18 ÷ 18 = 1 18 \div 18 = 1 18 ÷ 18 = 1 .

Exponent Sign Traps with Negative Bases

Base enclosed in parentheses: ( − 4 ) 2 = ( − 4 ) × ( − 4 ) = + 16 (-4)^2 = (-4) \times (-4) = +16 ( − 4 ) 2 = ( − 4 ) × ( − 4 ) = + 16

Base without parentheses: − 4 2 = − ( 4 × 4 ) = − 16 -4^2 = -(4 \times 4) = -16 − 4 2 = − ( 4 × 4 ) = − 16

Odd power with negative base: ( − 2 ) 3 = ( − 2 ) × ( − 2 ) × ( − 2 ) = − 8 (-2)^3 = (-2) \times (-2) \times (-2) = -8 ( − 2 ) 3 = ( − 2 ) × ( − 2 ) × ( − 2 ) = − 8

Step-by-Step Worked Solution: Evaluate: 24 − 3 ⋅ [ 4 + ( 2 − 6 ) 2 ] ÷ ( − 5 + 1 ) \text{Evaluate: } 24 - 3 \cdot [4 + (2 - 6)^2] \div (-5 + 1) Evaluate: 24 − 3 ⋅ [ 4 + ( 2 − 6 ) 2 ] ÷ ( − 5 + 1 )

Resolve inner grouping: 2 − 6 = − 4 2 - 6 = -4 2 − 6 = − 4 and − 5 + 1 = − 4 -5 + 1 = -4 − 5 + 1 = − 4

Expression simplifies to: 24 − 3 ⋅ [ 4 + ( − 4 ) 2 ] ÷ ( − 4 ) 24 - 3 \cdot [4 + (-4)^2] \div (-4) 24 − 3 ⋅ [ 4 + ( − 4 ) 2 ] ÷ ( − 4 )

  • Resolve exponent inside bracket: ( − 4 ) 2 = 16 (-4)^2 = 16 ( − 4 ) 2 = 16
  • Add inside bracket: 4 + 16 = 20 4 + 16 = 20 4 + 16 = 20
  • Expression simplifies to: 24 − 3 ⋅ 20 ÷ ( − 4 ) 24 - 3 \cdot 20 \div (-4) 24 − 3 ⋅ 20 ÷ ( − 4 )

Execute multiplication/division left-to-right: 3 ⋅ 20 = 60 3 \cdot 20 = 60 3 ⋅ 20 = 60 , then 60 ÷ ( − 4 ) = − 15 60 \div (-4) = -15 60 ÷ ( − 4 ) = − 15

  • Final subtraction: 24 − ( − 15 ) = 24 + 15 = 39 24 - (-15) = 24 + 15 = 39 24 − ( − 15 ) = 24 + 15 = 39
  • 5. Rules of Exponents, Radicals & Scientific Notation
  • Laws of Exponents Reference Table

Law Name Algebraic Rule Numerical Demonstration

Product Rule x a ⋅ x b = x a + b x^a \cdot x^b = x^{a+b} x a ⋅ x b = x a + b 3 2 ⋅ 3 4 = 3 2 + 4 = 3 6 = 729 3^2 \cdot 3^4 = 3^{2+4} = 3^6 = 729 3 2 ⋅ 3 4 = 3 2 + 4 = 3 6 = 729

Quotient Rule x a x b = x a − b \frac{x^a}{x^b} = x^{a-b} x b x a ​ = x a − b 4 7 4 4 = 4 7 − 4 = 4 3 = 64 \frac{4^7}{4^4} = 4^{7-4} = 4^3 = 64 4 4 4 7 ​ = 4 7 − 4 = 4 3 = 64

Power of a Power ( x a ) b = x a ⋅ b (x^a)^b = x^{a \cdot b} ( x a ) b = x a ⋅ b ( 2 3 ) 2 = 2 3 ⋅ 2 = 2 6 = 64 (2^3)^2 = 2^{3 \cdot 2} = 2^6 = 64 ( 2 3 ) 2 = 2 3 ⋅ 2 = 2 6 = 64

Power of a Product ( x y ) a = x a y a (xy)^a = x^a y^a ( x y ) a = x a y a ( 3 x ) 3 = 3 3 x 3 = 27 x 3 (3x)^3 = 3^3 x^3 = 27x^3 ( 3 x ) 3 = 3 3 x 3 = 27 x 3

Zero Exponent x 0 = 1 x^0 = 1 x 0 = 1 (for x ≠ 0 x \neq 0 x  = 0 ) ( − 58.2 ) 0 = 1 (-58.2)^0 = 1 ( − 58.2 ) 0 = 1

Negative Exponent x − n = 1 x n x^{-n} = \frac{1}{x^n} x − n = x n 1 ​ and 1 x − n = x n \frac{1}{x^{-n}} = x^n x − n 1 ​ = x n 5 − 2 = 1 5 2 = 1 25 5^{-2} = \frac{1}{5^2} = \frac{1}{25} 5 − 2 = 5 2 1 ​ = 25 1 ​

Fractional Exponent x m / n = x m n = ( x n ) m x^{m/n} = \sqrt[n]{x^m} = (\sqrt[n]{x})^m x m / n = n x m ​ = ( n x ​ ) m 64 2 / 3 = ( 64 3 ) 2 = 4 2 = 16 64^{2/3} = (\sqrt[3]{64})^2 = 4^2 = 16 6 4 2/3 = ( 3 64 ​ ) 2 = 4 2 = 16

Simplifying Radicals

To simplify square roots without a calculator, factor out the largest perfect square ( 1 , 4 , 9 , 16 , 25 , 36 , 49 , 64 , 81 , 100 , 121 , 144... 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144... 1 , 4 , 9 , 16 , 25 , 36 , 49 , 64 , 81 , 100 , 121 , 144... ): 180 = 36 × 5 = 36 × 5 = 6 5 \sqrt{180} = \sqrt{36 \times 5} = \sqrt{36} \times \sqrt{5} = 6\sqrt{5} 180 ​ = 36 × 5 ​ = 36 ​ × 5 ​ = 6 5 ​

Combining and Rationalizing Radicals

Combining Like Radicals: 4 18 − 2 50 + 72 = 4 ( 3 2 ) − 2 ( 5 2 ) + 6 2 = 12 2 − 10 2 + 6 2 = 8 2 4\sqrt{18} - 2\sqrt{50} + \sqrt{72} = 4(3\sqrt{2}) - 2(5\sqrt{2}) + 6\sqrt{2} = 12\sqrt{2} - 10\sqrt{2} + 6\sqrt{2} = 8\sqrt{2} 4 18 ​ − 2 50 ​ + 72 ​ = 4 ( 3 2 ​ ) − 2 ( 5 2 ​ ) + 6 2 ​ = 12 2 ​ − 10 2 ​ + 6 2 ​ = 8 2 ​

Rationalizing Denominators: Multiply numerator and denominator by the radical to eliminate square roots from the denominator: 15 5 = 15 ⋅ 5 5 ⋅ 5 = 15 5 5 = 3 5 \frac{15}{\sqrt{5}} = \frac{15 \cdot \sqrt{5}}{\sqrt{5} \cdot \sqrt{5}} = \frac{15\sqrt{5}}{5} = 3\sqrt{5} 5 ​ 15 ​ = 5 ​ ⋅ 5 ​ 15 ⋅ 5 ​ ​ = 5 15 5 ​ ​ = 3 5 ​

Scientific Notation Operations

Multiplication: ( 2.5 × 10 4 ) × ( 6.0 × 10 3 ) = ( 2.5 × 6.0 ) × 10 4 + 3 = 15.0 × 10 7 = 1.5 × 10 8 (2.5 \times 10^4) \times (6.0 \times 10^3) = (2.5 \times 6.0) \times 10^{4+3} = 15.0 \times 10^7 = 1.5 \times 10^8 ( 2.5 × 1 0 4 ) × ( 6.0 × 1 0 3 ) = ( 2.5 × 6.0 ) × 1 0 4 + 3 = 15.0 × 1 0 7 = 1.5 × 1 0 8

Division: 7.2 × 10 6 1.8 × 10 − 2 = ( 7.2 1.8 ) × 10 6 − ( − 2 ) = 4.0 × 10 8 \frac{7.2 \times 10^6}{1.8 \times 10^{-2}} = \left(\frac{7.2}{1.8}\right) \times 10^{6 - (-2)} = 4.0 \times 10^8 1.8 × 1 0 − 2 7.2 × 1 0 6 ​ = ( 1.8 7.2 ​ ) × 1 0 6 − ( − 2 ) = 4.0 × 1 0 8

6. Polynomial Operations, Special Products & Factoring Archetypes

Definitions:

Monomial: Single term (e.g., 7 x 3 7x^3 7 x 3 ).

Binomial: Two terms separated by + + + or − - − (e.g., 4 x 2 − 9 4x^2 - 9 4 x 2 − 9 ).

Trinomial: Three terms (e.g., 2 x 2 + 5 x − 12 2x^2 + 5x - 12 2 x 2 + 5 x − 12 ).

Degree: The highest exponent of the variable in the polynomial.

Polynomial Multiplication: The FOIL Method

To multiply two binomials ( a x + b ) ( c x + d ) (ax + b)(cx + d) ( a x + b ) ( c x + d ) , apply FOIL (First, Outer, Inner, Last): (3x - 4)(2x + 5)

  • ├── First: 3x · 2x = 6x²
  • ├── Outer: 3x · 5 = +15x
  • ├── Inner: (-4) · 2x = -8x
  • └── Last: (-4) · 5 = -20
  • Sum = 6x² + (15x - 8x) - 20 = 6x² + 7x - 20
  • Special Algebraic Products

Difference of Squares: ( a − b ) ( a + b ) = a 2 − b 2 (a - b)(a + b) = a^2 - b^2 ( a − b ) ( a + b ) = a 2 − b 2

Square of a Binomial: ( a + b ) 2 = a 2 + 2 a b + b 2 (a + b)^2 = a^2 + 2ab + b^2 ( a + b ) 2 = a 2 + 2 ab + b 2 and ( a − b ) 2 = a 2 − 2 a b + b 2 (a - b)^2 = a^2 - 2ab + b^2 ( a − b ) 2 = a 2 − 2 ab + b 2

5 Core ASVAB Factoring Archetypes +-----------------------------------------------------------------------------------------+

  • | 5 ASVAB FACTORING ARCHETYPES |
  • +------------------------------------+----------------------------------------------------+
  • | 1. Greatest Common Factor (GCF) | 12x⁴y² - 18x²y³ = 6x²y²(2x² - 3y) |
  • +------------------------------------+----------------------------------------------------+
  • | 2. Difference of Two Squares | a² - b² = (a - b)(a + b) |
  • | | 49x² - 36 = (7x - 6)(7x + 6) |
  • +------------------------------------+----------------------------------------------------+
  • | 3. Perfect Square Trinomials | a² ± 2ab + b² = (a ± b)² |
  • | | 9x² - 24x + 16 = (3x - 4)² |
  • +------------------------------------+----------------------------------------------------+
  • | 4. General Trinomials (ax²+bx+c) | 2x² + 7x + 3 = (2x + 1)(x + 3) |
  • +------------------------------------+----------------------------------------------------+
  • | 5. Factoring by Grouping | x³ + 2x² + 5x + 10 = x²(x + 2) + 5(x + 2) |
  • | | = (x² + 5)(x + 2) |
  • +------------------------------------+----------------------------------------------------+

ASVAB Trap: The sum of two squares ( a 2 + b 2 a^2 + b^2 a 2 + b 2 ) is prime and cannot be factored over the set of real numbers! For example, x 2 + 25 x^2 + 25 x 2 + 25 cannot be factored into ( x + 5 ) ( x + 5 ) (x+5)(x+5) ( x + 5 ) ( x + 5 ) or ( x − 5 ) ( x + 5 ) (x-5)(x+5) ( x − 5 ) ( x + 5 ) .

Loading diagram... Hierarchy of Real Numbers and Exponent Rules Flow