14.1 Algebraic Expressions, Patterns & Translating Verbal Phrases
Key Takeaways
- Algebraic expressions are composed of variables, numerical coefficients, constant terms, and arithmetic operations; like terms must possess identical variable bases raised to identical exponents to be combined.
- Evaluating expressions demands strict adherence to the Order of Operations (PEMDAS/GEMS), with particular care given to negative numbers: exponentiation of a negative base requires grouping parentheses, where (-x)^n != -x^n for even powers (e.g., (-3)² = 9 vs. -3² = -9).
- Simplifying multi-step expressions requires applying the Distributive Property, correctly propagating negative signs across parentheses, expanding polynomial products (FOIL), and resolving nested grouping symbols from the innermost layer outward.
- Numerical and geometric patterns are defined by explicit nth-term formulas: arithmetic sequences exhibit a constant common difference (a_n = a_1 + (n - 1)d), while geometric sequences exhibit a constant common ratio (a_n = a_1 · r^(n - 1)).
- Translating verbal phrases into algebraic statements requires precision with turnaround phrases such as 'less than' and 'subtracted from' (e.g., '8 less than x' translates to x - 8, never 8 - x), alongside inequality constraints like 'is at least' (≥) and 'is no more than' (≤).
Algebraic Expressions, Patterns & Translating Verbal Phrases
Quick Answer: On the WEST-B Mathematics subtest (Objective 0017), algebraic competence requires fluency in three interconnected skills: manipulating formal expressions, extending numerical patterns, and translating word phrases into mathematical syntax. Remember: Like terms require identical variable bases and exponents (3x² and -5x² can combine, but 3x² and 3x cannot). When evaluating expressions with negative values, remember that (-4)² = 16 while -4² = -16. For patterns, use explicit formulas: Arithmetic Sequences (a_n = a_1 + (n - 1)d) with common difference d, and Geometric Sequences (a_n = a_1 · r^(n - 1)) with common ratio r. In translation, watch out for turnaround words: "7 less than y" is y - 7, and "is at least 15" translates to ≥ 15.
1. Algebraic Terminology & Structural Anatomy
Algebra uses symbolic notation to represent quantitative relationships, generalizations, and unknowns. To solve problems efficiently, you must understand its standard structural components.
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| ALGEBRAIC EXPRESSION ANATOMY |
| |
| 5x³ - 7x² + 2x - 19 |
| | | | | | | | |
| Coefficient --+ | | | | | +-- Constant Term (fixed value) |
| Variable --+ | | | +------- Linear Term (degree 1) |
| Quadratic -+ +------- Exponent / Power (degree 2) |
| Term |
| |
| +-------------------+----------------------------+------------------------------------------+ |
| | ELEMENT | FORMAL DEFINITION | EXAMPLE / IDENTIFICATION | |
| +-------------------+----------------------------+------------------------------------------+ |
| | Variable | Symbol (letter) for unknown| x, y, n, θ (represents varying values) | |
| +-------------------+----------------------------+------------------------------------------+ |
| | Constant | Fixed numerical value | -19, 4, 3/4, π (does not change) | |
| +-------------------+----------------------------+------------------------------------------+ |
| | Coefficient | Numerical multiplier | In -7x², the coefficient is -7 | |
| +-------------------+----------------------------+------------------------------------------+ |
| | Term | Single number, variable, | 5x³, -7x², 2x, and -19 are 4 distinct | |
| | | or product of both | terms separated by + or - | |
| +-------------------+----------------------------+------------------------------------------+ |
| | Like Terms | Terms with identical | 4ab² and -9ab² (can be combined); | |
| | | variable bases & exponents | 4a²b and 4ab² are UNLIKE (cannot combine)| |
| +-------------------+----------------------------+------------------------------------------+ |
| | Degree of Term | Sum of exponents on vars | 5x³ has degree 3; 6x²y³ has degree 2+3=5 | |
| +-------------------+----------------------------+------------------------------------------+ |
| | Degree of Poly | Highest degree of any term | 5x³ - 7x² + 2x - 19 has degree 3 (cubic) | |
| +-------------------+----------------------------+------------------------------------------+ |
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Types of Polynomials by Term Count
- Monomial: An algebraic expression containing exactly one term (e.g., -12x⁴, 7xy).
- Binomial: An algebraic expression containing exactly two terms connected by addition or subtraction (e.g., 3x² - 8, 5a + 2b).
- Trinomial: An algebraic expression containing exactly three terms (e.g., x² - 6x + 9).
- Polynomial: An algebraic expression composed of one or more terms with non-negative integer exponents.
2. Evaluating Algebraic Expressions & Order of Operations
Evaluating an expression means replacing every variable with its assigned numerical value and simplifying using the strict Order of Operations (PEMDAS / GEMS).
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| ORDER OF OPERATIONS HIERARCHY |
| |
| 1. G - Grouping Symbols Parentheses ( ), Brackets [ ], Braces { }, Fraction Bars, Radicals |
| 2. E - Exponents Powers, Roots, Absolute Values (|x|) |
| 3. M/D - Multiply/Divide Strictly LEFT-TO-RIGHT in order of appearance |
| 4. A/S - Add/Subtract Strictly LEFT-TO-RIGHT in order of appearance |
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The High-Yield Negative Base Pitfall
On the WEST-B exam, sign errors with exponents are among the most common mistakes:
- (-a)ⁿ (Parentheses Present): The negative sign is part of the base being multiplied. (-3)² = (-3) × (-3) = +9 (-2)³ = (-2) × (-2) × (-2) = -8 (-2)⁴ = (-2) × (-2) × (-2) × (-2) = +16
- -aⁿ (No Parentheses): The negative sign denotes multiplication by -1 after exponentiation. -3² = -(3 × 3) = -9 -2⁴ = -(2 × 2 × 2 × 2) = -16
Negative Base vs. Negative Coefficient Comparison
(-4)² = (-4) × (-4) = +16
-4² = -(4 × 4) = -16
(-4)³ = (-4) × (-4) × (-4) = -64
-4³ = -(4 × 4 × 4) = -64
Multi-Variable Rational Evaluation
When substituting negative values into rational expressions, evaluate the entire numerator and the entire denominator separately before dividing:
Example: Evaluate (3x² - 4xy + y²)/(2x - y) for x = -2, y = 3
- Numerator: 3(-2)² - 4(-2)(3) + (3)² = 3(4) - (-24) + 9 = 12 + 24 + 9 = 45
- Denominator: 2(-2) - 3 = -4 - 3 = -7
- Result: -45/7
3. Simplifying Algebraic Expressions & Polynomial Operations
Simplification produces an equivalent expression with the minimum number of terms and operations.
The Distributive Property
Multiplication distributes over addition and subtraction: a(b + c) = ab + ac a(b - c) = ab - ac
- Distributing a Negative Sign: Distributing a negative multiplier reverses every sign inside the parentheses: -(3x - 7y + 4) = -3x + 7y - 4 -5(2a - 4b + 1) = -10a + 20b - 5
Nested Grouping Symbols
When expressions contain parentheses ( ), brackets [ ], and braces { }, work from the innermost layer outward:
Simplify: 7 - 3[2x - 4(x - 3) + 5]
- Distribute the innermost -4: 2x - 4x + 12 + 5 = -2x + 17
- Substitute back into the brackets: 7 - 3[-2x + 17]
- Distribute the -3 across the brackets: 7 + 6x - 51
- Combine constants: 6x - 44
Multiplying Binomials (FOIL & Box Method)
To multiply two binomials (ax + b)(cx + d): (ax + b)(cx + d) = (ac·x²) [First] + (ad·x) [Outside] + (bc·x) [Inside] + (bd) [Last]
- Special Product 1 (Square of Binomial): (a + b)² = a² + 2ab + b² and (a - b)² = a² - 2ab + b²
- Special Product 2 (Difference of Squares): (a + b)(a - b) = a² - b²
4. Number & Geometric Patterns: Arithmetic & Geometric Sequences
A sequence is an ordered list of numbers following a systematic mathematical rule. On the WEST-B, questions test your ability to determine pattern rules, find missing terms, and calculate the n-th term.
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| SEQUENCE CLASSIFICATION & FORMULAS |
| |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | SEQUENCE TYPE | DEFINING CHARACTERISTIC | FORMULAS & PROPERTIES | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Arithmetic Sequence | Constant ADDITION/SUBTRACTION| Common difference: d = a_n - a_(n-1)| |
| | (Linear Growth) | between consecutive terms | Explicit: a_n = a_1 + (n - 1)d | |
| | | | Recursive: a_n = a_(n-1) + d | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Geometric Sequence | Constant MULTIPLICATION | Common ratio: r = a_n / a_(n-1) | |
| | (Exponential Growth) | by a fixed scalar ratio | Explicit: a_n = a_1 · r^(n - 1) | |
| | | | Recursive: a_n = a_(n-1) · r | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Quadratic Pattern | Second differences are | Explicit: a_n = An² + Bn + C | |
| | (Non-linear Growth) | constant (triangular numbers)| 2nd difference = 2A | |
| +-----------------------+-----------------------------+-------------------------------------+ |
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Arithmetic Sequence Step-by-Step Analysis
Consider the sequence: 4, 11, 18, 25, 32, ...
- Find the common difference d: 11 - 4 = 7, 18 - 11 = 7 => d = 7.
- Write the explicit formula: a_n = a_1 + (n - 1)d = 4 + (n - 1)7 = 4 + 7n - 7 = 7n - 3.
- Find the 40th term (a_40): a_40 = 7(40) - 3 = 280 - 3 = 277.
- Find which term equals 179: 7n - 3 = 179 => 7n = 182 => n = 26.
Geometric Sequence Step-by-Step Analysis
Consider the sequence: 6, -18, 54, -162, 486, ...
- Find the common ratio r: -18 / 6 = -3, 54 / (-18) = -3 => r = -3.
- Write the explicit formula: a_n = a_1 · r^(n - 1) = 6 · (-3)^(n - 1).
- Find the 7th term (a_7): a_7 = 6 · (-3)^(7 - 1) = 6 · (-3)⁶ = 6 · 729 = 4,374.
Geometric / Visual Tile Growth Patterns
Visual pattern questions present consecutive geometric stages (e.g., toothpicks, grid tiles, border stones):
- Stage 1: 4 squares
- Stage 2: 7 squares
- Stage 3: 10 squares
- Stage 4: 13 squares Since the increase per stage is constant (+3), it forms an arithmetic sequence with a_1 = 4 and d = 3. The rule for Stage n is a_n = 4 + (n - 1)3 = 3n + 1.
5. Translating Verbal Phrases into Expressions, Equations & Inequalities
Converting everyday English descriptions into algebraic symbolism is one of the most heavily tested skills on the WEST-B exam.
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| VERBAL TRANSLATION LEXICON |
| |
| +------------------+-----------------------------------------------+------------------------+ |
| | OPERATION | COMMON ENGLISH KEYWORDS / PHRASES | ALGEBRAIC TRANSLATION | |
| +------------------+-----------------------------------------------+------------------------+ |
| | Addition (+) | sum of, plus, increased by, more than, | Sum of x and 9 -> x + 9| |
| | | total of, exceeds by, combined | 6 more than k -> k + 6 | |
| +------------------+-----------------------------------------------+------------------------+ |
| | Subtraction (-) | difference, minus, decreased by, diminished by| Diff of x and 4 -> x - 4|
| | | **less than**, **subtracted from**, fewer than| 5 LESS THAN y -> y - 5 | |
| +------------------+-----------------------------------------------+------------------------+ |
| | Multiplication(·)| product of, times, twice (2x), triple (3x), | Product of 7 and w-> 7w| |
| | | of (fractions/percents: 30% of x -> 0.30x) | 3/4 of a number -> 3/4n| |
| +------------------+-----------------------------------------------+------------------------+ |
| | Division (÷) | quotient of, ratio of, divided by, per | Quotient of x and 8->x/8|
| +------------------+-----------------------------------------------+------------------------+ |
| | Equality (=) | is, equals, is equal to, results in, yields | 2x plus 3 IS 11 -> 2x+3=11|
| +------------------+-----------------------------------------------+------------------------+ |
| | At Least (≥) | is at least, minimum of, no less than, ≥ | x is at least 50 -> x≥50|
| +------------------+-----------------------------------------------+------------------------+ |
| | At Most (≤) | is at most, maximum of, no more than, ≤ | y is at most 20 -> y≤20| |
| +------------------+-----------------------------------------------+------------------------+ |
| | Greater (>) | strictly greater than, more than, exceeds | k exceeds 12 -> k > 12 | |
| +------------------+-----------------------------------------------+------------------------+ |
| | Less (<) | strictly less than, fewer than, below | p is under 10 -> p < 10| |
| +------------------+-----------------------------------------------+------------------------+ |
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The Critical "Turnaround Words" Trap
In standard English, some subtraction and comparison phrases invert the stated order of elements:
- "A decreased by B" => A - B (standard order)
- "A less B" => A - B (standard order)
- "A less than B" => B - A (Turnaround!)
- "A subtracted from B" => B - A (Turnaround!)
- "A fewer than B" => B - A (Turnaround!)
Example: "12 less than four times a number n" translates to: 4n - 12 (NEVER 12 - 4n).
Parenthetical Grouping Verbal Signals
- "The square of the sum of x and y" => (x + y)²
- "The sum of the squares of x and y" => x² + y²
- "Twice the difference of n and 5" => 2(n - 5)
- "The difference of twice n and 5" => 2n - 5
6. Step-by-Step Worked Problems & Exact Derivations
Problem 1: Complex Expression Evaluation with Nested Operations
Problem: Evaluate the expression (4a² - 3ab + 2b³)/(a - b) when a = -3 and b = -2.
Step-by-Step Solution:
- Substitute a = -3 and b = -2 with grouping parentheses into the numerator: Numerator = 4(-3)² - 3(-3)(-2) + 2(-2)³
- Calculate exponents first: (-3)² = 9 (-2)³ = -8 Numerator = 4(9) - 3(-3)(-2) + 2(-8)
- Perform multiplications left to right: 4(9) = 36 -3(-3)(-2) = 9(-2) = -18 => - (18) = -18 2(-8) = -16
- Combine numerator terms: Numerator = 36 - 18 - 16 = 18 - 16 = 2
- Evaluate denominator: Denominator = a - b = (-3) - (-2) = -3 + 2 = -1
- Divide numerator by denominator: Final Value = 2 / (-1) = -2
Problem 2: Sequence Analysis and Term Derivation
Problem: In an arithmetic sequence, the 4th term is 19 and the 11th term is 54. Determine the first term a_1, the common difference d, and calculate the value of the 35th term a_35.
Step-by-Step Solution:
- Set up a system using the arithmetic sequence formula a_n = a_1 + (n - 1)d:
- For n = 4: a_4 = a_1 + 3d = 19
- For n = 11: a_11 = a_1 + 10d = 54
- Subtract the first equation from the second to eliminate a_1: (a_1 + 10d) - (a_1 + 3d) = 54 - 19 7d = 35 => d = 5
- Substitute d = 5 into the first equation to find a_1: a_1 + 3(5) = 19 => a_1 + 15 = 19 => a_1 = 4
- Formulate the explicit rule: a_n = 4 + (n - 1)5 = 5n - 1
- Calculate the 35th term (a_35): a_35 = 5(35) - 1 = 175 - 1 = 174
Problem 3: Multi-Step Verbal Translation with Inequality Constraints
Problem: A community center is renting a venue that charges a flat reservation fee of $250 plus $18 per attendee. The catering service charges an additional $14 per attendee. If the total event budget cannot exceed $1,450, write an algebraic inequality to model this situation and determine the maximum number of attendees that can be hosted.
Step-by-Step Solution:
- Identify the variable: Let n represent the total number of attendees.
- Identify the fixed and variable costs:
- Flat reservation fee = 250
- Venue fee per attendee = 18n
- Catering cost per attendee = 14n
- Combined variable cost = 18n + 14n = 32n
- Translate 'cannot exceed': 'Cannot exceed' means ≤ (is less than or equal to).
- Formulate the inequality: 250 + 32n ≤ 1450
- Solve for n: 32n ≤ 1450 - 250 32n ≤ 1200 n ≤ 1200 / 32 = 37.5
- Interpret the result in context: Since attendees must be a whole integer, round down to the nearest whole number: Maximum Attendees = 37
What is the value of the algebraic expression 3x² - 2xy + y³ when evaluated at x = -2 and y = -3?
An arithmetic sequence begins with the terms 7, 13, 19, 25, ... What is the 25th term (a₂₅) of this sequence?
Which of the following algebraic statements correctly translates the verbal phrase: 'Five less than three times the square of the sum of a number k and 4 is at most 70'?
Which expression represents the complete simplification of 4[2x - 3(x - 5)] - 2(x + 8)?