4.1 Algebraic Expressions, Combining Like Terms & Distributive Property

Anatomy of Algebraic Expressions

Algebraic expressions serve as the universal language of higher mathematics. An algebraic expression is a mathematical phrase composed of numbers, variables, and arithmetic operations ($+, -, \times, \div$) without an equality sign ($=$) or inequality symbol ($<, >, \le, \ge$).

Core Structural Components

To analyze and manipulate expressions accurately on the HiSET Mathematics subtest, you must understand each component part:

• Variable: A letter or symbol (such as $x, y, t, n$) that represents an unknown or changeable quantity.
• Constant: A fixed numerical value that does not change (e.g., $7, -12, \frac{3}{4}, \pi$).
• Term: A single number, a single variable, or a product of numbers and variables separated from other terms by addition or subtraction signs. For example, the expression $4x^3 - 7xy + 9$ contains three distinct terms: $4x^3$, $-7xy$, and $+9$.
• Coefficient (Numerical Coefficient): The numerical factor multiplying the variable part of a term. In $-8x^2$, the coefficient is $-8$. When a variable appears without a written number (such as $x$ or $-y$), its implied coefficient is $+1$ or $-1$ ($x = 1x$, $-y = -1y$).
• Degree of a Term: The sum of the exponents of all variables contained within that specific term. For example, the term $5x^3$ has degree $3$, while $4x^2y^3$ has degree $2 + 3 = 5$. A non-zero constant term has degree $0$.
• Degree of an Expression (Polynomial): The highest degree of any individual non-zero term in the expression.

                    Anatomy of an Algebraic Expression
                    
                         Coefficient     Exponent / Power
                             │               │
                             ▼               ▼
                          - 5       x        3
                          ──┬──   ─────    ─────
                            │       ▲
                            │       │
                            │    Variable
                            │
              ┌─────────────┴───────────────────────────┐
              │   -5x³   +   7x²y   -   4x   +   12     │
              └─────┬──────────┬─────────┬────────┬─────┘
                    │          │         │        │
                 Term 1     Term 2    Term 3   Constant
                 (deg 3)    (deg 3)   (deg 1)  (deg 0)

Expressions vs. Equations: A Critical Distinction

A common error on standardized math tests is confusing an expression with an equation:

Expression Classification Reference Table

Identifying and Combining Like Terms

Simplifying an algebraic expression involves transforming it into its most compact, equivalent form. The primary algebraic mechanism for simplification is combining like terms.

Defining Like Terms

Like terms are terms that have the exact same variable base(s) raised to the exact same exponent(s). Only the numerical coefficients may differ.

$\text{Like Terms Law: } ax^n + bx^n = (a + b)x^n$

• Like Terms:
  • $7x$ and $-12x$ (both have variable $x^1$)
  • $4x^2y$ and $-9x^2y$ (both have $x^2y^1$)
  • $3ab^2$ and $8b^2a$ (by the commutative property of multiplication, $ab^2 = b^2a$)
  • $14$ and $-29$ (all pure constants are like terms of degree 0)
• Unlike Terms (Cannot Be Combined):
  • $5x^2$ and $5x^3$ (exponents differ: $2 \neq 3$)
  • $6x$ and $6y$ (variable bases differ: $x \neq y$)
  • $3x^2y$ and $3xy^2$ (exponents on corresponding variables differ)
  • $4x$ and $4$ ($4x$ has variable $x$; $4$ is a constant)

The Combining Algorithm

To combine like terms in a multi-term expression:

1. Rearrange and Group: Use the commutative and associative properties of addition to group like terms adjacent to one another. Always keep each term attached to the sign ($+$ or $-$) immediately preceding it.
2. Add/Subtract Coefficients: Apply signed arithmetic to the numerical coefficients while keeping the variable part completely unchanged.
3. Order in Standard Form: Write the simplified polynomial in descending order of powers (highest degree term first, followed by lower degree terms, ending with constants).

      Combining Like Terms Step-by-Step
      
      Original:    8x²  -  5x  +  3  -  2x²  +  9x  -  11
                   │▲      │▲     │      │▲     │▲      │
                   ││      ││     │      ││     ││      │
      Grouped:    (8x² - 2x²)  +  (-5x + 9x)  +  (3 - 11)
                      │                │             │
      Combined:      6x²       +       4x     -      8

The Distributive Property and Signed Multipliers

The Distributive Property of Multiplication over Addition and Subtraction states that multiplying a sum or difference by a factor is equivalent to multiplying each addend individually by that factor and summing the products:

$a(b + c) = ab + ac$ $a(b - c) = ab - ac$ $a(b + c - d + e) = ab + ac - ad + ae$

The Negative Multiplier and the Invisible Minus Sign

One of the most frequent sources of algebraic errors on the HiSET is distributing a negative factor or a leading negative sign. When a negative sign precedes a set of parentheses without a written number, there is an implied multiplier of $-1$:

$-(3x - 5) \equiv -1(3x - 5) = (-1)(3x) + (-1)(-5) = -3x + 5$ $-4(2x^2 - 3x + 7) = (-4)(2x^2) + (-4)(-3x) + (-4)(7) = -8x^2 + 12x - 28$

Universal Rule: Distributing a negative number inverts the sign of every single term inside the parentheses: positive terms become negative, and negative terms become positive.

Simplifying Expressions with Nested Grouping Symbols

When expressions contain nested grouping symbols (parentheses $()$, brackets $[]$, or braces ${}$), simplify systematically from the innermost group outward:

$\text{Simplify: } 7x - 3[2(x - 4) - (5x - 6)] + 8$

1. Innermost Parentheses (Distribute): $2(x - 4) = 2x - 8$ $-(5x - 6) = -5x + 6$ $\text{Expression becomes: } 7x - 3[(2x - 8) + (-5x + 6)] + 8$
2. Combine Like Terms Inside Brackets: $[2x - 8 - 5x + 6] = [(2x - 5x) + (-8 + 6)] = [-3x - 2]$ $\text{Expression becomes: } 7x - 3[-3x - 2] + 8$
3. Distribute Outer Multiplier ($-3$): $-3[-3x - 2] = +9x + 6$ $\text{Expression becomes: } 7x + 9x + 6 + 8$
4. Combine Final Like Terms: $(7x + 9x) + (6 + 8) = 16x + 14$

Evaluating Expressions for Given Variable Values

Evaluating an algebraic expression means replacing each variable with a specified numerical value (called substitution) and computing the single resulting arithmetic value following the strict order of operations (PEMDAS).

The Mandatory Parentheses Substitution Technique

When substituting a number (especially a negative number or fraction) into an algebraic expression, always wrap the replacement value in parentheses. This prevents catastrophic sign and exponentiation errors:

$\text{Evaluate } -x^2 + 3xy - y^2 \quad \text{when } x = -4 \text{ and } y = 2$

• Correct Substitution (with parentheses): $-(-4)^2 + 3(-4)(2) - (2)^2$ $- (16) + 3(-8) - 4 = -16 - 24 - 4 = -44$
• Incorrect Substitution (without parentheses): Writing $--4^2$ often leads students to erroneously write $+16 - 24 - 4 = -12$, which violates the base negation rules of PEMDAS.

Evaluating Rational and Radical Algebraic Expressions

Standardized tests often require evaluating algebraic fractions where both numerator and denominator contain multi-term expressions.

Worked Example: Rational Algebraic Evaluation

$\text{Evaluate } E = \frac{2a^2 - 3b + c}{ab - 2c^2} \quad \text{for } a = -3, \, b = 4, \, c = -2$

1. Substitute values with parentheses: $E = \frac{2(-3)^2 - 3(4) + (-2)}{(-3)(4) - 2(-2)^2}$
2. Simplify Numerator: $(-3)^2 = 9 \implies 2(9) = 18$ $3(4) = 12$ $\text{Numerator} = 18 - 12 + (-2) = 6 - 2 = 4$
3. Simplify Denominator: $(-3)(4) = -12$ $(-2)^2 = 4 \implies 2(4) = 8$ $\text{Denominator} = -12 - 8 = -20$
4. Divide and Simplify Fraction: $E = \frac{4}{-20} = -\frac{1}{5} = -0.2$