2.1 Algebraic Expressions & Operations

2.1 Algebraic Expressions & Operations

The CLEP College Algebra outline lists operations with algebraic expressions inside the Algebraic Operations content area, which carries 25% of the exam. This section handles the mechanics of that bullet: naming the parts of an expression, evaluating it at given values, and simplifying it correctly. The real number system these expressions operate over — its subsets, its field axioms, and the full order-of-operations hierarchy — is developed in Section 5.1; the arithmetic used here assumes it.

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Vocabulary: Terms, Coefficients, Factors, and Degree

An algebraic expression is any legal combination of numbers, variables, and operations. It has no equals sign — that would make it an equation.

• Term: a constant, a variable, or a product of constants and variables. Terms are separated by $+$ and $-$ signs at the top level. The expression $5x^3 - 7xy + 2y - 9$ has four terms.
• Coefficient: the numerical multiplier of a term. In $-7xy$ the coefficient is $-7$; the sign always travels with the coefficient.
• Factor: a quantity being multiplied. In $-7xy$ the factors are $-7$, $x$, and $y$. Terms are separated by addition; factors are separated by multiplication, and confusing the two is the root of most cancelling errors.
• Constant term: a term with no variable, such as $-9$.
• Degree of a term: the sum of the exponents on its variables. In $-7xy$ each variable has an implied exponent of 1, so the degree is $1 + 1 = 2$. In $4x^2y^3$ the degree is $5$.
• Degree of an expression: the largest degree among its terms.

Like Terms

Like terms contain the exact same variables raised to the exact same powers. Only like terms may be added or subtracted.

• $6x^2y$ and $-11x^2y$ are like terms; they combine to $-5x^2y$.
• $6x^2y$ and $6xy^2$ are not like terms, because the exponents are attached to different variables.
• $6x^2$ and $6x^3$ are not like terms, even though the variable matches.

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Evaluating Expressions by Substitution

To evaluate an expression, replace each variable with its given value in parentheses, then simplify. The parentheses are not optional decoration: they preserve the sign of a negative input and keep the original grouping intact.

Worked Example 1: Evaluation With Negative Inputs

Problem: Evaluate $\frac{3a^2 - b^3}{2a + b}$ for $a = -2$ and $b = -4$.

1. Substitute with parentheses: $\frac{3(-2)^2 - (-4)^3}{2(-2) + (-4)}$
2. Evaluate the powers first. Note the difference the parentheses make: $(-2)^2 = 4$ because the base is $-2$, whereas $-2^2$ would be $-4$. $(-2)^2 = 4 \quad \text{and} \quad (-4)^3 = -64$
3. Simplify the numerator: $3(4) - (-64) = 12 + 64 = 76$.
4. Simplify the denominator: $2(-2) + (-4) = -4 - 4 = -8$.
5. Divide: $\frac{76}{-8} = -\frac{19}{2} = -9.5$

Exam Trap: Substituting a negative value without parentheses turns $3a^2$ into $3 \cdot -2^2 = -12$ instead of $12$. Sign errors from missing parentheses are the single most common arithmetic mistake on this content area.

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Simplifying: Distribute, Then Combine

The distributive property $a(b + c) = ab + ac$ is what clears grouping symbols. Work from the innermost grouping outward, then combine like terms, then write the result in descending order of degree.

A leading minus sign in front of a group distributes as $-1$ across every term inside: $-(3x - 5) = -3x + 5$. Dropping the sign on the second term is a routine source of wrong answers.

Worked Example 2: Multi-Variable Simplification

Problem: Fully expand and simplify: $4(3x^2 - 5xy + 2) - 3(2x^2 + 4xy - 9) - 2xy(3 - 4x)$

1. Distribute each multiplier across its parentheses:
   • $4(3x^2) + 4(-5xy) + 4(2) = 12x^2 - 20xy + 8$
   • $-3(2x^2) - 3(4xy) - 3(-9) = -6x^2 - 12xy + 27$
   • $-2xy(3) - 2xy(-4x) = -6xy + 8x^2y$
2. Write everything in one line: $12x^2 - 20xy + 8 - 6x^2 - 12xy + 27 - 6xy + 8x^2y$
3. Group like terms:
   • $x^2y$: $8x^2y$
   • $x^2$: $12x^2 - 6x^2 = 6x^2$
   • $xy$: $-20xy - 12xy - 6xy = -38xy$
   • constants: $8 + 27 = 35$
4. Write in descending degree order: $8x^2y + 6x^2 - 38xy + 35$

Worked Example 3: Nested Grouping Symbols

Problem: Simplify $7x - \left[ 4 - 2\left( 3x - 5 \right) \right]$.

1. Innermost group first: $-2(3x - 5) = -6x + 10$, so the bracket becomes $\left[ 4 - 6x + 10 \right] = \left[ 14 - 6x \right]$.
2. Distribute the leading minus across the bracket: $-\left[14 - 6x\right] = -14 + 6x$.
3. Combine like terms: $7x - 14 + 6x = 13x - 14$

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Translating Verbal Phrases into Algebraic Expressions

CLEP word problems test your ability to convert written descriptions into algebra. Accurate translation starts with recognising the operational keywords.

Critical Order Swap Alert: Pay close attention to "less than" and "subtracted from".

• "$A$ less than $B$" means $B - A$, NOT $A - B$.
• "$A$ subtracted from $B$" means $B - A$, NOT $A - B$.

Also watch the word "quantity": it signals that a whole sum or difference is being operated on, which means parentheses. "Twice the quantity $x$ plus 3" is $2(x + 3)$, while "twice $x$, plus 3" is $2x + 3$.

Worked Example 4: Multi-Step Verbal Translation

Problem: Translate and simplify: "Eight times the difference of a number $x$ and 3, subtracted from half the cube of $x$."

1. "difference of $x$ and 3": $(x - 3)$.
2. "eight times the difference": $8(x - 3)$.
3. "half the cube of $x$": $\frac{1}{2}x^3$.
4. "subtracted from" reverses the order: $\frac{1}{2}x^3 - 8(x - 3)$.
5. Expand and simplify: $\frac{1}{2}x^3 - 8x + 24$.