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Algebraic Expressions
and Simplification

Master the art of manipulating algebraic expressions: combining like terms, applying the distributive property, and simplifying with confidence for the GRE Quantitative section.

GRE Quantitative

What Is an Algebraic Expression?

An algebraic expression is a combination of constants, variables, and operations (addition, subtraction, multiplication, division, exponentiation). Unlike equations, expressions have no equals sign.

Variable
A symbol (like \(x\) or \(y\)) representing an unknown value
Coefficient
The numerical factor before a variable: in \(5x\), the coefficient is \(5\)
Constant
A fixed number with no variable attached, such as \(7\) or \(-3\)

Anatomy of an Expression

\(3x^2 + 5x - 7\)
 Term 1 coefficient variable exponent Term 2 Term 3 constant The + and − signs separate the terms

Why Simplification Matters

On the GRE, you are constantly racing the clock. Simplification skills let you reduce complex expressions to manageable forms quickly, whether you are solving equations, comparing quantities, or evaluating functions.

Speed
Simplified expressions are faster to evaluate and less error-prone under time pressure
Clarity
Reveals the structure of a problem: factored forms expose roots, combined terms expose patterns

Core Rule: Combining Like Terms

Like terms have identical variable parts (same variables, same exponents). You combine them by adding or subtracting their coefficients.

\(7x^2 + 3x^2\)
\(10x^2\)
\(7x^2 + 3x\)
 ✕ Cannot combine

The Distributive Property

The distributive property lets you multiply a factor across a sum or difference inside parentheses:

\(3(2x + 5)\)
 distribute =
\(6x + 15\)

Expression vs Equation

Expression Equation
Equals sign? No Yes
Example \(3x + 5\) \(3x + 5 = 14\)
Goal Simplify Solve for \(x\)
Result Simpler expression A value: \(x = 3\)

Special Product Identities

Memorize these three identities: they appear constantly on the GRE and save enormous time.

1
Perfect Square (sum): \((a+b)^2 = a^2 + 2ab + b^2\)
2
Perfect Square (diff): \((a-b)^2 = a^2 - 2ab + b^2\)
3
Difference of Squares: \((a+b)(a-b) = a^2 - b^2\)

Step-by-Step Simplification

1
Remove parentheses: Distribute any coefficients or negative signs across grouped terms.
2
Identify like terms: Group terms with identical variable parts (same variables, same exponents).
3
Combine coefficients: Add or subtract the numerical coefficients of like terms.
4
Apply exponent rules: Simplify any products or quotients of powers using \(x^a \cdot x^b = x^{a+b}\) and \(\frac{x^a}{x^b} = x^{a-b}\).
5
Write in standard form: Order terms by decreasing degree, with constants last.

Example 1: Combine Like Terms

1
Group \(x^2\) terms: \(4x^2 + 2x^2\) \(= 6x^2\)
2
Group \(x\) terms: \(3x - 5x\) \(= -2x\)
3
Group constants: \(-7 + 10\) \(= 3\)

Example 2: Distribution Then Combine

1
Distribute \(2\): \(2 \cdot 3x - 2 \cdot 4 = \) \(6x - 8\)
2
Distribute \(-5\): \(-5 \cdot x + (-5) \cdot 2 = \) \(-5x - 10\)
3
Combine: \(6x - 5x = x\) and \(-8 - 10 = -18\)

Example 3: Special Products

1
Expand \((2x+3)^2\): \(4x^2 + 12x + 9\)
2
Expand \((2x-3)^2\): \(4x^2 - 12x + 9\)
3
Subtract: \((4x^2 + 12x + 9) - (4x^2 - 12x + 9)\)
4
Simplify: \(4x^2 - 4x^2 + 12x + 12x + 9 - 9 = 24x\)

Exponent Rules Refresher

Rule Formula Example
Product \(x^a \cdot x^b = x^{a+b}\) \(x^3 \cdot x^4 = x^7\)
Quotient \(\frac{x^a}{x^b} = x^{a-b}\) \(\frac{x^5}{x^2} = x^3\)
Power of a Power \((x^a)^b = x^{ab}\) \((x^2)^3 = x^6\)
Zero Exponent \(x^0 = 1\) \(5^0 = 1\)
Negative Exponent \(x^{-a} = \frac{1}{x^a}\) \(x^{-2} = \frac{1}{x^2}\)

Simplifying Rational Expressions

A rational expression is a fraction of polynomials. Simplify by factoring numerator and denominator, then cancelling common factors.

1
Factor numerator: \(x^2 - 9 = (x+3)(x-3)\)
2
Factor denominator: \(x^2 + 5x + 6 = (x+2)(x+3)\)
3
Cancel \((x+3)\): \(\dfrac{(x+3)(x-3)}{(x+2)(x+3)} = \dfrac{x-3}{x+2}\), where \(x \neq -3\)

Common Mistakes to Avoid

✕ Sign Errors
Forgetting to distribute the negative: \(-(x+3) = -x - 3\), not \(-x + 3\)
✕ Unlike Terms
Combining \(x^2\) with \(x\): they have different exponents and cannot be combined
✕ Squaring Sums
Writing \((a+b)^2 = a^2 + b^2\). The correct expansion is \(a^2 + 2ab + b^2\)

Summary of Key Techniques

Technique When to Use Key Formula
Combine Like Terms Same variable, same exponent \(ax^n + bx^n = (a+b)x^n\)
Distributive Property Factor outside parentheses \(a(b+c) = ab + ac\)
Difference of Squares \(A^2 - B^2\) pattern \(a^2 - b^2 = (a+b)(a-b)\)
Perfect Square Squaring a binomial \((a \pm b)^2 = a^2 \pm 2ab + b^2\)
Cancel Common Factors Rational expressions Factor, then cancel

Question 1 of 5

Simplify: \(5x^2 + 3x - 2x^2 + 7 - x\)

Question 2 of 5

Simplify: \(3(2x - 1) - 4(x + 5)\)

Question 3 of 5

Which expression is equivalent to \((x+4)(x-4)\)?

Question 4 of 5

Simplify: \(\dfrac{x^2 - 4}{x + 2}\) for \(x \neq -2\)

Question 5 of 5

What is \((3x + 2)^2\) expanded and simplified?