Algebraic Expressions
Almost nothing in GRE Algebra asks you to "simplify this expression" for its own sake. Instead, expanding, factoring and combining like terms are the quiet machinery you run inside every equation, inequality and Quantitative Comparison. The test rewards people who can look at (x + 3)(x − 3) and write x² − 9 without pausing, or spot that x² − 6x + 9 is really (x − 3)² before touching the calculator. This chapter builds that reflex: collecting like terms cleanly, expanding with FOIL and distribution, recognising the three or four factoring patterns the GRE reuses endlessly, and handling algebraic fractions without the sign slips that cost easy points. The basic on-screen calculator will not factor anything for you, so this is the skill that stays purely mental.
Topics
⚡ GRE shortcuts & speed methods
The fastest ways to crack this chapter under time pressure — the techniques that separate a 95+ percentiler from the rest.
- Before you compute anything, simplify: a distributed, combined expression is far faster on the basic calculator than the raw one.
- Memorise the three special products cold. Seeing x² − 9 as (x + 3)(x − 3) or x² − 6x + 9 as (x − 3)² saves whole steps.
- When a minus sign leads a bracket, write the flipped signs immediately — treat −(…) as ×(−1) on every term.
- To factor x² + bx + c, list factor pairs of c and read off the pair that sums to b. It is faster than trial expansion.
- On Quantitative Comparison, factor both quantities before comparing — a shared factor often cancels and settles the question at a glance.
- For algebraic fractions, always factor first, then cancel factors (not terms). If nothing factors, you likely mis-copied a sign.
⚠️ Common mistakes & traps
GRE is designed so that careless errors here cost you marks. Internalise each trap before the exam.
- Combining unlike terms — adding 3x and 5x² as if the powers matched.
- Forgetting the middle term in a squared binomial: writing (x + 5)² as x² + 25 instead of x² + 10x + 25.
- Not distributing a leading minus across every term inside the bracket.
- Cancelling individual terms in a fraction instead of common factors — "cancelling" the x in (x + 2)/x.
- Losing a sign when factoring a trinomial with a negative constant (mixing up which factor is negative).
- Ignoring excluded values — treating a cancelled fraction as equal to the original even where the denominator was zero.
📈 GRE exam insight & question patterns
Quantitative Comparison — Quantity A: (x + 3)(x − 3). Quantity B: x² − 9. Which is greater, or is more information needed?
They are equal for every x (difference of squares), so the answer is "the two quantities are equal". Recognising the identity avoids testing numbers.
A "which of the following equals (a + b)²?" multiple-answer item lists a² + b², a² + 2ab + b², and (a + b)(a + b). Which apply?
a² + 2ab + b² and (a + b)(a + b) — both equal (a + b)². a² + b² is the classic trap; it omits the 2ab middle term.
Numeric Entry — if x² − 5x + 6 = (x − 2)(x − k), enter k.
Expanding (x − 2)(x − k) gives x² − (2 + k)x + 2k; matching the constant, 2k = 6, so k = 3.
A question hides division: it asks the value of (x² − 4)/(x − 2) when x = 100. Fastest route?
Factor and cancel first: (x + 2)(x − 2)/(x − 2) = x + 2 = 102. Never plug 100 into the unfactored form.
🎴 Flashcards — instant recall
Tap a card to reveal the answer. Drill these until they are automatic.
📌 Quick revision
Chapter test
🏆 Vidaara GRE success checklist
You have truly mastered Algebraic Expressions when you can tick every box below.
- Recall every formula in this chapter without looking them up
- Solve each topic’s practice set with at least 80% accuracy
- Use the chapter shortcuts to cut your solving time in half
- Spot and avoid every common trap listed above
- Score 80%+ on the timed chapter test
📋 Chapter mastery scorecard
Track where you stand. Aim for the target before moving to the next chapter.
| Skill checkpoint | Target |
|---|---|
| Concept theory & formulas understood | 100% |
| Topic practice sets attempted (4 topics) | 4/4 |
| Best topic-test score | — → 80%+ |
| Chapter test score | — → 80%+ |
| Flashcards drilled to instant recall | 8 cards |
Formula Reference Sheet
Expansion identities
| Distributive law | a(b + c) = ab + ac |
|---|---|
| FOIL | (a + b)(c + d) = ac + ad + bc + bd |
| Square of a sum | (a + b)² = a² + 2ab + b² |
| Square of a difference | (a − b)² = a² − 2ab + b² |
| Difference of squares | (a + b)(a − b) = a² − b² |
Factoring & fractions
| Common factor | ab + ac = a(b + c) |
|---|---|
| Trinomial x² + (p+q)x + pq | = (x + p)(x + q) |
| Multiply fractions | (a/b)·(c/d) = ac / bd |
| Divide fractions | (a/b) ÷ (c/d) = a d / b c |
| Add fractions | a/b + c/d = (ad + bc) / bd |