Simplify: (x² - 9) ÷ (x - 3)

The expression simplifies to x + 3 when the numerator x² - 9 is recognized as a difference of squares factoring into (x - 3)(x + 3), allowing cancellation of the common factor (x - 3) with the denominator, yielding x + 3 for all x ≠ 3 where the original expression is defined.

A. x - 3

This expression would result from incorrectly factoring x² - 9 as (x - 3)² = x² - 6x + 9 rather than the correct difference of squares (x - 3)(x + 3), then canceling one (x - 3) factor to leave x - 3. Students likely confused difference of squares (a² - b² = (a-b)(a+b)) with perfect square trinomial (a - b)² = a² - 2ab + b² a critical algebraic misconception where learners recognize factoring is required but misapply the specific pattern based on sign configuration. Another plausible error pathway involves executing polynomial long division correctly but misreading the quotient as x - 3 instead of x + 3 through sign omission during transcription. Students might compute x² ÷ x = x correctly but then execute -9 ÷ -3 = 3 correctly yet combine as x - 3 instead of x + 3 due to cognitive interference from the denominator's negative sign. The consistent sign error (minus instead of plus) reveals procedural rather than conceptual deficiency students understand cancellation mechanics but lack precision in sign management during factor combination, a transitional error requiring explicit emphasis on difference of squares structure with visual reinforcement showing symmetric factors (a-b) and (a+b) producing the minus sign in a² - b² through middle term cancellation.

B. x + 3

This expression correctly represents the simplified form through multiple verification pathways demonstrating algebraic manipulation mastery. Difference of squares factorization: x² - 9 = x² - 3² = (x - 3)(x + 3); cancellation: [(x - 3)(x + 3)] ÷ (x - 3) = x + 3 for x ≠ 3. Polynomial long division: dividing x² + 0x - 9 by x - 3 yields quotient x + 3 with remainder 0, confirming exact divisibility. Numerical verification: substitute x = 5 → (25 - 9) ÷ (5 - 3) = 16 ÷ 2 = 8; x + 3 = 8 ✓. Substitute x = 4 → (16 - 9) ÷ (4 - 3) = 7 ÷ 1 = 7; x + 3 = 7 ✓. Graphical verification: the rational function y = (x² - 9)/(x - 3) has a removable discontinuity at x = 3 but otherwise coincides exactly with the line y = x + 3. Domain awareness: explicitly noting x ≠ 3 preserves mathematical precision while recognizing simplification holds for all other real values. This solution demonstrates comprehensive understanding of rational expression simplification including pattern recognition (difference of squares), factor cancellation with domain restriction awareness, alternative methodology verification (long division), numerical substitution for concrete validation, and graphical interpretation reinforcing algebraic results integrated competencies essential for calculus (limit evaluation at removable discontinuities), engineering (transfer function simplification), physics (dimensional analysis with rational expressions), and computer algebra systems where accurate simplification enables efficient computation and pattern recognition across increasingly complex symbolic manipulations.

C. x² - 3

This expression appears disconnected from standard simplification pathways involving the given rational expression, lacking direct mathematical relationship through factoring or division. Potential error origins include: misreading the numerator as x² - 3x then canceling x to get x - 3 but mis-transcribing as x² - 3; executing (x² ÷ x) - (9 ÷ 3) = x - 3 then erroneously squaring the x term; or performing term-by-term division without recognizing polynomial structure (x² ÷ x = x, -9 ÷ -3 = 3) but then combining as x² - 3 through unstructured adjustment. Students might recognize x and 3 as relevant components but combine them incorrectly through exponent preservation error treating the x² term as maintaining its exponent after division rather than reducing to x¹. The value x² - 3 also equals the original numerator minus 6 suggesting possible arithmetic adjustment without algebraic justification. This distractor functions primarily as a random distractor without strong foundation in typical student error patterns for this question type, though it may trap learners employing unstructured term manipulation rather than systematic factoring protocols a deficiency requiring explicit emphasis on polynomial structure recognition before simplification attempts to prevent arbitrary term combinations that violate algebraic equivalence principles.

D. x² + 3

This expression equals the numerator plus 12 (x² - 9 + 12 = x² + 3), suggesting students likely added 12 to the numerator without denominator interaction or executed (x² ÷ 1) + (9 ÷ 3) = x² + 3 through invalid term separation. Another plausible pathway involves misreading the operation as multiplication instead of division: (x² - 9)(x - 3) expanded yields x³ - 3x² - 9x + 27, not x² + 3, but students might approximate this expansion downward to x² + 3 through unstructured reduction. Students might recognize 3 as related to the constant term 9 (since 9 = 3²) but then incorrectly add rather than incorporate through factoring. The preservation of the x² term reveals fundamental misunderstanding of polynomial division where learners fail to recognize that dividing a degree-2 polynomial by a degree-1 polynomial must yield a degree-1 quotient a critical degree-awareness deficiency where students lack the conceptual framework that deg(numerator) - deg(denominator) = deg(quotient) for polynomial division without remainder. This distractor effectively identifies learners who approach rational expressions through arithmetic intuition rather than algebraic structure analysis a foundational gap requiring explicit degree analysis protocols before simplification attempts to build awareness that division reduces polynomial degree predictably, preventing solutions that violate fundamental algebraic properties.

Conclusion

The simplified expression x + 3 emerges through rigorous application of difference of squares factorization recognizing x² - 9 as (x - 3)(x + 3), followed by cancellation of the common linear factor with explicit domain restriction acknowledgment (x ≠ 3). This problem reinforces critical algebraic manipulation competencies essential across mathematical domains: recognizing special factoring patterns (particularly difference of squares with its distinctive minus sign between perfect squares), executing factor cancellation while maintaining awareness of domain restrictions at removable discontinuities, verifying simplifications through multiple independent methods (factoring, long division, numerical substitution), and understanding degree relationships in polynomial division (quotient degree equals numerator degree minus denominator degree). Mastery of these integrated skills proves indispensable for calculus (evaluating limits at points of discontinuity), differential equations (simplifying rational differential expressions), control theory (transfer function reduction), and symbolic computation where accurate simplification enables efficient algorithm execution and pattern recognition. The distractors strategically target pervasive misconceptions including pattern confusion (difference of squares versus perfect square trinomials), degree preservation errors, and invalid term-by-term operations highlighting the necessity of explicit pattern recognition protocols with visual scaffolding contrasting different factoring structures, and mandatory degree verification before accepting any simplification to prevent solutions violating fundamental polynomial properties that compromise algebraic equivalence in increasingly complex rational expression manipulations.