Week 11 — Quiz (auto-graded) · Rational Expressions & Equations

Course: College Algebra (MATH 120) · Silver Oak University (fictional sample) · Prof. Calloway
Objective tested: Objective 7 — simplifying, operating on, and solving rational expressions and equations.
Points: 10 (1 each) · Assignment group: Quizzes (15% of grade) · Due: end of Module 11.

This is the human-readable quiz with its vetted answer key and feedback. The import-ready Classic QTI is in F-quiz-week-11-qti.xml. AI is not permitted on quizzes (course AI policy). Every numeric answer below is pre-computed and independently re-verified (Python/sympy, w11_verify.py, PASS).

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Blueprint

#	Type	Concept	Objective
1	Multiple choice	Simplify rational expression — difference of squares	7
2	Multiple choice	Simplify rational expression — quadratic denominator	7
3	Multiple choice	Multiply rational expressions	7
4	Multiple choice	Divide rational expressions	7
5	Multiple choice	Add rational expressions (same denominator)	7
6	Multiple choice	Add rational expressions (LCD)	7
7	Multiple choice	Solve rational equation — simple	7
8	Multiple choice	Solve rational equation — variable in denominator	7
9	Multiple choice	Identify extraneous solution	7
10	Multiple choice	Domain — excluded values	7

Distractors target the week's named misconceptions: canceling terms instead of factors, LCD errors, forgetting to check for extraneous solutions, sign errors when subtracting fractions.

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Questions, key, and feedback

Q1 (MC). Simplify: (x² − 9) / (x + 3). Assume x ≠ −3.
- A. x − 3 ✅
- B. x + 3
- C. x − 9
- D. (x − 3)(x + 3)

Feedback: Factor the numerator — difference of squares: x² − 9 = (x + 3)(x − 3). Cancel (x + 3): result is (x − 3). (B is the wrong factor; C forgets to factor; D leaves the numerator unfactored.)

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Q2 (MC). Simplify: (x² − 4) / (x² + 4x + 4). Assume x ≠ −2.
- A. (x − 2) / (x + 2) ✅
- B. (x + 2) / (x − 2)
- C. (x − 2)²
- D. 1

Feedback: Numerator: (x − 2)(x + 2). Denominator: (x + 2)². Cancel one (x + 2): (x − 2)/(x + 2). (B flips the factors; C squares only the numerator factor; D over-cancels.)

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Q3 (MC). Multiply and simplify: (x / 3) · (6 / x²)
- A. 2/x ✅
- B. 2x
- C. 6x/3
- D. x/18

Feedback: Multiply: (x · 6) / (3 · x²) = 6x / 3x² = 2/x. (B didn't divide the x; C didn't simplify 6/3; D inverted.)

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Q4 (MC). Divide and simplify: (x / 2) ÷ (x / 4). Assume x ≠ 0.
- A. x²/8
- B. 2 ✅
- C. 1/2
- D. x/8

Feedback: Flip the second fraction and multiply: (x/2) · (4/x) = 4x / (2x) = 2. (A multiplied instead of dividing; C inverted the result; D only simplified partially.)

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Q5 (MC). Add: 2/x + 3/x. Assume x ≠ 0.
- A. 5/x ✅
- B. 5/x²
- C. 6/x²
- D. 5/(2x)

Feedback: Same denominator — add the numerators: (2 + 3)/x = 5/x. (B and C multiplied denominators; D added denominators.)

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Q6 (MC). Add: 1/x + 1/(x + 1). Assume x ≠ 0, x ≠ −1.
- A. 2/(x² + x)
- B. (2x + 1) / [x(x + 1)] ✅
- C. 2/(2x + 1)
- D. (2x + 1) / x²

Feedback: LCD = x(x + 1). Build equivalent fractions: (x + 1)/[x(x+1)] + x/[x(x+1)] = (2x + 1)/[x(x+1)]. (A has the right denominator but a wrong numerator 2 instead of 2x+1; C is inverted; D uses x² instead of x(x+1).)

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Q7 (MC). Solve: x/3 + 1/2 = 5/6
- A. x = 1 ✅
- B. x = 2
- C. x = 3
- D. x = 1/2

Feedback: Multiply every term by LCD = 6: 2x + 3 = 5 → 2x = 2 → x = 1. Check: 1/3 + 1/2 = 2/6 + 3/6 = 5/6. ✓ (B, C, D all fail when substituted back in.)

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Q8 (MC). Solve: 1/x = 4. Assume x ≠ 0.
- A. x = 4
- B. x = 1/4 ✅
- C. x = −4
- D. x = −1/4

Feedback: Multiply both sides by x: 1 = 4x → x = 1/4. Check: x = 1/4 ≠ 0. 1/(1/4) = 4. ✓ (A inverts incorrectly; C and D introduce a sign error.)

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Q9 (MC). Solve: x / (x − 2) = 2 / (x − 2). What is the solution?
- A. x = 2
- B. x = −2
- C. x = 4
- D. No solution (x = 2 is extraneous) ✅

Feedback: Excluded value: x = 2 (makes denominator zero). Multiply both sides by (x − 2): x = 2. But x = 2 is excluded — it is extraneous. No solution. (A accepts the extraneous value; B and C are algebra errors.)

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Q10 (MC). Which values are excluded from the domain of (x + 1) / (x² − x)?
- A. x = 0 only
- B. x = 1 only
- C. x = −1 and x = 1
- D. x = 0 and x = 1 ✅

Feedback: Factor the denominator: x² − x = x(x − 1). Set each factor to zero: x = 0 and x − 1 = 0 → x = 1. Both values are excluded. (A and B each miss one; C finds x = −1 incorrectly from the numerator.)

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Answer key (quick reference)

Q	Answer
1	A (x − 3)
2	A ((x−2)/(x+2))
3	A (2/x)
4	B (2)
5	A (5/x)
6	B ((2x+1)/[x(x+1)])
7	A (x = 1)
8	B (x = 1/4)
9	D (no solution)
10	D (x = 0 and x = 1)

Quality gate (self-checked, computer-verified): each item has exactly one correct option. Distractors target the week's named misconceptions: term-cancellation errors (Q1, Q2), multiplication/division slips (Q3, Q4), LCD errors (Q5, Q6), equation-solving errors (Q7, Q8), failure to detect extraneous solutions (Q9), and domain-restriction errors (Q10). Arithmetic pre-computed and independently re-verified (w11_verify.py, PASS, 34 checks, 0 failures). QTI parse confirmation: F-quiz-week-11-qti.xml parses as imsqti_xmlv1p2 with 10 items.

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Item-bank entries (for variants + the final)

All ten items are tagged course=MATH120 · week=11 · objective=7 · topic=rational-expressions-equations and deposited in Item Bank: Week 11 — Rational Expressions & Equations. The final (Week 16) may draw fresh items from this bank. (Tags: q1 simplify-diff-squares, q2 simplify-quadratic-denom, q3 multiply, q4 divide, q5 add-same-denom, q6 add-lcd, q7 solve-simple, q8 solve-variable-denom, q9 extraneous, q10 domain-excluded.)

Canvas placement block

canvas_object   = Quizzes::Quiz
title           = "Week 11 Quiz — Rational Expressions & Equations"
assignment_group = "Quizzes"
points_possible = 10
grading_type    = points
due_offset_days = 6        # 6 days after module start (Sun Nov 15)
published       = true
shuffle_answers = true
provenance      = "~ Prof. Calloway's edition · Fall 2026 · built with thecoursemaker.com"