Week 6 — Quiz (auto-graded) · Polynomials & Factoring

Course: College Algebra (MATH 120) · Silver Oak University (fictional sample) · Prof. Calloway
Objective tested: Objective 5 — polynomial operations, special products, factoring (GCF, trinomials, difference of squares, grouping).
Points: 10 (1 each) · Assignment group: Quizzes (15% of grade) · Due: end of Module 6.

This is the human-readable quiz with its vetted answer key and feedback. The import-ready Classic QTI is in F-quiz-week-06-qti.xml. AI is not permitted on quizzes (course AI policy). Every numeric answer below is pre-computed and independently re-verified (Python w06_verify.py, PASS — 43 checks, 0 failures).

Blueprint

No trick questions; distractors target the Week 6 misconceptions named in the lecture outline ((a+b)² missing the 2ab; sign errors in trinomial factoring; forgetting the GCF; believing a sum of squares factors over the reals).

Questions, key, and feedback

Q1 (MC). Expand using FOIL: (x+3)(x−5)
- A. x²−2x−15 ✅
- B. x²+2x−15
- C. x²−8x−15
- D. x²−2x+15
Feedback: FOIL: First x², Outer −5x, Inner +3x, Last −15. Collect: x²+(−5x+3x)−15 = x²−2x−15. (B adds instead of subtracts the x terms; C ignores the Inner; D flips the sign of the constant.)

Q2 (MC). Expand using the perfect-square formula: (2x+3)²
- A. 4x²+12x+9 ✅
- B. 4x²+9
- C. 4x²+6x+9
- D. 2x²+12x+9
Feedback: (a+b)² = a²+2ab+b² with a = 2x, b = 3: (2x)²+2(2x)(3)+3² = 4x²+12x+9. (B drops the middle term 2ab entirely — the classic error; C uses 2·3 = 6 instead of 2·(2x)·3 = 12x; D forgets to square the coefficient 2.)

Q3 (MC). Expand: (x+4)(x−4)
- A. x²−16 ✅
- B. x²+16
- C. x²−8x−16
- D. x²+8x−16
Feedback: This is a difference of squares: (a+b)(a−b) = a²−b² with a = x, b = 4 → x²−16. The middle terms (+4x and −4x) cancel perfectly. (B misses the minus; C and D include a middle term that cancels.)

Q4 (MC). Add the polynomials: (3x²−2x+1) + (x²+5x−4)
- A. 4x²+3x−3 ✅
- B. 4x²−7x+5
- C. 4x²+3x+5
- D. 2x²+3x−3
Feedback: Combine like terms: (3+1)x² = 4x², (−2+5)x = 3x, (1−4) = −3 → 4x²+3x−3. (B subtracts instead of adding the x-terms; C uses 1+4 instead of 1−4; D subtracts the x²-terms.)

Q5 (MC). Factor by pulling out the GCF: 6x³−9x²
- A. 3x²(2x−3) ✅
- B. 3x(2x²−3x)
- C. 3x²(2x+3)
- D. x²(6x−9)
Feedback: GCF of 6 and 9 is 3; GCF of x³ and x² is x² → 3x²(2x−3). Check: 3x²·2x = 6x³, 3x²·3 = 9x². (B uses x instead of x²; C has a sign error; D fails to take the 3 from the coefficients.)

Q6 (MC). Factor the trinomial: x²+7x+12
- A. (x+3)(x+4) ✅
- B. (x+2)(x+6)
- C. (x+1)(x+12)
- D. (x−3)(x−4)
Feedback: Need pq = 12, p+q = 7 → p = 3, q = 4 → (x+3)(x+4). FOIL check: x²+4x+3x+12 = x²+7x+12 ✓. (B: 2·6 = 12 but 2+6 = 8 ≠ 7; C: 1·12 = 12 but 1+12 = 13; D: (−3)(−4) = 12 but gives x²−7x+12.)

Q7 (MC). Factor the trinomial: x²−5x−14
- A. (x−7)(x+2) ✅
- B. (x+7)(x−2)
- C. (x−2)(x−7)
- D. (x+14)(x−1)
Feedback: Need pq = −14, p+q = −5 → p = −7, q = 2 → (x−7)(x+2). Check: x²+2x−7x−14 = x²−5x−14 ✓. (B: (x+7)(x−2) = x²+5x−14 — middle term sign flipped; C: (−2)(−7) = +14, not −14; D: sums don't work.)

Q8 (MC). Factor the difference of squares: 9x²−25
- A. (3x−5)(3x+5) ✅
- B. (3x−5)²
- C. (9x−5)(x+5)
- D. (3x+5)²
Feedback: a²−b² = (a+b)(a−b) with a = 3x, b = 5 → (3x+5)(3x−5). (B squares instead of using opposite signs; C splits 9x incorrectly; D is a perfect square, not a difference of squares.)

Q9 (MC). Factor by grouping: x³+2x²+3x+6
- A. (x+2)(x²+3) ✅
- B. (x+3)(x²+2)
- C. x(x²+2x+3)+6
- D. (x+6)(x²+1)
Feedback: Pair: (x³+2x²)+(3x+6) → x²(x+2)+3(x+2) → (x+2)(x²+3). (B swaps which factor gets which constant and expands to x³+3x²+2x+6 ≠ original; C is a half-step that is not fully factored; D expands to x³+6x²+x+6 ≠ original. Only A multiplies back to x³+2x²+3x+6.)

Q10 (True / False). The statement "x²+16 factors as (x+4)(x+4)" is true.
- True
- False ✅
Feedback: False. (x+4)(x+4) = x²+8x+16 ≠ x²+16. A sum of squares is prime over the real numbers — no real binomial factors exist. (The trap is confusing (x+4)² with x²+4², and forgetting that (a+b)² always includes the middle term 2ab = 8x here.)

Answer key (quick reference)

Quality gate (self-checked, computer-verified): each single-answer item has exactly one correct option; no multi-answer items this week; the True/False item keys False. Arithmetic pre-computed and independently re-verified (w06_verify.py, PASS — 43 checks, 0 failures): Q1 x²−2x−15; Q2 4x²+12x+9; Q3 x²−16; Q4 4x²+3x−3; Q5 3x²·(2x−3) = 6x³−9x²; Q6 (x+3)(x+4) = x²+7x+12; Q7 (x−7)(x+2) = x²−5x−14; Q8 (3x+5)(3x−5) = 9x²−25; Q9 (x+2)(x²+3) = x³+2x²+3x+6; Q10 (x+4)²= x²+8x+16 ≠ x²+16. QTI parse confirmation: F-quiz-week-06-qti.xml parses as imsqti_xmlv1p2 with 10 items.

Item-bank entries (for variants + the midterm/final)

All ten items are tagged course=MATH120 · week=6 · objective=5 · topic=polynomials-factoring and deposited in Item Bank: Week 6 — Polynomials & Factoring. The midterm (Week 8) and per-term variant updates draw fresh items from this bank. (Tags: q1 FOIL, q2 special-product-square, q3 diff-of-squares-expand, q4 poly-add, q5 GCF, q6 trinomial-lc1, q7 trinomial-lc1-negative-c, q8 diff-of-squares-factor, q9 grouping, q10 sum-of-squares-tf.)

Canvas placement block

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