Week 8 · Study guide

Midterm Study Guide · Weeks 1–7 (Objectives 1–6)

College Algebra · MATH 120 Fall 2026 · Prof. Calloway Fictional sample

Course: College Algebra (MATH 120) · Silver Oak University (fictional sample) · Prof. Calloway
This is a student-facing review page. Read it, work the fresh practice, and follow the dated plan. Then run the paired Exam-Prep Tutorial and take the Practice Exam for active recall. (This guide points to those two — it does not repeat them.)

Integrity note for students. Every practice item on this page is a fresh variant — new numbers and contexts — with a pre-computed, vetted answer. None of these are the live midterm questions. Working them builds the skill the midterm tests, the honest way. No AI on the midterm itself — but use anything you like to study.

What the midterm covers (read this first)


Exam	Midterm — cumulative, Weeks 1–7, Objectives 1–6
Format	20 items, 100 points (5 each). Application-skewed: most items ask you to do something — simplify, solve, evaluate, factor, find a slope — not just recite a rule. Auto-gradable types only (multiple-choice, multiple-answer, matching).
Coverage (where the points are)	Obj 1 ≈ 3 (real numbers & exponents) · Obj 2 ≈ 3 (linear equations & inequalities) · Obj 3 ≈ 3 (functions) · Obj 4 ≈ 4 (lines & systems — the biggest slice) · Obj 5 ≈ 3 (polynomials & factoring) · Obj 6 ≈ 4 (quadratics — tied for the biggest). Study Objectives 4 and 6 hardest.
Weight	The midterm is 20% of your course grade.
When it opens / where	Opens in the Week 8 module (the review-and-exam week). The exam window and the room/timing are posted with the exam itself in Canvas; this guide and the exam-prep tutorial post before it so you can prepare. There is no Quiz 8 or Assignment 8 — the midterm replaces them.
What to bring	A calculator and the one-page formula list you build from this guide. No AI / no chatbot on the exam (course AI policy). Desmos is for practice at home; the exam is do-it-by-hand.

How to use this guide. Each objective below has the same four parts: (A) the key ideas in plain language, (B) the definitions / procedures, (C) the predictable mistakes and their cures, and (D) where to review in the module. After all six objectives come fresh worked examples + self-check questions (with answers), a dated study plan, and how it's graded + test strategy.

Objective 1 — Real Numbers, Exponents & Expressions (Week 1) · ~3 items

(A) Key ideas, plain language

Algebra starts with doing arithmetic honestly — in the right order, with the right signs — and rewriting expressions without changing their value. The two things that quietly cost the most points: the placement of a negative sign around an exponent, and distributing a negative across parentheses.

(B) Definitions, procedures

Order of operations (PEMDAS): Parentheses, Exponents, Mult/Div (left→right), Add/Sub (left→right).
A negative in front of a power: −2³ means −(2³) = −8. Only (−2)³ cubes the negative. The exponent attaches to whatever is directly under it; a bare minus sign is multiplication by −1 applied last.
Even vs. odd powers of a negative: (−3)² = +9 (even → positive); (−2)³ = −8 (odd → negative).
Exponent rules: product xᵃ·xᵇ = xᵃ⁺ᵇ; power of a power (xᵃ)ᵇ = xᵃᵇ; power of a product (xy)ⁿ = xⁿyⁿ — every factor, including the coefficient, gets the exponent. So (2x²y)³ = 2³x⁶y³ = 8x⁶y³.
Distributing: a(b + c) = ab + ac. When the multiplier is negative, the sign flips on every term: −2(x − 5) = −2x + 10.

(C) Predictable mistakes → cures

❌ "−2³ = (−2)³, and both could be anything." → ✅ Without parentheses the minus is outside: −2³ = −8. Decide what the exponent is attached to first.
❌ "(2x²y)³ = 2x⁶y³." → ✅ Cube the coefficient too: 2³ = 8. Power of a product hits every factor.
❌ "−2(x − 5) = −2x − 10." → ✅ Two negatives make a positive: −2 · −5 = +10. Write the +10 deliberately.
❌ Adds exponents when they should multiply (or vice-versa). → ✅ (x²)³ multiplies (x⁶); x²·x³ adds (x⁵). Name the rule before you move.

(D) Review in the module

Week 1 → Lecture Outline, Slides (Deck 1), Readings (order of operations, exponent rules, simplifying), and Lecture Tutorial 1.

Objective 2 — Linear Equations & Inequalities (Week 2) · ~3 items

(A) Key ideas, plain language

Solving means isolating x by undoing operations in reverse. Two wrinkles change the rules: inequalities flip when you multiply/divide by a negative, and absolute-value equations always split into two cases.

(B) Definitions, procedures

Linear equation: collect variable terms on one side, constants on the other, divide out the coefficient. Always check by substituting back.
Inequalities: solve like an equation, but flip the inequality sign whenever you multiply or divide both sides by a negative number. Interval notation: x ≥ −3 is [−3, ∞); x > −2 is (−2, ∞).
Absolute-value equation |expr| = k (k > 0): split into expr = k OR expr = −k, then solve each. (If k < 0 there is no solution — absolute value is never negative.)

(C) Predictable mistakes → cures

❌ "−3x ≤ 9, so x ≤ −3." → ✅ Dividing by −3 flips the sign: x ≥ −3. The flip is mandatory on a negative divide.
❌ "|2x − 3| = 7, so x = 5." → ✅ That's only one case. Also solve 2x − 3 = −7 → x = −2. Two cases, almost always two answers.
❌ Forgets to check, keeps an extraneous value. → ✅ Substitute every answer back; absolute-value and (later) radical/rational equations can produce values that don't actually work.

(D) Review in the module

Week 2 → Lecture Outline, Slides (Deck 2), Lecture Tutorial 2 (solving equations, inequalities & interval notation, absolute-value equations).

Objective 3 — Functions: Notation, Domain & Operations (Week 3) · ~3 items

(A) Key ideas, plain language

A function is a rule: put a number in, get exactly one number out. f(x) is "the output when the input is x." Three moves: evaluate (substitute), find the domain (which inputs are legal), and compose (feed one function's output into another).

(B) Definitions, procedures

Evaluate f(a): replace every x with a — in parentheses — and simplify, watching signs. f(x) = x² − 2x, f(−3) = (−3)² − 2(−3) = 9 + 6 = 15.
Domain (the two usual restrictions): you can't divide by zero (so set any denominator ≠ 0) and you can't take an even root of a negative (so set the inside of a square root ≥ 0). Domain of √(x − 4): x − 4 ≥ 0 → x ≥ 4, i.e. [4, ∞).
Composition (f ∘ g)(x) = f(g(x)): work inside-out — do g first, then f. (f ∘ g)(2) with f(x) = 2x + 1, g(x) = x²: g(2) = 4, then f(4) = 9. Order matters: (g ∘ f) ≠ (f ∘ g) in general.

(C) Predictable mistakes → cures

❌ Sign slip on evaluation: f(−3) = 9 − 6 = 3. → ✅ −2(−3) = +6, not −6. Put the input in parentheses to protect the signs.
❌ "√ domain is x > 4." → ✅ Use ≥: √0 = 0 is defined, so x = 4 is allowed → [4, ∞).
❌ Composes in the wrong order. → ✅ (f ∘ g) means g first. Read it right-to-left.

(D) Review in the module

Week 3 → Lecture Outline, Slides (Deck 3), Lecture Tutorial 3 (function notation, domain & range, composition).

Objective 4 — Linear Functions, Graphs & Systems (Weeks 4–5) · ~4 items — STUDY HARD

(A) Key ideas, plain language

Lines are the workhorse of algebra: a slope (steepness) and a point pin a line down completely. Systems ask where two lines meet. This is the biggest single slice of the midterm — budget time here.

(B) Definitions, procedures

Slope = rise/run = (y₂ − y₁)/(x₂ − x₁). Through (−1, 2) and (3, 10): (10 − 2)/(3 − (−1)) = 8/4 = 2.
Forms of a line: slope-intercept y = mx + b; point-slope y − y₁ = m(x − x₁). Slope −2 through (1, 5): y − 5 = −2(x − 1) → y = −2x + 7.
Parallel vs. perpendicular: parallel lines share the same slope; perpendicular slopes are negative reciprocals (flip and change the sign). Perpendicular to slope 1/4 is −4.
Systems (elimination): add or subtract the equations to cancel a variable. x + y = 7 and x − y = 1 → add → 2x = 8 → x = 4, then y = 3 → (4, 3). (Substitution works too.) A system can have one solution (lines cross), none (parallel), or infinitely many (same line).

(C) Predictable mistakes → cures

❌ Inverts slope: (x₂ − x₁)/(y₂ − y₁). → ✅ It's rise over run — y's on top. 8/4 = 2, not 4/8.
❌ "Slope −2 through (1, 5), so y = −2x + 5." → ✅ 5 is the point's y, not the intercept. Use point-slope and simplify → y = −2x + 7.
❌ "Perpendicular slope is just the reciprocal, 4." → ✅ Negative reciprocal — flip and change the sign → −4.
❌ Sign error when eliminating. → ✅ Line up like terms; adding x − y to x + y cancels y because +y and −y sum to 0.

(D) Review in the module

Week 4 → Lecture Outline, Slides (Deck 4), Lecture Tutorial 4 (slope, line forms, parallel/perpendicular). Week 5 → Lecture Outline, Slides (Deck 5), Lecture Tutorial 5 (systems by substitution & elimination).

Objective 5 — Polynomials & Factoring (Week 6) · ~3 items

(A) Key ideas, plain language

Multiplying polynomials is bookkeeping (every term times every term). Factoring is that run in reverse — and it's the engine that solves quadratics next. The points-loser here is the middle term of a squared binomial.

(B) Definitions, procedures

Multiply binomials (FOIL): (x − 5)(x + 2) = x² + 2x − 5x − 10 = x² − 3x − 10. Combine the middle terms; watch their signs.
Special products: (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b² (don't forget 2ab); (a + b)(a − b) = a² − b² (difference of squares). (2x + 3)² = 4x² + 12x + 9.
Factoring toolkit: pull out the GCF first (x² − 4x = x(x − 4)); difference of squares x² − 9 = (x − 3)(x + 3); perfect-square trinomial x² + 6x + 9 = (x + 3)²; general trinomial x² − x − 6 → two numbers multiplying to −6, adding to −1 → −3 and +2 → (x − 3)(x + 2).

(C) Predictable mistakes → cures

❌ "(2x + 3)² = 4x² + 9." → ✅ The middle term 2ab = 2·2x·3 = 12x is required → 4x² + 12x + 9.
❌ Wrong middle sign in FOIL. → ✅ Track outer + inner: (x − 5)(x + 2) → +2x − 5x = −3x.
❌ Forgets the GCF, or stops factoring too early. → ✅ Always check for a common factor first; then see if what's left factors further.

(D) Review in the module

Week 6 → Lecture Outline, Slides (Deck 6), Lecture Tutorial 6 (polynomial operations, special products, factoring methods).

Objective 6 — Quadratic Equations (Week 7) · ~4 items — STUDY HARD

(A) Key ideas, plain language

A quadratic can have two, one, or no real solutions. You have four methods to find them — and one number, the discriminant, that tells you how many to expect before you start. Tied with Objective 4 for the largest exam slice.

(B) Definitions, procedures

Standard form: ax² + bx + c = 0 (zero on one side first).
Factor + zero-product: if A·B = 0 then A = 0 or B = 0. x² + 2x − 8 = 0 → (x + 4)(x − 2) = 0 → x = −4 or x = 2.
Square root property: if u² = k then u = ±√k (the ± is mandatory).
Completing the square and the quadratic formula x = (−b ± √(b² − 4ac)) / (2a) always work; mind the −b (if b = −4, then −b = +4).
Discriminant b² − 4ac: positive → two real solutions; zero → one repeated solution; negative → no real solutions. For 2x² + 3x + 5: 9 − 40 = −31 → no real solutions.
Perfect-square quadratic: x² − 6x + 9 = (x − 3)² = 0 → x = 3 (repeated; discriminant 0).

(C) Predictable mistakes → cures

❌ Solves x² = 5x by dividing by x. → ✅ You lose x = 0. Move everything over and factor: x(x − 5) = 0 → x = 0 or 5.
❌ Drops the ± in the square root property. → ✅ Both signs are solutions — plus-or-minus is the whole point.
❌ Uses +b instead of −b in the formula. → ✅ Write −b = ___ on its own line before substituting.
❌ "Negative discriminant = I made an error." → ✅ No — it's a valid conclusion: no real solutions.

(D) Review in the module

Week 7 → Lecture Outline, Slides (Deck 7), Lecture Tutorial 7 (four solving methods + the discriminant).

Representative practice (all fresh — vetted answers)

None of these are live midterm items. New numbers, new contexts. Each answer is pre-computed; the one-line why names the idea it tests. Cover the answers, work each one, then check.

Objective 1 practice

Worked example 1 — order of operations & exponent rules.
- (a) Evaluate −4² + (−2)⁴. (b) Simplify (3a²b)².
Answer. (a) −4² = −(16) = −16; (−2)⁴ = +16 (even power). −16 + 16 = 0. (b) 3² = 9, (a²)² = a⁴, b² = b² → 9a⁴b². Why: the minus sits outside −4²; even powers of a negative are positive; power of a product hits every factor including the 3.

Worked example 2 — distribute and combine.
Simplify 5(2x − 1) − 3(x − 4).
Answer. 10x − 5 − 3x + 12 = 7x + 7. Why: −3 · −4 = +12; combine like terms.

Self-check (Obj 1).
1. −5² = ? → −25 (minus outside). 2. (−5)² = ? → 25. 3. (2x³)⁴ = ? → 16x¹². 4. −2(3x − 7) = ? → −6x + 14.

Objective 2 practice

Worked example 1 — linear equation.
Solve 3(x − 2) = x + 8.
Answer. 3x − 6 = x + 8 → 2x = 14 → x = 7. Check: 3(7 − 2) = 15 = 7 + 8 ✓. Why: collect, isolate, verify.

Worked example 2 — inequality (flip) and absolute value.
- (a) Solve −4x + 1 > 9. (b) Solve |x + 2| = 6.
Answer. (a) −4x > 8 → divide by −4 and flip: x < −2, i.e. (−∞, −2). (b) x + 2 = 6 → x = 4; or x + 2 = −6 → x = −8 → x = 4 or x = −8. Why: negative divide flips the inequality; absolute value splits into two cases.

Self-check (Obj 2).
1. Solve 2x + 5 = 3 → x = −1. 2. Solve −2x ≥ 10 → x ≤ −5 (flip). 3. |3x| = 12 → x = 4 or −4. 4. Write x > 7 in interval notation → (7, ∞).

Objective 3 practice

Worked example 1 — evaluate & domain.
- (a) f(x) = x² − 4x, find f(−2). (b) Domain of √(x + 5)?
Answer. (a) (−2)² − 4(−2) = 4 + 8 = 12. (b) x + 5 ≥ 0 → x ≥ −5, i.e. [−5, ∞). Why: substitute in parentheses; the radicand must be ≥ 0.

Worked example 2 — composition.
f(x) = x + 3, g(x) = 2x. Find (a) (f ∘ g)(4) and (b) (g ∘ f)(4).
Answer. (a) g(4) = 8, f(8) = 11. (b) f(4) = 7, g(7) = 14. Why: do the inside function first; order matters — the two answers differ.

Self-check (Obj 3).
1. f(x) = 5 − x², f(3) = ? → −4. 2. Domain of √(2x − 8)? → x ≥ 4. 3. With f(x)=x², g(x)=x−1, (f ∘ g)(3) = ? → f(2) = 4. 4. Is the domain of √x all reals? → No — x ≥ 0.

Objective 4 practice — largest slice; work all of these

Worked example 1 — slope & line.
Through (−2, 1) and (2, 9): (a) slope, (b) equation in slope-intercept form.
Answer. (a) (9 − 1)/(2 − (−2)) = 8/4 = 2. (b) Use (2, 9): y − 9 = 2(x − 2) → y = 2x − 4 + 9 = 2x + 5. Check (−2, 1): 2(−2) + 5 = 1 ✓. Why: rise over run; point-slope then simplify.

Worked example 2 — parallel/perpendicular.
A line has slope −3/5. Give (a) a parallel slope, (b) a perpendicular slope.
Answer. (a) −3/5 (same). (b) negative reciprocal → flip to 5/3, change sign → +5/3. Why: parallel = same slope; perpendicular = negative reciprocal.

Worked example 3 — system by elimination.
Solve 3x + y = 10 and x − y = 2.
Answer. Add: 4x = 12 → x = 3; then 3 − y = 2 → y = 1 → (3, 1). Check: 3(3) + 1 = 10 ✓, 3 − 1 = 2 ✓. Why: adding cancels y; back-substitute.

Self-check (Obj 4).
1. Slope through (0, 0) and (4, 6)? → 3/2. 2. Perpendicular to slope 2? → −1/2. 3. Lines y = 3x + 1 and y = 3x − 4 — do they intersect? → No (parallel, same slope). 4. Solve x + y = 5, x − y = 3 → (4, 1).

Objective 5 practice

Worked example 1 — multiply.
- (a) (x + 6)(x − 2). (b) (x − 4)².
Answer. (a) x² − 2x + 6x − 12 = x² + 4x − 12. (b) x² − 2(4)x + 16 = x² − 8x + 16. Why: FOIL and combine; the square needs the −8x middle term.

Worked example 2 — factor.
Factor (a) x² − 16, (b) x² + 7x + 12, (c) 5x² − 20x.
Answer. (a) difference of squares → (x − 4)(x + 4). (b) 3 and 4 → (x + 3)(x + 4). (c) GCF 5x → 5x(x − 4). Why: recognize the pattern, pull the GCF first.

Self-check (Obj 5).
1. (2x + 1)² = ? → 4x² + 4x + 1. 2. Factor x² − 36 → (x − 6)(x + 6). 3. Factor x² − 2x − 8 → (x − 4)(x + 2). 4. (x − 3)(x + 3) = ? → x² − 9.

Objective 6 practice — tied for largest; work all of these

Worked example 1 — factor & solve.
Solve x² − x − 12 = 0.
Answer. Two numbers multiplying to −12, adding to −1: −4 and +3 → (x − 4)(x + 3) = 0 → x = 4 or x = −3. Why: zero-product property after factoring.

Worked example 2 — repeated root & square root property.
- (a) Solve x² + 8x + 16 = 0. (b) Solve (x − 1)² = 9.
Answer. (a) (x + 4)² = 0 → x = −4 (repeated; discriminant 64 − 64 = 0). (b) x − 1 = ±3 → x = 4 or x = −2 → x = 4 or x = −2. Why: perfect square → one root; square root property → keep the ±.

Worked example 3 — discriminant.
For 3x² + 2x + 4 = 0, compute the discriminant and say how many real solutions.
Answer. b² − 4ac = 4 − 48 = −44 < 0 → no real solutions. Why: a negative discriminant means the parabola never crosses the x-axis.

Self-check (Obj 6).
1. Solve x² − 9 = 0 → x = 3 or −3. 2. Solve x² + 6x + 9 = 0 → x = −3 (repeated). 3. Discriminant of x² + x + 1 = 0? → −3 → no real solutions. 4. Solve x² = 7x (don't divide by x!) → x = 0 or 7.

Study plan — a dated countdown (sized to 2 sessions/week)

Built for the Week 8 midterm (module starts Mon Oct 19; exam due Sun Oct 25). Adjust the exact dates to your section's posted exam day; the rhythm is what matters. Do a little every day rather than one long cram.

When	Do this (≈45–75 min)
~6 days out (Mon Oct 19)	Read this guide's Objectives 1–3. Work the Obj 1 & 2 practice. Build your one-page formula sheet (order-of-ops/exponent rules, inequality-flip, absolute-value cases, slope & line forms).
~5 days out (Tue Oct 20, review session)	Come to the review session. Then read Objective 4 carefully (it's ~4 of 20 items) and work all of its practice — slope, line equations, perpendicular slopes, systems.
~3 days out (Thu Oct 22, review session)	Read Objectives 5–6 and work their practice — special products, factoring, quadratics, the discriminant. Then run the paired Exam-Prep Tutorial (`N-exam-prep-tutorial-week-08`) in an approved chatbot — it diagnoses your weak spots across all six objectives and drills them.
~2 days out (Fri Oct 23)	Take the Practice Exam (`O-practice-exam-week-08`) under timed, closed-note, no-AI conditions (like the real thing). Score it; list every missed idea.
~1 day out (Sat Oct 24)	Re-teach only the topics you missed on the practice exam (use this guide's mistake-cures and the relevant Lecture Tutorial). Re-do those specific self-checks. Sleep.
Exam day (by Sun Oct 25)	Skim your one-page formula sheet. Read each item twice. Sit the Midterm (no AI), then post Discussion 8 (the debrief) while it's fresh.

Two paired tools — use both (don't skip):
- Exam-Prep Tutorial (N-exam-prep-tutorial-week-08) — a copy/paste chatbot tutor that diagnoses, re-teaches, and drills you across all of Objectives 1–6, ending with a readiness summary. Best for active recall and shoring up weak spots.
- Practice Exam (O-practice-exam-week-08) — a full, fresh, timed run that mirrors the real format. Best for pacing and a final readiness check.

(This guide points to both on purpose — it doesn't duplicate them.)

How the midterm is graded + test-taking strategy

How it's graded.
- 100 points across 20 items (5 each), weighted toward doing (solve/simplify/factor/evaluate), with auto-gradable item types only.
- The midterm is 20% of your course grade. It replaces Week 8's quiz and assignment (there are none that week).
- Coverage matches this guide: Obj 1 ≈ 3 · Obj 2 ≈ 3 · Obj 3 ≈ 3 · Obj 4 ≈ 4 · Obj 5 ≈ 3 · Obj 6 ≈ 4. Objectives 4 and 6 carry the most weight — drill them until the procedures are automatic.

Honest test-taking strategies for this material.
1. Find both answers when a problem can have two. Absolute-value equations, the square root property, and most quadratics give two solutions. Half-finished answers are where points vanish.
2. Protect your signs. Put inputs in parentheses when you evaluate; write −b = ___ before you use the quadratic formula; remember a negative times a negative is positive when you distribute.
3. Flip the inequality on a negative divide. The single most common Objective-2 miss. Circle the negative coefficient as a reminder.
4. Get to zero before you factor a quadratic. The zero-product property needs 0 on one side — rearrange first.
5. Compute the discriminant first. Five seconds of b² − 4ac tells you whether to expect two roots, one, or none — and a negative result is a valid answer (no real solutions).
6. Read slope as rise over run (y's on top), and use point-slope for line equations so you don't mistake a point's y for the intercept.
7. Sanity-check each answer: substitute back when you can. If a "solution" doesn't satisfy the original equation, it's extraneous.
8. Do the easy items first, flag the hard ones, budget your time — 20 items, a few minutes each. Don't sink ten minutes into one while four quick ones wait.
9. Read each item twice and answer exactly what's asked (the slope vs. the equation, all solutions vs. one).

Canvas placement block

canvas_object   = Page
title           = "Midterm Study Guide — Weeks 1–7 (Objectives 1–6)"
module          = "Week 8 — Midterm Review & Exam"
grading_type    = not_graded
available_from  = 2026-10-15      # posts before the Week 8 exam window opens
published       = true
provenance      = "~ Prof. Calloway's edition · Fall 2026 · built with thecoursemaker.com"

Term-update note: each term's $39 update regenerates fresh practice variants from this same scope — the live midterm is never reproduced here.

The per-term $39 update (fresh assessment variants, re-paced to your next calendar) referenced above is on the roadmap — coming soon. Today's download is yours to keep, but it doesn't refresh itself.

~ Prof. Calloway's edition · Fall 2026 · built with thecoursemaker.com