Week 6 — Build It, Take It Apart: A Polynomial Check Report

Assignment 6 of the term — subtract, multiply, and factor four polynomials, check every result yourself, and explain to someone new to algebra why a square of a sum keeps its middle term.

This adaptive assignment uses a coaching dialogue to help you turn a first attempt into a clear, checkable report. You remain the author. The coach asks questions, checks the reasoning you show, and waits for you to revise before it produces the required report. In Week 1 you learned the distributive property and the product rule for exponents, and in Week 3 you evaluated expressions at a number; this week those two skills become the way you build a polynomial and the way you check one. This week's lecture tutorial, quiz, discussion, and Model/Build Studio practise the same ideas from other angles; this assignment is where you put them together and explain them.

The task

You will work with this week's shared polynomial x² − 6x + 5 and its partner x² − 2x − 15, a poster x inches on a side in a frame 3 inches wide all around (so the framed square is x + 6 inches on a side), and two trinomials to factor, x² − 5x − 14 and 2x² + 7x + 3.

Work these four problems, showing your arithmetic:

1. Subtract (x² − 6x + 5) − (x² − 2x − 15) and simplify. What is the coefficient of x in the result?
2. A poster x inches on a side sits in a frame 3 inches wide all around, so the framed square is x + 6 inches on a side. Expand (x + 6)² as x² + bx + 36. What is b?
3. Factor x² − 5x − 14 into two binomials, then use your factors to find the value of x² − 5x − 14 at x = 9.
4. Factor 2x² + 7x + 3 by the ac method, then use your factors to find the value of 2x² + 7x + 3 at x = 4.

Beside each answer, add one brief interpretation: what the result shows and how you checked it. For problem 1, say what the minus sign in front of the second polynomial did and why the x² terms disappear. For problem 2, say what the middle term stands for in the picture of the framed poster. For problems 3 and 4, check your factored value by evaluating the original trinomial at the same x, and say what a mismatch would have told you.

Explain it to a non-expert

A friend is cutting mat board for a campus art show. They say the framed area of a poster x inches on a side, with 3 inches of frame all around, is x² + 36 square inches, "because you square the x and square the 6." Write a 3–5 sentence message to your friend: what (x + 6)² really is, what the missing piece stands for on the mat board, and one quick way to check any formula like this before cutting. Explain any technical term in everyday words. More than one good check can be defended.

For a different, ungraded example, imagine a square tile x inches on a side that is replaced by a tile 5 inches longer on every edge, so the new area is (x + 5)². Model answer: "The new tile is x plus 5 inches on each side, so its area is (x + 5) times (x + 5), not x squared plus 25. Multiplying it out gives x² + 10x + 25: the old tile, two strips that are x by 5 along two edges, and a 5-by-5 corner. A fast check is to try a small number, like x = 1: the new tile is 6 by 6, which is 36, and 1 + 10 + 25 is also 36, but 1 + 25 is only 26. The formula only describes square tiles, so a rectangular tile needs two different side lengths."

Use that pattern for the framed poster, with your own explanation.

Part 1 — Student instructions

1. Draft all four answers, the four brief interpretations, and your message to the friend cutting mat board before opening a chatbot.
2. Open any course-approved chatbot. Copy the entire Coach Prompt box, including its first and last marker lines, and paste it as one message. The coach will ask for your draft and then work one question at a time.
3. Answer in your own words. Show calculations when asked. Use the coach's feedback to revise at least one calculation or explanation, even if your first attempt was correct.
4. Continue until the prompt's exit condition is met.
5. Save the chat's share link and copy the final report exactly as the coach gives it.

The chatbot is a coach, not a ghostwriter. It must draw out your reasoning rather than write the report for you.

Coach Prompt

COPY EVERYTHING BELOW THIS LINE
ROLE: You are a careful learning coach. Draw out and challenge MY thinking about polynomials: subtracting with the minus sent to every term, the square of a sum, factoring a trinomial by a product-and-sum pair, the ac method, and checking every product or factoring by evaluating at a number. Do not write my assignment for me and do not invent work I did not show.

ASSIGNMENT: Work through these four problems in order, one at a time. Do not invent, replace, or reorder a problem. The data: the shared polynomial x^2 - 6x + 5 and its partner x^2 - 2x - 15; a poster x inches on a side in a frame 3 inches wide all around, so the framed square is x + 6 inches on a side; and the trinomials x^2 - 5x - 14 and 2x^2 + 7x + 3.
PROBLEM P1: Subtract (x² − 6x + 5) − (x² − 2x − 15) and simplify. What is the coefficient of x in the result?
PROBLEM P2: A poster x inches on a side sits in a frame 3 inches wide all around, so the framed square is x + 6 inches on a side. Expand (x + 6)² as x² + bx + 36. What is b?
PROBLEM P3: Factor x² − 5x − 14 into two binomials, then use your factors to find the value of x² − 5x − 14 at x = 9.
PROBLEM P4: Factor 2x² + 7x + 3 by the ac method, then use your factors to find the value of 2x² + 7x + 3 at x = 4.
After the four answers, I must briefly interpret each one: what the result shows and how I checked it; for P1, what the minus sign did and why the x^2 terms disappear; for P2, what the middle term stands for in the picture of the framed poster; for P3 and P4, a check of my factored value against the original trinomial at the same x. I must then write a 3-5 sentence message to a friend cutting mat board who thinks (x + 6)^2 is x^2 + 36, explaining what (x + 6)^2 really is, what the missing piece stands for, and one quick way to check such a formula. I must explain technical terms in everyday words.

SCORING CRITERIA: Score only evidence I supply: correct polynomial computations (40 points), reasoning about products, factors, and checks (32 points), clarity for a non-expert reader (16 points), and showing work so it can be checked (12 points). For each of P1-P4, give 2, 1, or 0 evidence units per criterion; one unit is worth 5 computation points, 4 reasoning points, 2 clarity points, or 1.5 showing-work points. Sum all four problems within each criterion. Count my final message only with P4, not again with P1-P3.
COMPUTATION UNITS: A correct coefficient or value = 2; the right values and operations with only an arithmetic slip = 1; a missing result or the wrong values or operations = 0. Do not penalize an alternative valid method or equivalent correct notation.
REASONING UNITS: For P1-P3 separately, an accurate account of what the result shows (the sign flip, the side strips of the frame, or the product-and-sum pair) plus a check or a use or limit = 2; exactly one of the two = 1; neither = 0. For P2, a value of b given without the middle-term meaning earns at most 1. For P4, BOTH accurate ac-method reasoning with a check AND a final message that explains why (x + 6)^2 is not x^2 + 36 with a way to check it = 2. If full P4 evidence is absent but at least one accurate substantive part appears, give 1. No accurate reasoning = 0. An arithmetic slip alone is not a reasoning error.
CLARITY UNITS: For each problem, labeled evidence understandable to someone new to algebra, with technical terms explained = 2; intelligible evidence that needs rereading because of ambiguity or jargon = 1; absent or unintelligible evidence = 0. Include the final message with P4 if submitted. Missing reasoning alone does not lower clarity for otherwise clear submitted work.
SHOWING-WORK UNITS: For each problem, checkable values and steps = 2; a relevant but incomplete value or step = 1; a result alone or no method = 0. A visible method stays checkable despite an arithmetic slip.

Do NOT invent grading rules.

DIALOGUE RULES: Ask exactly one question per message and wait for my answer. Start by asking for my complete first attempt. Build each next question from my exact words. Probe an unsupported claim and ask me to examine its evidence. Keep the conversation focused on questions rather than completed work.

ENGAGEMENT GUARDS: A greeting, copied prompt, yes/no reply, or one-word answer is not a substantive exchange. Do not flatter or accept agreement as evidence. Praise only a concrete move I made. If evidence is missing, say what is missing and ask for it. Require a real revision plus a short reflection on why the revision is better. Do not issue the report early.

EXIT CONDITION: Continue until at least five substantive exchanges have occurred AND I have supplied all four answers and their brief interpretations AND written my message to the friend cutting mat board AND revised at least one calculation or explanation AND reflected on the revision. Stop only after whichever requirement happens last.

REPORT: After the exit condition is satisfied, output exactly these labelled lines in this order. Replace X with the exact earned score; it may end in .5 and must not be rounded to a whole number. Base every line only on evidence in this chat; write "Not supplied" for missing evidence and never raise the score for polished chatbot prose.
STUDENT'S SCORE: X/100
EVIDENCE FOR THE SCORE: [specific evidence from my calculations and reasoning]
REVISION MADE: [what I changed and why]
STRENGTH: [one earned strength]
NEXT STEP: [one concrete improvement]
Then tell me to submit this report, my revised polynomial report, and the share link.
COPY EVERYTHING ABOVE THIS LINE

Deliverables

Submit these three items in the assignment entry:

• your revised polynomial report with all four answers, their brief interpretations, and your message to the friend cutting mat board — written, or recorded as audio or video with a transcript, or as a labelled area sketch of the framed poster with a short note; every form is graded by the same scoring criteria;
• the coach's five-line report; and
• the conversation share link.

Submit in any accessible file type or as a link; a recorded response may be submitted as audio or video with a transcript, or as text.

Equivalent route (no AI required)

This route assesses the same Week 6 skills — subtracting polynomials with the minus sent to every term, expanding the square of a sum without dropping the middle term, factoring a trinomial with and without a leading coefficient, and checking every product or factoring by evaluating at a number — for comparable time and effort, about 60 to 90 minutes, the same as the coached route. Complete the same four problems, the four brief interpretations, and the message to the friend cutting mat board. Then audit your own first draft against the attached scoring criteria, criterion by criterion, and write a second draft. Add a 150–200 word revision memo that names one calculation or explanation you changed, the evidence that prompted the change, and one question you still have. Submit the revised polynomial report, your completed criteria checklist, and the revision memo in the assignment entry; together they replace the share link and the coach's report. This route uses the same scoring criteria and the same points, with no penalty for choosing it.

How you'll be graded

The full analytic rubric is attached to this assignment in Canvas.

Academic integrity and AI use

The calculations, reasoning, revision, and final report must be your own. Do not ask the chatbot to create the assignment or add evidence you did not produce. Submit an honest share link from your own dialogue. The equivalent route keeps the same workload and evidence and requires no AI account.

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