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Week 6 · Assignment: Traditional alternative (no AI)

Week 6 — Build It, Take It Apart: A Polynomial Check Report — Traditional alternative (no AI)

College Algebra Free sample edition
Traditional alternative (no AI), unpublished. In the course this is a separate assignment that ships unpublished beside the adaptive assignment: same objective, no AI. An instructor who prefers it publishes it and unpublishes or deletes the adaptive one, as the note at its top says.

Instructors: publish this instead of the adaptive version if you prefer, and unpublish or delete the adaptive one.

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 week you learned to name a polynomial's parts, to add and subtract polynomials with the minus sent to every term, to multiply with FOIL and the three special products, and to factor by the greatest common factor, by product-and-sum pairs, by the ac method, by the difference of squares, and by grouping. 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 skills become the way you build a polynomial and the way you check one. In this assignment you work four short problems, check each one, and explain to someone who has never studied algebra why (x + 6)² is not x² + 36. This week's lecture tutorial, quiz, discussion, and Model/Build Studio practise the same ideas from other angles.

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.

Show your arithmetic. A reader should be able to reproduce every number you report from the values above without guessing which values or steps you used.

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.

Deliverables

Submit a single polynomial report containing:

  • the four answers, each with the arithmetic shown and a brief interpretation of what the result shows and how you checked it;
  • your message to the friend cutting mat board, explaining what (x + 6)² really is, what the missing piece stands for, and one quick way to check such a formula; and
  • a one-line restatement of the polynomials and values you used, so your results are reproducible.

Keep it to roughly one page. Clear and correct beats long.

Choose how you show it. Present your report as written text, or as a recorded audio or video explanation of about three minutes 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. 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.

How you'll be graded

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

Academic integrity and AI use

This version is designed to be completed without AI: do the computations, the interpretations, and the writing yourself, without a chatbot or other AI tool. You may use a calculator or a free graphing calculator such as Desmos to check your arithmetic, for example by graphing a polynomial and your factored form together. Submitting AI-generated work as your own, or copying another student's numbers, is an integrity violation under the course policy.

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