Ray's New Elementary Algebra · Art. 72 · Unit 1: Chapter I: Fundamental rules

8. Signs in multiplication

Goal: Your student uses the rule of signs (like signs give plus, unlike give minus) and multiplies any two polynomials.

⏱ About 45 minutes · 8 steps

Open it at the Internet Archive ↗

You'll need

  • The book, open to the page
  • Pencil and plenty of paper (or a whiteboard)
  • A card to cover printed answers

What's in the book

Why minus times minus gives plus, shown first with plain numbers like (8 − 3) × 5, then the rule for the signs, the general rule, and thirty exercises, the last few with three or four factors. Answers are printed.

The lesson, step by step

  1. Warm-up

    Multiply 3a²b by 4ab, and 2x + 3y by 4x. (12a³b², 8x² + 12xy.)

  2. See it in numbers

    Work the first numerical exercises together:

    1. Multiply 8—3 by 5 Ans. 40—15=25.

    Then 9 − 5 by 8 − 2: 72 − 18 − 40 + 10 = 24, and 4 × 6 is 24. Say: The minus times minus product, +10, had to be plus to make it come out right.

  3. The rule for the signs

    Plus multiplied by plus, or minus by minus, gives plus.

    Plus multiplied by minus, or minus by plus, gives minus.

    Have your student copy it into the rule notebook.

  4. Work an example together

    Exercise 9:

    9. Multiply m+n by m — n Ans. m²—n².

    Write the four products: m², −mn, +mn, −n². The middle two cancel. Say: Remember this one; it comes back as a theorem.

  5. Guided practice

    Your student does these while you watch:

    17. Multiply 3a+5b by 3a— 5b. . . Ans. 9a²—25b².

    26. (x— 3) (x— 3) (x— 3). . . Ans. x³— 9x²+27x— 27.

    For the three factors, multiply the first two, then multiply that result by the third.

  6. Independent practice

    The rest of the exercises in Art. 72, from 1 to 30. For a 45-minute session, do 7 to 20 today and 21 to 30 next time.

  7. Check

    After working exercise 27, (x − 4)(x − 5)(x + 4)(x + 5), by hand, your student may confirm it on the site's calculator: expand((x-4)*(x-5)*(x+4)*(x+5)) gives x⁴ − 41x² + 400.

  8. Wrap-up

    Ask: What sign does −2a × −3b × −c have? (Minus: two minuses make plus, and one more minus makes minus.)

Tip

Have your student write every partial product with its sign before combining anything. Skipped signs cause most wrong answers.

Answers to the book's problems

The book prints the answer after each exercise.

Worked out for this site from the scan. If a number in your copy differs, trust the book. Every answer is in the Key ↗.

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