Ray's New Elementary Algebra · Arts. 94–96 · Unit 2: Chapter II: Theorems, factoring, etc.
13. Factoring with the theorems; quadratic trinomials
Goal: Your student factors perfect squares, differences of squares, sums and differences of cubes, and trinomials like x² − 5x + 6.
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
Factoring by reversing the theorems of Arts. 80 to 86, with twelve exercises whose answers are printed together below them; then quadratic trinomials split into two binomials by inspection, and the use of factoring to shorten work by canceling.
The lesson, step by step
Warm-up
Ask your student to say Theorems I, II and III from memory, with a and b.
Reverse the theorems
Read Art. 94.
No general rule can be given for this case.
Read the five hints (1st to 5th). Write the examples: a² + 2ab + b² = (a + b)(a + b), a² − b² = (a + b)(a − b), a³ − b³ = (a − b)(a² + ab + b²).
Guided practice
Cover the answers. Your student factors exercises 1, 7 and 10 while you watch: x² + 2xy + y², 9m² − 16n², and x³ − 1. (The book gives (x + y)(x + y), (3m + 4n)(3m − 4n), and (x − 1)(x² + x + 1).)
Quadratic trinomials
Read Art. 95.
Such quantities may be resolved into two binomial factors by inspection.
SayFor x² − 5x + 6, find two numbers that multiply to 6 and add to −5: −2 and −3. So (x − 2)(x − 3).
Work together:
1. x²+5x+6 Ans. (x+2)(x+3).
8. x²— x— 30 Ans. (x— 6)(x+5).
Common factor first
Read the note before exercise 9: 5ax² − 10ax − 40a = 5a(x² − 2x − 8) = 5a(x − 4)(x + 2). Your student does exercise 9, 3x² + 12x − 15. (3(x + 5)(x − 1).)
Canceling
Read Art. 96.
The principal use of factoring is to shorten algebraic operations by canceling common factors.
Do exercise 2 together: (x − 3)(x² − 1) ÷ (x − 1) = (x − 3)(x + 1)(x − 1) ÷ (x − 1) = (x − 3)(x + 1) = x² − 2x − 3.
Independent practice
Art. 94 exercises 2 to 12 (except those done), Art. 95 exercises 2 to 11, and Art. 96 exercises 3 and 4. Afterwards, check two on the site's calculator with factor(x^2-13*x+40) and similar.
Wrap-up
Ask: Why can't x² + 4 be split like x² − 4?
Tip
For trinomials, write the factor pairs of the last number in a little list. Guessing without the list wastes time.
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 ↗.