Ray's New Elementary Algebra · Arts. 80–86 · Unit 2: Chapter II: Theorems, factoring, etc.
11. The first algebraic theorems
Goal: Your student squares a sum or a difference and multiplies a sum by a difference at sight, and knows what negative and zero exponents mean.
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
Theorems I to III (square of a sum, square of a difference, sum times difference), each with six examples written out; Theorems IV and V on negative and zero exponents; and Theorems VI to VIII (top of p. 50) on dividing differences and sums of powers.
The lesson, step by step
Warm-up
Multiply a + b by a + b the long way. (a² + 2ab + b².)
Theorem I
Read Art. 80.
The square of the sum of two quantities is equal to the square of the first, plus twice the product
of the first by the second, plus the square of the second.
Work the number example: (2 + 3)² = 4 + 12 + 9 = 25. Then (2a + b)² = 4a² + 4ab + b².
Theorems II and III
Read Arts. 81 and 82.
The product of the sum and difference of two quantities is equal to the difference of their squares.
Do the number examples: (5 − 4)² = 25 − 40 + 16 = 1, and (5 + 3)(5 − 3) = 25 − 9 = 16.
Guided practice
Cover the right-hand sides. Your student writes the answers at sight, saying the theorem aloud each time:
3. (2x+3y)²=4x²+12xy+9y².
4. (5a+4b)(5a— 4b)=25a²— 16b².
Then (3x − 2y)², which is 9x² − 12xy + 4y².
Negative and zero exponents
Read Arts. 83 and 84.
The reciprocal of a quantity is equal to the same quantity with the sign of its exponent changed.
Saya⁻² means 1 over a². And a⁰ is 1, because a³ ÷ a³ is 1 and by the exponent rule it is also a⁰.
Differences and sums of powers
Read Arts. 85 and 86 (Theorems VI and VII) and Theorem VIII at the top of p. 50. Show them with numbers: 5² − 3² = 16, which divides by 5 − 3 = 2 and by 5 + 3 = 8; 5³ − 3³ = 98, which divides by 2; and 5³ + 3³ = 152, which divides by 8.
Independent practice
Cover and redo all six examples under each of Theorems I, II and III. Your student can confirm one afterwards on the site's calculator, for example expand((2*x+3*y)^2), but the point is to write these at sight.
Wrap-up
Your student states Theorems I, II and III from memory, with a + b and a − b.
Tip
These three theorems are used constantly in factoring, which comes next. A few minutes of drill at the start of the next several sessions pays off.
Answers to the book's problems
The book writes out the answer to each example under Theorems I to III.
Worked out for this site from the scan. If a number in your copy differs, trust the book. Every answer is in the Key ↗.