Lesson 15 of 17
The big picture
Show on a big screenMost of these rewrites run both ways. You can multiply the brackets out. You can also write the result back as brackets.
| Idea | Remember | Example |
|---|---|---|
| Cutting down | (a − b)² = a² − 2ab + b². The corner is taken away twice. | side 7, strip 2: leftover 25 |
| One from another | a² − b² = (a − b)(a + b). Gap times total. | 101² − 99² = 400 |
| Three parts | (a+b+c)² = a²+b²+c²+2ab+2bc+2ca. Nine pieces, not three. | 213² = 45369 |
| A cube | (a+b)³ = a³+3a²b+3ab²+b³. Two kinds of block come three at a time. | 2 and 1: 27 bricks |
| The pair | Add to the middle, multiply to the last. | x²+9x+20 = (x+4)(x+5) |
| A number in front | The pair multiplies to first times last. | 3x²+11x+6 = (x+3)(3x+2) |
| Cancelling | Strike out only a shared factor, never a term that is added on. | (x²−9)/(x+3) = x−3, if x ≠ −3 |
| Adding two | Build one shared bottom, then add the tops. | 1/x + 1/(x+1) = (2x+1)/(x²+x) |
| Your own claim | One disagreement settles it. A few agreements do not. | (a+b)²−(a−b)² = 4ab |
| Two letters | Square each part. The middle is twice their product, with the sign between the parts. | (8a+3b)² = 64a²+48ab+9b² |
| Two cubes | x³ − y³ = (x − y)(x² + xy + y²). x³ + y³ = (x + y)(x² − xy + y²). | 5³ − 3³ = 2 × 49 = 98 |
| Round numbers | a² = (a + b)(a − b) + b². Choose b to make a factor round. | 47² = 50 × 44 + 9 = 2209 |
Now open Chapter 4 in your book. It is Exploring Algebraic Identities.