THE COMPOUND // FIELD NOTES // WEEK 29
Cubes, exponent cases, and quadratic inequalities
Field Book
Lessons 108 to 111
Due
Test 26 due ยท Supplement 29
Method
Paper first, every step
The cube patterns come with the Compound's favorite mnemonic: same, opposite, always positive. Say it while you write.
What you have to be able to do
- Factor sums and differences of two cubes
- Manage parentheses with exponents in both cases
- Solve quadratic inequalities of the greater than kind and write three variable systems from words
The procedures
Sum and difference of cubesField Book Lesson 108
- A3 + B3 = (A + B)(A2 - AB + B2), and the difference flips the first sign.
- Signs run same, opposite, always positive across the two factors.
Parentheses with exponentsField Book Lesson 109
- An exponent outside parentheses hits every factor inside.
- Without parentheses it binds only to the nearest base. Read carefully before touching.
Quadratic inequalities, greater thanField Book Lesson 110
- Solve the matching equation to find the boundary points.
- Greater than selects outside the boundaries: two rays. Confirm with a test point.
Three variable word systemsField Book Lesson 111
- Name all three unknowns, write three independent equations.
- Then run the five column elimination from Lesson 90.
Worked: a sum of cubes
- Factor x3 + 8.
- A = x and B = 2: (x + 2)(x2 - 2x + 4).
- Signs check: same, opposite, always positive.
Terms on the record
| Term | Meaning |
|---|
| sum of cubes | A cubed plus B cubed, with its fixed factoring pattern |
| boundary points | the roots of the matching equation that split the number line |
| test point | a sample value used to check which region satisfies an inequality |