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Algebra 2 9-12 / Week 03 / Friday
5/6
Week 03 · Dividing Polynomials

Friday

Identities and structure
// Rows, quotients and what is left over
⏱ about 20 min

Friday: Identities and Structure

"Fifty-three times forty-seven," Wren says. "No calculator."

Comet reaches for a pencil. Nova flashes a hint: (50 + 3)(50 - 3).

"a plus b times a minus b," Comet says. "That is a² - b². So 2500 - 9. 2491."

"An identity," Wren says. "True for every a and b, so it works for 50 and 3. What do you notice about x⁴ - 16?"

"It is a square minus a square too," Comet says. "(x²)² - 4². So (x² + 4)(x² - 4). And x² - 4 splits again."

"Reading the structure," Wren says. "You saw a² - b² hiding inside."

"Can we prove the identities?" Comet asks.

"Multiply out," Wren says. "If both sides match for every letter, it is proved."

Five identities, and why they hold

IdentityProof by expanding
(a + b)(a - b) = a² - b²a² - ab + ab - b² = a² - b²
(a + b)² = a² + 2ab + b²a² + ab + ab + b² = a² + 2ab + b²
(a - b)² = a² - 2ab + b²a² - ab - ab + b² = a² - 2ab + b²
a³ + b³ = (a + b)(a² - ab + b²)expand the right: a³ - a²b + ab² + a²b - ab² + b³ = a³ + b³
a³ - b³ = (a - b)(a² + ab + b²)expand the right: a³ + a²b + ab² - a²b - ab² - b³ = a³ - b³

An identity holds for every value of its letters. That is why it works as a numerical shortcut and as a factoring tool.

Check one with numbers. (2 + 3)² = 25. And 2² + 2 × 2 × 3 + 3² = 4 + 12 + 9 = 25. They match.

The derivation, read first

Why is (a + b)² = a² + 2ab + b², and not a² + b²? Read, then put the steps in order below.

  1. Write (a + b)² as (a + b)(a + b).
  2. Multiply every term by every term: a × a, a × b, b × a, b × b.
  3. That gives a² + ab + ba + b².
  4. ab and ba are the same product, so combine them: 2ab.
  5. So (a + b)² = a² + 2ab + b². The middle term is the one people forget.
PROVE (A + B)² = A² + 2AB + B²
  • ?Write (a + b)² as (a + b)(a + b)
  • ?Get a² + ab + ba + b²
  • ?Conclude (a + b)² = a² + 2ab + b²
  • ?Combine ab and ba into 2ab
  • ?Multiply every term by every term
WHY THIS EXERCISEProving an identity is one careful expansion. After that you may use it without thinking.

Reading structure

Ask: is this a square minus a square? A perfect square trinomial? A sum or difference of cubes?

x⁴ - 16 = (x²)² - 4² = (x² + 4)(x² - 4) = (x² + 4)(x + 2)(x - 2). Three factors from one pattern used twice.

x³ - 8 = x³ - 2³ = (x - 2)(x² + 2x + 4). Check: expand the right side and get x³ - 8.

SPOT THE PATTERN
  • Read the question.
  • Tap your answer.
Which expression equals x² - 25?
Which expression equals 9x² - 16?
Which expression equals (2x + 3)²?
Which expression equals x³ - 8?
EXPAND AND CHECK
  • Read the question.
  • Tap your answer.
Which expression equals (x - 3)(x² + 3x + 9)?
Which expression equals (2x + 3)²?
Which expression equals x³ + 1?
Which identity turns 29 × 31 into 30² - 1²?
Use a² - b²: 53 × 47 = 50² - 3² = ____. Type the number.
Use (a + b)²: 61² = 60² + 2 × 60 × 1 + 1² = ____. Type the number.
An equation that is true for every value of its letters is called an ____. Type one word.
StatementTrue or false?
53 × 47 = 2491?
50 × 50 - 3 × 3 = 2491?
(a + b)² = a² + b² for every a and b.?
(2 + 3) × (2 + 3) = 2 × 2 + 2 × 2 × 3 + 3 × 3?
(3 - 2) × (9 + 6 + 4) = 19?
WHY THIS EXERCISEOne numerical check cannot prove an identity, but one numerical failure disproves a false one.
Try it
Multiply 48 × 52 and 72 × 68 in your head with a² - b². Check one with a pencil.
Write x⁶ - 64 on a card. Is it a difference of squares, a difference of cubes, or both? Factor it one way.

A full week. Tomorrow is Garden Day: rows and remainders at home, and one identity for your family.

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