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Algebra 2 9-12 / Week 03 / Garden Day
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Week 03 · Dividing Polynomials

Garden Day

Garden Day: rows at home
// Rows, quotients and what is left over
⏱ about 20 min

Garden Day: Rows at Home

Comet spreads a bag of dried beans on the kitchen table. "Count with me," she tells her brother.

"Eighty-five," he says. "Now what?"

"Rows of six." They lay them out. "14 rows, 1 left over. The leftover is the remainder."

"Rows of seven?" he asks, and rebuilds. "Twelve rows, one left."

Nova hovers at the window. "Would you like a hint? Ask which row size leaves nothing."

"Rows of five," Comet says. "85 is 5 × 17. Five is a factor, so the remainder is 0."

"Factor," her brother repeats. Comet writes it in the Glass House Log for him.

"One more trick," she says. "Fifty-three times forty-seven. Watch."

  1. Polynomial long division repeats four moves: divide, multiply, subtract, bring down. It ends with a quotient and a remainder.
  2. The remainder theorem: the remainder on dividing p(x) by x - a is p(a).
  3. The factor theorem: x - a is a factor of p(x) exactly when p(a) = 0.
  4. Identities such as a² - b² = (a + b)(a - b) hold for every value of the letters.
  5. An identity is proved by expanding both sides and seeing them match.
  6. Read the structure of an expression: a square minus a square, a perfect square, a sum or difference of cubes.
◇ ROWS AND A SHORTCUT, TOGETHER
Count out a pile of dried beans or buttons, between 40 and 100, with a family member.
Make rows of 4, then 6, then 7. Record the full rows and the leftover each time. Nothing here is eaten.
Find a row size that leaves nothing. Say the word factor and explain what it means.
Show the shortcut 53 × 47 = 50² - 3². Then ask your partner for two numbers that straddle a round number.
Together, divide x² + 7x + 10 by x + 2 on paper. Check the remainder with p(-2).

Comet's kitchen numbers

Comet's counts are made up for this lesson. The pile is 85.

Row sizeFull rowsLeft over
6141
7121
5170
COMET'S PILE
  • Read the question.
  • Tap your answer.
Comet's pile is 85 beans in rows of 6. How many are left over?
The partner division: what is the remainder when x² + 7x + 10 is divided by x + 2?
Which row size shares Comet's 85 beans with nothing left over: 5, 6 or 7? Type the number.
WHY THIS EXERCISEA zero remainder means a factor, for piles of beans and for polynomials alike.
Draw your pile as rows for one row size, with the leftover beside it. Write the quotient and remainder.

Well done this week. Next week the crew mixes plant food into a tank, and the concentration is a rational expression.

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