Week 02 · Every Polynomial Factors
Dock Day
Dock Day: the coin triangle
// Planks in Pascal's rows
⏱ about 20 min
Dock Day: The Coin Triangle
Saturday a jar of coins and a stack of index cards come to the kitchen table.
"One card on top, labelled 1," Comet tells the family. "Each new card gets the sum of the two above it."
"Row 5 reads 1, 5, 10, 10, 5, 1," someone says. "And it adds to 32."
Nova dims to a hint. "Would you like a hint? Drop a coin from the top and count the ways down."
"Three ways to the second card of row 3," Wren says. "The card says 3. The paths and the planks agree."
"Now a binomial," Comet says. "(x + 1)³. Read the row, write the terms, and check with x = 1."
"1 + 3 + 3 + 1 is 8, and 2³ is 8," comes the answer. "It checks."
- Identities like a² - b² extend to complex numbers: x² + 4 = (x + 2i)(x - 2i).
- The Fundamental Theorem of Algebra: a polynomial of degree n has exactly n complex zeros, counting multiplicity.
- For a quadratic, completing the square proves it. The discriminant decides real, double or conjugate-pair zeros.
- The Binomial Theorem: the coefficients of (x + y)ⁿ are row n of Pascal's triangle, and the powers add to n.
- Rational expressions form a system like fractions, closed under the four operations with a nonzero bottom.
◇ FAMILY COIN TRIANGLE
Lay index cards in rows: one, then two, then three, up to row 5. Label the ends 1 and each inside card with the sum above it.
Drop a coin from the top card, moving down-left or down-right each step. Count the paths to each card in row 4.
Pick a row and expand (x + 1) to that power together. Check by putting x = 1: the terms should add to the row sum.
Trade roles so everyone labels a row and everyone checks an expansion.
Drop a coin from the top card, moving down-left or down-right each step. Count the paths to each card in row 4.
Pick a row and expand (x + 1) to that power together. Check by putting x = 1: the terms should add to the row sum.
Trade roles so everyone labels a row and everyone checks an expansion.
For a grown-up
This is a counting activity with coins and cards on a table. Keep coins away from small children and off the floor.
A wrong label is a good moment: check the two cards above it and add again.
The rows and expansions are the family's own math game, not a count of any real plank.
A wrong label is a good moment: check the two cards above it and add again.
The rows and expansions are the family's own math game, not a count of any real plank.
DOCK DAY CHECK
- Read the question.
- Tap your answer.
The family's row 4 reads 1, 4, 6, 4, 1. In (x + y)⁴, what is the coefficient of x²y²?
Use row 3, which reads 1, 3, 3, 1. Expand (x + 1)³.A family member asks about Monday's problem. Factor x² + 9 over the complex numbers.
How many downhill paths reach the middle card of row 4? Type the number.
The path count is the number written on the card: C(4, 2).
WHY THIS EXERCISEPaths, planks and binomial coefficients are one count seen three ways.
Draw the family triangle to row 5 with the coin paths to one card traced in color.