Monday
Monday: Trays in Rows
Seedlings cover the long bench, 61 of them in tiny pots. Comet counts twice to be sure.
"Rows of 4 for the Open House," she says. "What can we make? How many full rows?"
Wren divides. "61 divided by 4. 15 rows, and 1 left over. What do you notice about the leftover?"
"It is smaller than a row," Comet says. "If it were not, we would make another row."
Nova projects the crew's tally rule, p(x) = x³ + 6x² + 11x + 7. It counts the seedlings when each cell holds x.
"Would you like a hint? Divide the rule by x + 2 the same way you divided 61 by 4."
"Divide a polynomial?" Comet says. "Like long division?"
"Exactly like it," Wren says, and writes the first step on the chalkboard.
Division with a remainder
61 ÷ 4 gives quotient 15 and remainder 1. Check: 4 × 15 + 1 = 61.
The remainder is always smaller than the divisor. Otherwise another full row fits.
Polynomials divide the same way. The remainder has a smaller degree than the divisor.
A solved problem to study
Divide the crew's rule p(x) = x³ + 6x² + 11x + 7 by x + 2. Watch the four moves repeat: divide, multiply, subtract, bring down.
- Divide the leading terms: x³ ÷ x = x². Write x² on top.
- Multiply back: x² × (x + 2) = x³ + 2x². Subtract from the first two terms: 6x² - 2x² = 4x². Bring down 11x.
- Divide again: 4x² ÷ x = 4x. Multiply back: 4x² + 8x. Subtract: 11x - 8x = 3x. Bring down 7.
- Divide again: 3x ÷ x = 3. Multiply back: 3x + 6. Subtract: 7 - 6 = 1.
- The degree of 1 is less than the degree of x + 2, so stop. Quotient x² + 4x + 3, remainder 1.
- Check: (x + 2)(x² + 4x + 3) + 1 = x³ + 6x² + 11x + 7. Multiply it out to be sure.
Now put x = 2 into everything. p(2) = 61 seedlings, rows of 4, quotient 15 rows, remainder 1.
The polynomial division and the number division tell the same story. That is why the method is the same.
Side by side
| Numbers | Polynomials |
|---|---|
| 61 ÷ 4 | (x³ + 6x² + 11x + 7) ÷ (x + 2) |
| quotient 15 | quotient x² + 4x + 3 |
| remainder 1 | remainder 1 |
| 4 × 15 + 1 = 61 | (x + 2)(x² + 4x + 3) + 1 = x³ + 6x² + 11x + 7 |
- Read the question.
- Tap your answer.
| Statement | True or false? |
|---|---|
| 4 × 15 + 1 = 61 | ? |
| 8 + 24 + 22 + 7 = 61 | ? |
| A remainder can be larger than the divisor. | ? |
| When dividing by x + 2, the remainder must have degree less than 1, so it is a number. | ? |
| Polynomial long division repeats: divide, multiply, subtract, bring down. | ? |
Match each step of one to a step of the other. Circle the two remainders.
A clean start. Tomorrow you find a remainder without dividing at all, and compare the two ways.