"I want the remainder when p(x) is divided by x - 1," Wren says, "without the long division."
"Is that allowed?" Comet asks.
He writes p(x) = (x - 1)q(x) + r. "The division always looks like this. Now, what do you notice if x is 1?"
"x - 1 becomes zero," Comet says. "The whole first part vanishes. So p(1) equals r."
"p(1) = 1 + 6 + 11 + 7 = 25," Wren says. "That is the remainder. No dividing."
Nova projects the long division beside it. "Would you like a hint? Run the division and see if they agree."
Comet works for a minute. "Quotient x² + 7x + 18, remainder 25. They agree."
"Two ways, one remainder," Wren says. "Put the number in, or divide. Pick the quicker one."
Divide x³ + 6x² + 11x + 7 by x - 1 with long division. The quotient is x² + 7x + 18 and the remainder is 25.
Every division by x - a can be written p(x) = (x - a)q(x) + r, where r is a number.
Put x = a. Then x - a = 0, the first part is 0, and p(a) = r.
This is the remainder theorem: the remainder on dividing by x - a is p(a).
For x - 1, a = 1 and p(1) = 25. For x + 2, a = -2 and p(-2) = 1. Monday's remainder again.
| Divisor | a | p(a) | Remainder by division |
|---|---|---|---|
| x - 1 | 1 | 25 | 25 |
| x + 1 | -1 | 1 | 1 |
| x + 2 | -2 | 1 | 1 |
| x - 2 | 2 | 61 | 61 |
Watch the sign. Dividing by x + 2 means a = -2, because x + 2 = x - (-2).
Compare the ways. Putting in a number takes a line. Division takes several, but it also gives the quotient.
| Statement | True or false? |
|---|---|
| The remainder when p(x) is divided by x - a equals p(a). | ? |
| 1 + 6 + 11 + 7 = 25 | ? |
| 2 - 3 + 4 - 5 = -2 | ? |
| To divide by x + 3, the remainder theorem uses a = 3. | ? |
| The remainder theorem also gives the quotient. | ? |
Two ways, one answer. Tomorrow is Lab day: egg cartons, dried beans, rows and remainders.