"Four quadratics on the chalkboard," Wren says. "Two have real zeros. One has a double zero. One looked like it had none."
"x² + 9 has zeros now," Comet says. "-3i and 3i. What do you notice about the counts?"
"Two, two, two and two," Wren says. "Every quadratic has exactly two zeros, if you count 3 twice for x² - 6x + 9."
Nova projects a cubic beside them. "Would you like a hint? Try the same count for degree 3."
"x³ - x is x(x - 1)(x + 1). Three zeros," Comet says. "Degree 3, three zeros. Is that always true?"
"That is the Fundamental Theorem of Algebra," Wren says. "Degree n means n complex zeros, counting repeats."
"Then nothing is unfactorable," Comet says. "Every polynomial splits all the way down into linear pieces."
The Fundamental Theorem of Algebra: every polynomial of degree n with complex coefficients has exactly n complex zeros, counting multiplicity.
Multiplicity means a repeated zero is counted each time it repeats. The zero 3 in (x - 3)² counts twice.
So every polynomial of degree n factors into n linear factors (x - z₁)(x - z₂)...(x - zₙ), times a constant.
The theorem does not say the zeros are real. It says they exist, somewhere on the complex plane.
x² - 5x + 6 = (x - 2)(x - 3). Two real zeros, 2 and 3.
x² - 6x + 9 = (x - 3)². One zero, 3, with multiplicity 2. Still two zeros when counted.
x² + 9 = (x + 3i)(x - 3i). Two complex zeros, -3i and 3i, from Monday.
For x² - 4x + 13, factoring by eye is hard. The quadratic formula gives x = (4 ± √(16 - 52))/2 = (4 ± √(-36))/2.
√(-36) = 6i, so x = (4 ± 6i)/2 = 2 - 3i and 2 + 3i. Two zeros again, a conjugate pair.
The discriminant b² - 4ac decides the kind. Positive gives two real zeros, zero gives one double zero, negative gives a conjugate pair.
Whatever the sign, the count is two. That is the theorem for degree 2.
| Quadratic | Discriminant | Zeros | Count with multiplicity |
|---|---|---|---|
| x² - 5x + 6 | 1 | 2 and 3 | 2 |
| x² - 6x + 9 | 0 | 3 (twice) | 2 |
| x² + 9 | -36 | -3i and 3i | 2 |
| x² - 4x + 13 | -36 | 2 - 3i and 2 + 3i | 2 |
| Statement | True or false? |
|---|---|
| Every quadratic has exactly two complex zeros, counting multiplicity. | ? |
| A polynomial of degree 4 can have five zeros. | ? |
| The zeros of a quadratic with a negative discriminant are a conjugate pair. | ? |
| x² + 9 has no zeros. | ? |
| 4 × 4 - 4 × 13 = -36 | ? |
Excellent counting. Tomorrow is Boathouse Lab: the planks come off the hull and stack into Pascal's triangle.