High School: Number and Quantity
The Complex Number System HSN-CN.C.9
9. Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
Once upon a time, students were given easy equations to solve, equations with one variable to the first power. They found one answer, checked it to make sure it was correct, and all was right with the world.
Then came the "F" word. Factoring. And along with it came quadratic equations and their two answers. Then they got into cubics, with three answers and… wait. Do we sense a pattern, here?
Well, the Fundamental Theorem of Algebra certainly thinks so. It says that the degree of a polynomial function is the same as the number of roots. So a linear equation (one with x to the first power) has 1 root. Quadratics (with x to the second power) have 2 roots. Cubics (with x to the third power) have 3 roots. And so it goes.
Regardless of how high the degree, the degree matches the number of roots.
But… (yeah, you knew there had to be a "but") it doesn't say they have to be different roots, necessarily. For example, if our equation is x2 + 4x + 4 = 0, we're going to get (x + 2)(x + 2) = 0, and our roots will both be -2. Sure there are 2 of them, but they're identical.
The Fundamental Theorem of Algebra also doesn't say that the roots have to be real, so make sure your students know how to work in the realm of the imaginary.
The one good thing about complex roots (and irrational roots, too): they come in conjugate pairs. So if x = 7i is one root, we automatically know that x = -7i is another. So if we have an odd number of roots, at least one of them has to be rational.