Chapter 7
Zeros of Polynomials
Hunting for the roots of a polynomial can feel like luck. Sometimes a candidate works on the first try; sometimes every candidate fails and the search ends with nothing to show. Each time that happened before, the fix was to widen the number system: fractions, then irrational numbers, then a number whose square is negative one. So the honest question is whether the widening ever ends. Answer it, and a second question is waiting: how much can a polynomial's roots tell you before you have found even one of them?
What You'll Explore
4 lessons.
- The Fundamental Theorem of Algebra
Nothing you have learned so far promises that a polynomial has a root. A list of candidates can come up empty, and a familiar curve can miss the axis entirely. This lesson asks what has to be true for that search to be guaranteed to succeed, and how many roots a polynomial of a given degree is entitled to.
- Complex and Irrational Roots
Solve a quadratic with no real roots and its two answers mirror each other. Roots carrying a square root seem to pair up too. The plus or minus in the formula looks like the cause, until the same pairing shows up where no formula produced it. The hunt here is for the real cause, and for when it fails.
- Vieta's Formulas
Back in the quadratics chapter, two facts about the roots could be read straight off the coefficients, and they looked like coincidences. If they are not coincidences, there should be relatives at every degree. Here you will go looking for the whole family, and for what coefficients reveal about roots nobody has found yet.
- Graphing Polynomial Functions
The two ends of a polynomial graph have long been readable, but the middle stayed out of reach. Now that the roots are in hand, the middle may be too. This lesson asks what a factored polynomial tells you about the shape of its curve, what happens at a root that repeats, and where a nonreal root shows up.