Zeroes of Polynomials

A zero of a polynomial is simply an input that makes it equal $0$; on a graph, it's where the curve meets the x-axis. (They're also called roots.) The formula up top looks busy, but it just says a polynomial can be written as a list of factors: $a$ (a number out front that stretches it), each $(x - r)$ (one factor for each zero $r$), and the little powers $m$ (how many times that factor repeats). So zeros are what you need to know to solve $P(x) = 0$, and to sketch the graph. It's mostly bookkeeping.

Why each zero gives a factor. This comes straight from last lesson. If $P(r) = 0$, then dividing by $x - r$ leaves remainder $0$, so $P(x) = (x - r)Q(x)$. Do the same with $Q$, and keep going, and you peel off one factor per zero.

Fundamental theorem of algebra. A degree-$n$ polynomial has exactly $n$ complex zeros, counted with multiplicity, so $m_1 + m_2 + \cdots + m_k = n$. It makes sense when you think about multiplying out: $n$ factors of the form $(x - r)$ give you $x^n$ as the top term, no more and no less.

Multiplicity is how many times a factor repeats. In $(x - 2)^3(x + 1)$, $x = 2$ has multiplicity 3 and $x = -1$ has multiplicity 1. You'll see in the next lesson that it changes how the graph looks at that zero.

Complex zeros pair up. If the coefficients are real and $a + bi$ is a zero, so is its conjugate $a - bi$. That's why a cubic always has at least one real zero: it has three zeros in all, the complex ones come in pairs, so at least one is left over and has to be real.

Building a polynomial from its zeros is the easy direction: multiply the factors $(x - r)$ together.

The zeros of a system's characteristic polynomial decide whether a bridge oscillates, a circuit rings or a control system is stable.