Polynomial factors and graphs

For a polynomial \(\mathrm{P}\), \(\mathrm{P}(r)=0\) means \(r\) is a root and \((x-r)\) is a factor. Dividing by \((x-r)\) leaves remainder \(\mathrm{P}(r)\).

Example

For \(\mathrm{P}(x)=x^3-4x^2+x+6\), \(\mathrm{P}(2)=0\) and \(\mathrm{P}(x)=(x-2)(x-3)(x+1)\). Its zeros are \(-1,2,3\), its \(y\)-intercept is \(6\), and its ends follow the positive cubic leading term.

An odd-multiplicity real root crosses the axis; an even-multiplicity root touches it without a sign change. For \((x+1)(x-2)^2\), the graph crosses at \(-1\) and touches at \(2\).

Use algebra for exact factors when feasible and graphing technology to locate additional features or roots when an analytic route is unsuitable.