Polynomial factors and graphs

For a polynomial P, P(r)=0 means r is a root and (x-r) is a factor. Dividing by (x-r) leaves remainder P(r). Example For P(x)=x3-4x2+x+6, P(2)=0 and 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.