Zeros and Multiplicity

Pre-Calculus · Axiom Academy

Whether a polynomial crosses the x-axis or just touches it is decided by one number: the multiplicity of the zero. 1. A Zero Is Where the Graph Meets the Axis Read the factored form like a list of meeting points. Every factor (x-r) forces p(r)=0 , so the curve must touch the x -axis at x=r . Watch the tracer sweep across p(x)=(x+2)(x-1)^2(x-3)^3 : each time it reaches a factor's root, a marker drops onto the axis. Those marks — at x=-2 , x=1 , x=3 — are the zeros. A factor of (x-r) means r is a zero zero = root of p(x)=0 = x -intercept Zoom in on two of the zeros. At x=-2 the factor (x+2) appears once — an odd multiplicity — and the curve drives straight through the axis, swapping sign. At x=1 the factor (x-1)^2 appears twice — an even multiplicity — and the curve only touches the axis and bounces back, keeping the same sign. The two tracers below come from the same p(x) . (x+2)^1 at x=-2 . The sign of p(x) flips from one side to the other, so the graph passes through the axis. (x-1)^2 at x=1 . The sign of p(x) is the same on both sides, so the graph kisses the axis and turns back. Near a zero r , . An odd power changes sign across r ; an even power stays positive — a perfect square is never negative. Odd multiplicity cross. Even multiplicity touch and bounce. The exponent is the whole story.

This is the written version of the interactive lesson above. See the full Pre-Calculus course.