Polynomial Behavior & Degree

Calculus Readiness · Axiom Academy

How a polynomial's degree, leading coefficient, and root multiplicities completely determine the shape of its graph. The degree n is the highest power of x . It is a hard ceiling on the graph's complexity: a degree- n polynomial has at most n real roots (places it crosses the x -axis) and at most n-1 turning points (local peaks and valleys). Watch the curve climb from a line to a quartic — each bump in degree buys exactly one more root and one more turn. at most n roots, at most n-1 turning points Far from the origin, the leading term dominates — every lower term is dust by comparison. So only two facts decide where the arms go: the parity of the degree (even or odd) and the sign of the leading coefficient . Watch the same wiggly middle while the leading term cycles through all four cases and flips the arms. Both arms rise. — like y=x^2 scaled up. Both arms fall. — the parabola flipped. Down on the left, up on the right. — like y=x^3 . Up on the left, down on the right. . For p(x)=2x^3-100x , at x=50 the leading term is 2(50)^3=250 , 000 while -100x=-5 , 000 . The cube has already swamped everything — and the gap only widens. That is why end behavior ignores all but the highest power. 3. Multiplicity: Cross or Touch

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