Polynomial functions: ends, zeros and terms
Read the behaviour at the ends from degree and leading coefficient, tell simple from double zeros, solve a biquadratic equation and find a term from a graph.
- Germany: grade 10
- Ukraine: grade 8
Guide
A polynomial function f(x) = anxn + … + a0 behaves like its leading term anxn for large |x|. Even degree: both ends go the same way; odd degree: the ends go opposite ways. The sign of an says whether the right end goes up or down.
If (x − x0) appears twice as a factor, x_0 is a double zero: the graph touches the x-axis and does not cross it. A simple zero is crossed.
An equation like x4 − 5x2 + 4 = 0 is biquadratic. Substitute z = x2, solve z2 − 5z + 4 = 0, then go back: x2 = z.
With the zeros known, write f(x) = a·(x − x1)(x − x2)… and find a from one more point.