Unit 2 · Core unit
Zeros, multiplicity, and end behavior. Together these three facts determine the shape of any polynomial graph before a single point is plotted.
What a strong answer looks like
A strong answer reads the graph from the factored form and the leading term rather than from plotted points.
Know
A zero of even multiplicity touches the axis and turns; odd multiplicity crosses. Multiplicity also flattens the approach.
Apply
Factor completely, then read crossing behavior from each exponent.
Watch out
Counting distinct zeros as the degree, which misses repeated factors.
Study move
Sketch a polynomial from its factored form alone.
Know
Degree and the sign of the leading coefficient fully determine both ends. Nothing else matters at the extremes.
Apply
Identify degree and leading coefficient, then state both ends.
Watch out
Letting a large constant term influence an end-behavior answer.
Study move
State end behavior for four polynomials without graphing.
Know
A negative discriminant gives a conjugate pair of complex solutions, which correspond to a graph that does not meet the axis.
Apply
Use the discriminant to predict whether real zeros exist before solving.
Watch out
Saying an equation has no solutions when it has no real ones.
Study move
Solve a quadratic with a negative discriminant and describe its graph.
Modeling drill
Given a degree-4 polynomial in factored form with one repeated zero, sketch it and describe both ends.
