Unit 2
Zeros, multiplicity, and end behavior. Together these three facts determine the shape of any polynomial graph before a single point is plotted.
Lesson 2.1
A factor x - r produces a zero at r. Its exponent is the zero multiplicity. Odd multiplicity crosses the x-axis; even multiplicity touches and turns. Multiplicity helps connect factored equations to a graph without expanding the polynomial.
Lesson 2.2
The leading term controls polynomial end behavior because it eventually dominates all lower-degree terms. Even degree gives matching end directions; odd degree gives opposite directions. The sign of the leading coefficient determines the right-hand end.
Lesson 2.3
Complex numbers extend real numbers by defining i squared to be -1. Negative radicands in the quadratic formula then give complex conjugate roots. When completing a square or taking a square root, include both signs unless a domain restriction says otherwise.
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