1.2 · Polynomial zeros, multiplicities, and end behavior

The leading term controls distant behavior

For large $\displaystyle |x|$, a polynomial behaves like its highest-degree term. An even degree with positive leading coefficient rises at both ends; an odd degree with positive leading coefficient falls to the left and rises to the right.

Factored form makes zeros visible, while expanded form makes degree and leading coefficient visible. Choose the form that answers the question rather than expanding automatically.

Multiplicity describes local behavior at a zero

A factor $\displaystyle (x-r)^m$ gives zero $\displaystyle r$ with multiplicity $\displaystyle m$. Odd multiplicity crosses the axis; even multiplicity touches and turns. Larger multiplicities flatten the curve near the intercept.

The degree counts complex zeros with multiplicity. For real coefficients, nonreal zeros occur in conjugate pairs. A graph shows real intercepts, not every complex zero.

Evaluate remainders and verify factors

The remainder from division by $\displaystyle x-c$ is $\displaystyle p(c)$. Thus $\displaystyle x-c$ is a factor exactly when $\displaystyle p(c)=0$. Synthetic division is a compact form of polynomial division by a linear factor.

The Rational Zero Theorem supplies candidates, not guaranteed roots. Substitute or divide to test candidates, then solve the lower-degree factor that remains.

$$p(c)=0\quad\Longleftrightarrow\quad(x-c)\text{ is a factor}$$