Family Overview

In Plain Language

A number $a$ is a zero of polynomial $p$ when $p(a)=0$. If $a$ is real, then $(x-a)$ is a factor and the graph has an $x$-intercept at $(a,0)$.

• A factor $(x-a)$ repeated $n$ times means zero $a$ has multiplicity $n$.
• A degree-$n$ polynomial has exactly $n$ complex zeros counting multiplicities.
• For real-coefficient polynomials, non-real zeros come in conjugate pairs: if $a+bi$ is a zero, so is $a-bi$.
• Even multiplicity at a real zero: graph is tangent to the $x$-axis (same sign of outputs nearby). Odd multiplicity: graph crosses the $x$-axis.
• Degree can also be read from tables: the least $n$ for which successive $n$th differences (equal input steps) are constant.

Why It Matters

Zeros drive factoring, graph intercepts, and inequality endpoints. Conjugate pairs explain why non-real zeros "use up" an even count of the degree. Multiplicity explains touch vs cross at the axis—high-frequency AP graph questions.

How Families Can Help

• Ask: "How many zeros total for this degree?"
• If they hear $2+3i$, ask: "What's the partner zero?"
• At an intercept: "Does it cross or just kiss the axis?"
• On a table with equal $x$-steps: help build difference rows until they go constant.

What Not to Do

• Do not forget to count multiplicity toward the degree total.
• Do not assume conjugate pairs if coefficients are not real.
• Do not confuse complex (non-real) zeros with "no zeros"—degree still counts them.
• Do not treat even/odd multiplicity as the same as even/odd functions.