Family Overview

In Plain Language

For a rational function $r(x)=\dfrac{p(x)}{q(x)}$, a real zero is an $x$-value where $r(x)=0$. That happens when the numerator is zero, but only if that $x$ is still in the domain (the denominator is not zero there).

So: solve $p(x)=0$, then throw out any solutions that make $q(x)=0$. The keepers are the real zeros (and $x$-intercepts) of $r$.

When solving inequalities like $r(x)\ge 0$ or $r(x)\le 0$, the real zeros of both $p$ and $q$ matter: they act as endpoints or asymptote barriers that split the number line into test intervals.

Why It Matters

Zeros locate intercepts and feed sign charts for rational inequalities—core AP Precalculus skills before vertical asymptotes (1.9) and holes (1.10). Students who list every numerator root without checking the domain invent fake zeros.

How Families Can Help

• Ask: “Does that $x$ make the bottom zero too?”
• Practice: numerator zero → candidate; denominator zero → not a zero of $r$.
• For inequalities: mark all num and den zeros, then test intervals.

What Not to Do

• Do not call a shared num/den zero a zero of $r$ without checking (often a hole, later topic).
• Do not ignore domain when listing zeros.
• Do not use only numerator zeros as the full sign-chart split set for inequalities.
• Do not confuse zeros of $r$ with vertical asymptotes.