The Zeros You See, and the Ones You Cannot

You'll learnto account for every zero of a polynomial, including the ones the graph never shows.

A degree-four polynomial has exactly four zeros. Its graph may show two. Nothing has gone wrong: multiplicity hides some of them on top of each other, and conjugate pairs hide the rest off the real line entirely. This is the dictionary that keeps the count at four.

8 STEPS · 6 QUICK CHECKS · ZEROS · FACTORS · MULTIPLICITY · CONJUGATE PAIRS · v2

zero ⇄ factor ⇄ intercept

Before you start

What you're looking at

A degree-four graph that crosses the x-axis only twice, with a running count of the zeros still unaccounted for.

The question

If the degree guarantees four zeros but you can see only two crossings, where are the other two hiding?

Watch for

Two hiding places: repeated zeros stacked at one point, and non-real zeros that never touch the real graph.

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