Topic 1.5 Polynomial Functions and Complex Zeros: relate the real and non-real complex zeros of a polynomial to its factored form, degree and graph, including the effect of multiplicity.

What this topic is asking

The College Board (Topic 1.5) wants you to connect a polynomial's zeros, both real and non-real complex, to its factored form, its degree, and the shape of its graph. You must use multiplicity to predict whether the graph crosses or touches the $x$-axis at each real zero, and you must know that non-real complex zeros come in conjugate pairs.

Zeros and factors

So the factored form of a polynomial is a product of linear factors $(x - r)$, one for each zero (with repeats for multiplicity), times the leading coefficient.

Multiplicity and graph behavior

This is the single most examined fact in the topic: even multiplicity means a touch, odd multiplicity means a cross. You can read multiplicities directly from a graph by watching how the curve meets each intercept.

Non-real complex zeros come in pairs

Because each non-real pair contributes a quadratic factor with no real roots, complex zeros never appear as $x$-intercepts. A polynomial with all real coefficients and an odd degree must therefore have at least one real zero, since complex zeros pair up and cannot account for an odd count.

Reading zeros from a graph and back

The exam moves freely between three views: a graph (where real zeros are intercepts and multiplicities show as crosses or bounces), a factored formula, and a list of zeros. To go from a graph to a formula, read each $x$-intercept and decide its multiplicity from the cross-or-touch behavior, then multiply the corresponding factors and fix the leading coefficient using one extra known point. To go from a formula to a graph, expand the factored form into intercepts and end behavior.

A point that catches students is that the degree counts zeros with multiplicity and includes complex zeros, so a degree-$4$ polynomial showing only two $x$-intercepts is entirely normal: the other two zeros may be a complex conjugate pair, or one intercept may have multiplicity $2$. Reconciling the visible intercepts with the stated degree, by accounting for multiplicity and hidden complex pairs, is the reasoning that ties this topic together and recurs whenever the exam gives partial information about a polynomial.

Try this

Q1. A polynomial has the factor $(x + 5)^2$. Does its graph cross or touch the axis at $x = -5$? [1 point]

• Cue. Multiplicity $2$ is even, so the graph touches and turns back at $x = -5$.

Q2. A degree-$5$ polynomial with real coefficients has zeros $1$, $2i$ and $-2i$. How many more real zeros must it have, counted with multiplicity? [1 point]

• Cue. Two complex zeros plus one real zero account for $3$; degree $5$ needs $2$ more zeros, and since complex ones pair up the remaining ones can be real (or another complex pair). At least the real count must total an odd number, so there is at least one more real zero.