Home / Courses / Precalculus / Unit 2

Precalculus Unit 2: Polynomial functions

Polynomial functions are smooth and continuous. Their degree controls end behavior and the number of zeros and turning points.

Covers: Zeros, multiplicity, end behavior, complex zeros, polynomial inequalities

Study guide

A polynomial function is a sum of power terms with whole-number exponents. Its degree and leading coefficient determine end behavior: even degree sends both ends the same direction, odd degree sends them opposite directions, and a positive leading coefficient makes the right end rise. A degree-n polynomial has at most n − 1 turning points.

Zeros are inputs where the polynomial equals 0. Each factor (x − a) gives a real zero at a, and if it appears k times that zero has multiplicity k: the graph crosses the axis at odd multiplicity and bounces at even multiplicity. The Fundamental Theorem of Algebra says a degree-n polynomial has exactly n complex zeros counting multiplicity. Non-real zeros of polynomials with real coefficients come in conjugate pairs a ± bi, so if 2 + 3i is a zero, so is 2 − 3i.

Between consecutive real zeros, a polynomial keeps the same sign. That's the key to polynomial inequalities: move everything to one side, find the zeros, mark them on a number line to create intervals, test one value in each interval (or use multiplicity to track sign changes), and select the intervals that satisfy the inequality. Include the zeros themselves for ≤ or ≥.

End behavior is written with limits: lim(x→∞) f(x) = ∞ means the outputs grow without bound as x increases. A polynomial's end behavior matches its leading term, so for −2x³, the limit as x → ∞ is −∞ and the limit as x → −∞ is ∞. Complex numbers a + bi follow ordinary algebra with i² = −1: add like parts, and multiply by distributing. To divide, multiply the numerator and denominator by the conjugate of the denominator, a − bi, which makes the denominator the real number a² + b².

Long or synthetic division rewrites p(x)/(x − c) as a quotient plus a remainder. The remainder equals p(c), so x − c is a factor exactly when p(c) = 0.

Key ideas

• A degree-n polynomial has at most n real zeros and at most n − 1 turning points.
• Fundamental Theorem of Algebra: exactly n complex zeros, counting multiplicity.
• Complex zeros of real polynomials come in conjugate pairs a ± bi.
• Odd multiplicity crosses the axis; even multiplicity bounces.
• Solve polynomial inequalities with a sign chart built from the zeros.

Formulas and rules

Common mistakes

• Forgetting complex zeros when counting.
• Assuming the number of turning points equals the degree.

Words to know

Turning points

A degree-n polynomial has at most n − 1.

Fundamental Theorem of Algebra

A degree-n polynomial has exactly n complex zeros, counting multiplicity.

Conjugate root theorem

Complex zeros of real polynomials come in pairs a ± bi.

Even multiplicity zero on a graph

The graph touches the x-axis and bounces back.

End behavior of an even-degree polynomial with positive leading coefficient

Up on both ends.

Polynomial function

A sum of power terms with whole-number exponents.

Leading coefficient

The coefficient of the highest-degree term.

Multiplicity

How many times a zero repeats; odd crosses, even bounces.

Complex zero

A zero of the form a + bi.

Polynomial inequality

Find zeros, test intervals, keep those that satisfy the inequality.

Sign chart

A number line showing where a function is positive or negative.

Sample practice questions

21 more questions like these, from level 1 to 5, are in the practice set.