# 3: Polynomial and Rational Functions

- Page ID
- 34887

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In this chapter, we will learn about these concepts and discover how mathematics can be used in such applications.

- 3.1: Graphs of Quadratic Functions
- Identify features of a parabola from equations or graphs: orientation, vertex, axis of symmetry, min/max point, domain and range, intercepts. Convert between general and vertex forms by multiplication, a formula or complete the square. Construct an equation from a graph of a parabola.

- 3.2: Circles
- Graph or construct an equation for a circle. Use complete the square to write the equation of a circle in standard form. In a similar fashion, graph or construct an equation for an ellipse.

- 3.3: Power Functions and Polynomial Functions
- End Behaviour of Power Functions. Degree, Leading Term, and End Behaviour of Polynomials. Find intercepts by factoring. Identify intercepts, possible degree of polynomial and sign of leading coefficient from a graph.

- 3.4: Graphs of Polynomial Functions
- Characteristics of polynomial graphs. Identify Zeros and their multiplicity from a graph and a (factorable) equation. Turning points. Graph factorable polynomials using end behavior and multiplicity. Construct an equation from a graph.

- 3.5: Dividing Polynomials
- Polynomial long division. Synthetic division.

- 3.6: Zeros of Polynomial Functions
- Use the Remainder Theorem to evaluate a polynomial. Factor Theorem. Rational Zero Theorem. Find all rational zeros of a polynomial. Find all zeros of up to fourth degree polynomials. Write as a product of linear and irreducible quadratic factors. Upper and Lower Bound Theorem. Intermediate Value Theorem. Descartes Rule of Signs. Fundamental Theorem of Algebra. Construct a polynomial given its zeros. Complex Conjugate Theorem.

- 3.7: The Reciprocal Function
- Graph 1/x and 1/x^2 and translations of those graphs. Use polynomial division to rewrite a rational function with linear numerator and denominator.

- 3.8: Polynomial and Rational Inequalities
- Solve polynomial and rational inequalities by using a sign chart.

## Contributors

Jay Abramson (Arizona State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at https://openstax.org/details/books/precalculus.