## Calculus AB and Calculus BC

## CHAPTER 1 Functions

### C. POLYNOMIAL AND OTHER RATIONAL FUNCTIONS

**C1. Polynomial Functions.**

A *polynomial function* is of the form

*f* (*x*) = *a*_{0} *x ^{n}* +

*a*

_{1}

*x*

^{n}^{− 1}+

*a*

_{2}

*x*

^{n}^{− 2}+ · · · +

*a*

_{n}_{− 1}

*x*+

*a*,

_{n}where *n* is a positive integer or zero, and the *a _{k}*’s, the

*coefficients*, are constants. If

*a*

_{0}≠ 0, the degree of the polynomial is

*n*.

A *linear* function, *f* (*x*) = *mx* + *b*, is of the first degree; its graph is a straight line with slope *m*, the constant rate of change of *f* (*x*) (or *y*) with respect to *x*, and *b* is the line’s *y*-intercept.

A *quadratic* function, *f* (*x*) = *ax*^{2} + *bx* + *c*, has degree 2; its graph is a parabola that opens up if *a* > 0, down if *a* < 0, and whose axis is the line

A *cubic, f* (*x*) = *a*_{0} *x*^{3} + *a*_{1} *x*^{2} + *a*_{2} *x* + *a*_{3}, has degree 3; calculus enables us to sketch its graph easily; and so on. The domain of every polynomial is the set of all reals.

**C2. Rational Functions.**

A *rational function* is of the form

where *P*(*x*) and *Q*(*x*) are polynomials. The domain of *f* is the set of all reals for which *Q*(*x*) ≠ 0.