Chapter 4. Polynomial and Rational Functions

4.1 Polynomial Functions and Their Graphs

A polynomial function of degree n is a function of the form

P(x) = anxn + an−1x

n−1 + … + a2x

2 + a1x + a0

Where a’s are constants, an ≠ 0; n is a nonnegative integer.

The number a0 is the constant coefficient, or the constant term. Note that a

polynomial can be of degree zero: it is just a constant function P(x) = a0.

The number an, the coefficient of the highest power term, is called the

leading coefficient. The term anxn is the leading term. (It doesn’t matter

whether the term is actually written first, last, or anywhere in between.)

End Behavior (What happens when x become extremely large or small?)

Notation: “x → ∞”, reads “as x approaches infinity”, means x becomes

(very) large in the positive direction. Similarly, “x → −∞”, reads “as x

approaches negative infinity”, means x becomes (very) large in the negative

direction.

The end behavior of each particular polynomial function depends only on its

leading term (both the degree and the leading coefficient):

When n is even:

If an > 0, then y → ∞ as x → ∞ and x → −∞.

If an < 0, then y → −∞ as x → ∞ and x → −∞.

When n is odd:

If an > 0, then y → ∞ as x → ∞, and y → −∞ as x → −∞.

If an < 0, then y → −∞ as x → ∞, and y → ∞ as x → −∞.

Real Zeros of Polynomials

If P is a polynomial and c is a real number, then the following properties are

equivalent (i.e., either they are all true, or none of them is true).

1. c is a zero of P.

2. x = c is a solution of the equation P(x) = 0.

3. x − c is a factor of P(x).

4. x = c is an x-intercept of the graph of P(x).

Behavior Near an x-intercept / Shape of the Graph Near a Zero

The behavior of a polynomial’s graph near each of its zeros, c, depends on

the multiplicity of that particular root (that is, the number of times c repeats

as a root, or, equivalently, the number of time (x − c) appear in the

factorization of the polynomial). If c is a root of multiplicity m, then the

graph takes the general shape of the graph of y = (x − c)m

near c. To wit

m = 1, “the usual”.

m is even, the curve behaves similar to a parabola/

m > 1, m is odd, the curve behaves similar to the graph of y = x3 near

the origin.

Intermediate Value Theorem for Polynomials

If P is a polynomial function and P(a) and P(b) have opposite signs, then

there exists at least one value c between a and b for which P(c) = 0.

In other words, between a positive point and a negative point on the graph of

a polynomial, there must be at least one root / x-intercept.

Comment: This really is just a special case of the general Intermediate

Value property, which is possessed by every continuous function.

An important consequence, for our purpose, of this theorem is that the curve

of a polynomial’s graph always resides on one side of the x-axis between its

x-intercepts.

Polynomial functions are very easy and quick to graph by hand, especially if

all the roots of a polynomial are real numbers.

Guidelines for Graphing Polynomial Functions

Locating Zeros Find all real roots of the polynomial, they are the x-

intercepts.

Test Points Break up the real line into intervals using the real roots as

endpoints. Test a point from each interval to determine if

the graph is above or below the x-axis.

End Behavior Determined the end behavior as x → ∞ and x → −∞.

Graph Sketch a smooth continuous curve that obeys the end

behavior and that passes through each zero exhibiting the

correct behavior according to the zero’s multiplicity.

Example: P(x) = x3 − 2x

2

Example: P(x) = (x + 2)(x − 1)2(x − 4)

Example: P(x) = x3(x − .01)

2

Example: P(x) = x3(x

2 − .01)

4.2 Dividing Polynomials

Division Algorithm

Suppose P(x) and D(x) are polynomials, with D(x) ≠ 0, then there exist

unique polynomials Q(x) and R(x) such that

P(x) = D(x) Q(x) + R(x).

The remainder, R(x), is either 0 or is a polynomial with degree strictly less

than the degree of the divisor, D(x).

Equivalently, it says that the following equality:

)(

)()(

)(

)()()(

)(

)(

xD

xRxQ

xD

xRxQxD

xD

xP+=

+=

How to Divide Polynomials: Polynomial Long Division

Example: (#11) (2x4 − x

3 + 9x

2) ÷ (x

2 + 4)

Example: (#20) Find the quotient and remainder of

864

13722

45

+−

−−

xx

xx

Synthetic Division

When it works, it is faster than long division. But it works only when the

divisor is in the form x − c.

Remainder Theorem

If the polynomial P(x) is divided by (x − c), then the remainder is the value

P(c).

Factor Theorem

c is a zero of polynomial P if and only if (x − c) is a factor of P(x).

Example: Given that x = 1 is a root, completely factor x3 − 6x

2 + 11x − 6.

4.5 Rational Functions

A rational function is a function of the form

)(

)()(

xQ

xPxr =

where P and Q are polynomials.

A Simple (the Simplest?) Rational Function

x

xf1

)( =

[See next page]

Indeed, using transformations we have learned, any rational function of the

form dcx

baxxr

+

+=)( can be obtained from the graph of

xxf

1)( = .

Example: 2

4)(

+=x

xr

Graph of x

y1

= .

Domain: All real numbers except 0.

Range: All real numbers except 0.

Vertical asymptote: x = 0.

Horizontal asymptote: y = 0.

If a ≠ 0, the expression can be simplified by polynomial division first.

Example: 1

52)(

−

−=x

xxr

Asymptotes of Rational Functions

A rational function has a vertical asymptote at each zero of its denominator,

after simplification to cancel out any common factor(s) shared by its

numerator and denominator. For an example of what happens when there is

an un-cancelled common factor, see the graph of y = (x2 − 1)/(x − 1):

There are as many vertical asymptotes as the denominator has distinct real

roots, if any. Complex roots do not result in vertical asymptotes.

The horizontal asymptote occurs when y approaches a finite value as x

approaches ±∞. No function could have more than 2 horizontal asymptotes

(one in each direction, if exist and if different). A rational function,

however, can have at most one horizontal asymptote.

In general, the asymptotes of rational functions can be summarized in the

following set of rules (you will learn about them in the calculus class):

Let )(

)()(

xQ

xPxr = be a rational function where P(x) is a polynomial of

degree m and Q(x) is a polynomial of degree n. Further, assume that r(x)

has been simplified such that P and Q share no common factors other than

perhaps a constant.

Vertical Asymptote(s):

The vertical asymptotes are the lines x = a, where a is a zero of the

denominator Q(x).

Horizontal Asymptote:

(i.) If m < n, then r(x) has horizontal asymptote y = 0.

(ii.) If m = n, then r(x) has horizontal asymptote b

ay =

where a and b are, respectively, the leading coefficients of P(x) and Q(x).

(iii.) If m > n, then r(x) has no horizontal asymptote.

Guidelines for Graphing Rational Functions

Factor Factor both the numerator and denominator.

Intercepts Find all real roots of the numerator, they are the x-

intercepts. The y-intercept is r(0).

Vertical Asymp. The zeros of the denominators are the vertical

asymptotes. Plot test points to determine the behavior of

the curve (whether y → ∞ or y → −∞) on either side of

each asymptote.

Horiz. Asymp. Determined the end behavior as x → ∞ and x → −∞, to

find the existence and location of the horizontal

asymptote.

Graph Sketch the graph using the information gathered in the

previous steps. Plot additional points as needed to fill in

the detail.

Example: 32

2)(

2−+

=xx

xxr

Example: 65

43)(

2

2

−+

−−=

xx

xxxr

Graph of 32

43)(

2

2

−+

+−=

xx

xxxr :

The graph has no x-intercepts. What is the mathematical significance of this

fact?

Note that the curve crosses over the horizontal asymptote just right of x = 1.

While the graph of a function can never cross a vertical asymptote, it can

and does cross a horizontal asymptote. (Recall that the horizontal asymptote

is determined only by the behavior of the function as x → ∞ and/or x →

−∞. Therefore, the behavior of the function elsewhere has no bearing on the

existence of a horizontal asymptote.)

Slant Asymptotes

Asymptotes need not to be vertical or horizontal. For example the ones

possessed by the hyperbola x2 − y

2 = 1. An asymptote that is neither

vertical nor horizontal is called a slant asymptote (or oblique asymptote). For

rational functions, it exists on the graph whenever the degree of the

numerator is exactly one higher than the degree of the denominator.

Example: x

xxxf

12)(

2++

=

Dividing x2 + 2x + 1 by x, we get

xxxf

1)2()( ++= . Therefore, the

graph has an oblique asymptote y = x + 2. It also has a vertical asymptote x

= 0. Its graph: