Mathematics 116 Chapter 4 Bittinger

• Polynomial

• and

• Rational Functions

Newt Gingrich

• “Perseverance is the hard work you do after you get tired of doing the hard work you already did.”

Definition of a Polynomial Function

• Polynomial function of x with degree n.

11

2 2 11 0

( ) n nn nf x a x a x

a x a x a

Joseph De Maistre (1753-1821 – French Philosopher

• “It is one of man’s curious idiosyncrasies to create difficulties for the pleasure of resolving them.”

Mathematics 116

• Polynomial Functions of Higher Degree

Continuous

• The graph has no breaks, holes, or gaps.

• Has only smooth rounded turns, not sharp turns

• Its graph can be drawn with pencil without lifting the pencil from the paper.

Leading Coefficient Test

• The leading term determines the “end behavior” of graphs.

• Very Important!

Objective

• Use the Leading Coefficient Test to determine the end behavior of graphs of polynomial functions.

Intermediate Value Theorem

• Informal – Find a value x = a at which a polynomial function is positive, and anther value x = b at which it is negative, the function has at least one real zero between these two values.

• Use numerical zoom with table or

• Use [CAL] [1:zero]

Real Zeros of Polynomial Functions

• x = a is a zero of function f• x = a is a solution of the polynomial

equation f(x)=0• (x-a) is a factor of the polynomial

f(x)• (a,0) is an x-intercept of the graph

of f.

Repeated Zeros• For a polynomial function, a factor

• Yields a repeated zero x = a of multiplicity k

• If k is odd, the graph crosses at x = a• If k is even, the graph touches at x=a (not

cross)

, 1k

x a k

Objective

• Find and use zeros of polynomial functions as sketching aids.

Chinese Proverb:

• “A journey of a thousand miles must begin with a single step.”

Mathematics 116

• Real Zeros

• of

• Polynomial Functions

Objective

• Use long division to divide polynomials by other polynomials.

Objective

• Use synthetic division to divide polynomials by

binomial of the form (x – k)

Reminder Theorem

• If a polynomial f(x) is divided by x – k, the reminder is r = f(k)

Factor Theorem

• A polynomial f(x) has a factor

• (x – k) if and only if f(k) = 0

Using the remainder• A reminder r obtained by dividing f(x)

by x – k

• 1. The reminder r gives the value of f at x = k that is r = f(k)

• 2. If r = 0, (x – k) is a factor of f(x)

• 3. If r = 0, the (k,0) is an x intercept of the graph of f

• 4. If r = 0, then k is a root.

Rational Roots Test

• Possible rational zeros =

• factors of constant term factors of leading coefficient

• Possible there are no rational roots.

Descarte’s Rule of Signs

• Provides information on number of positive roots and number of negative roots.

William Cullen Bryant (1794-1878) U.S. poet, editor

• “Difficulty, my brethren, is the nurse of greatness – a harsh nurse, who roughly rocks her foster-children into strength and athletic proportion.”

Mathematics 116

• The

• Fundamental Theorem

• of

• Algebra

Number of roots

• A nth degree polynomial has n roots.

• Some of these roots could be multiple roots.

Linear Factorization Theorem

• Any nth-degree polynomial can be written as the product of n linear factors.

Objective

• Use the fundamental Theorem of Algebra to determine the number of zeros (roots) of a polynomial function.

Objective

• Find all zeros of polynomial functions including complex zeros.

Conjugate Roots

• If a + bi, where b is not equal to 0 is a zero of a function f(x)

• the conjugate a – bi is also zero of the function.

John F. Kennedy

•“We must use time as a tool, not as a couch.”

Mathematics 116

• Rational Functions

• and

• Asymptotes

Rational Function

( )( )

( )

N xf x

D x

Graph – domain, range, intercepts, asymptotes

1( )f x

x

Graph – domain, range, intercepts, asymptotes

2

1( )f x

x

Asymptotes

• Vertical

• Horizontal

• Slant

Objective

• Find the domains of rational functions.

Objective

• Find horizontal and vertical asymptotes of graphs of rational functions.

Objective

• Use rational functions to model and solve real-life problems.

George S. Patton

•“Accept the challenges, so you may feel the exhilaration of victory.”

Mathematics 116

•Graphs of a Rational Function

Graphing Rational Function• 1. Simplify f if possible – reduce• 2. Evaluate f(0) for y intercept and plot• 3. Find zeros or x intercepts – set

numerator = 0 & solve• 4. Find vertical asymptotes – set

denominator = 0 and solve• 5. Find horizontal / slant asymptotes• 6. Find holes

Dan Rather

–“Courage is being afraid but going on anyhow.”

College Algebra 116

•Quadratic Inequalities

Sample Problem quadratic inequalities #1

2 2 8 0x x 2,4

Sample Problem quadric inequalities #2

26 2x x 1 2

, ,2 3

Sample Problem quadratic inequalities #3

2 6 9 0x x ,R

Sample Problem quadratic inequalities #4

40

1

x

x

( 1,4]

Sample Problem quadratic inequalities #5

3 20

2 3x x

2,3 5,

Everette Dennis – Media professor

• “There’s a compelling reason to master information and news. Clearly there will be better job and financial opportunities. Other high stakes will be missed by people if they don’t master and connect information.”