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Unformatted text preview: Polynomial Functions
Polynomial
Section 4.1 Defn. of a Polynomial Function
Defn. A polynomial function is a function whose rule
polynomial
is given by a polynomial
is f ( x) = an x n + an −1 x n −1 + L + a1 x + a 0
an , an −1 ,K , a1 , a0 where
where
with an ≠ 0
with are real numbers
and n is a nonnegative number. A polynomial that consists of only a constant
polynomial
term is called a constant polynomial.
constant The zero polynomial is the constant
The zero
polynomial 0.
polynomial The exponent of the highest power of x that
The
appears with nonzero coefficient is the
degree of the polynomial.
degree The nonzero coefficient of the highest power
The
of the variable is the leading coefficient.
leading Firstdegree polynomials are called linear
Firstdegree
functions.
functions. Seconddegree polynomial functions are
Seconddegree
called quadratic functions.
quadratic Thirddegree polynomial functions are called
Thirddegree
cubic functions.
cubic Fourthdegree polynomial functions are
Fourthdegree
called quartic functions.
quartic The Division Algorithm
The If a polynomial f(x) is divided by a nonzero
If
f(x)
polynomial h(x) then there is a quotient
h(x)
polynomial q(x) and a remainder polynomial
q(x and
r(x) such that
r(x)
Dividend f ( x ) = h( x ) • q ( x ) + r ( x )
Divisor Quotient Remainder where r(x)=0 or r(x) has degree less than the
degree of the divisor, h(x).
h(x). Remainders and Factors
Remainders If the remainder is 0, the divisor and the
If
quotient are factors of the dividend.
quotient If a polynomial f(x) is divided by x  c, then
If
f(x)
then
the remainder is f(c).
f(c). A polynomial function f(c) has a linear factor
polynomial
x – a iff f(a) = 0.
f(a) Zeros, xintercepts, solutions,
and factors
and Let f(x) be a polynomial. If r is a real number
f(x)
that satisfies any of the following statements,
then r satisfies all the statements.
satisfies
r is a zero of the function f
r is an xintercept of the graph of f
intercept
x = r is a solution, or root, of the equation f(x)
=0
x – r is a factor of f Number of Zeros
Number A polynomial of degree n has at most n
polynomial
distinct real zeros.
distinct Joke Time
Joke What would you say if
What
someone took your playing
cards?
cards?
decagon What kind of insect is good
at math?
at
an accountant ...
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 Winter '11
 stuff
 Calculus

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