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Unformatted text preview: Polynomial Functions
Section 4.1 Defn. of a Polynomial Function
Defn. A polynomial function is a function whose rule
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
with an ≠ 0
with are real numbers
and n is a nonnegative number. A polynomial that consists of only a constant
term is called a constant polynomial.
constant The zero polynomial is the constant
polynomial The exponent of the highest power of x that
appears with nonzero coefficient is the
degree of the polynomial.
degree The nonzero coefficient of the highest power
of the variable is the leading coefficient.
leading First-degree polynomials are called linear
functions. Second-degree polynomial functions are
called quadratic functions.
quadratic Third-degree polynomial functions are called
cubic Fourth-degree polynomial functions are
called quartic functions.
quartic The Division Algorithm
The If a polynomial f(x) is divided by a nonzero
polynomial h(x) then there is a quotient
polynomial q(x) and a remainder polynomial
r(x) such that
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
quotient are factors of the dividend.
quotient If a polynomial f(x) is divided by x - c, then
the remainder is f(c).
f(c). A polynomial function f(c) has a linear factor
x – a iff f(a) = 0.
f(a) Zeros, x-intercepts, solutions,
and Let f(x) be a polynomial. If r is a real number
that satisfies any of the following statements,
then r satisfies all the statements.
r is a zero of the function f
r is an x-intercept of the graph of f
x = r is a solution, or root, of the equation f(x)
x – r is a factor of f Number of Zeros
Number A polynomial of degree n has at most n
distinct real zeros.
distinct Joke Time
Joke What would you say if
someone took your playing
dec-a-gon What kind of insect is good
an account-ant ...
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- Winter '11