POLYNOMIALS
Polynomials
A polynomial is a function of the form
f ( x)  an x n  an1 x n1  an2 x n2  ...  a2 x 2  a1 x  a0
where the
an , an1, ...a1, a0
are real numbers and n is a nonnegative integer.
The domain of a polynomial function is the set of
real numbers
The Degree of Polynomial Functions
The Degree, of a polynomial function in one
variable is the largest power of x
Example
Below is a polynomial of degree 2
2
f ( x)  3x  6x  2
See Page 183 for a summary of the properties of polynomials of
degree less than or equal to two
Properties of Polynomial Functions
The graph of a polynomial function is a smooth and
continuous curve
A smooth curve is one that contains no Sharp corners or
cusps
A polynomial function is continuous if its graph has
no breaks, gaps or holes
Power Functions
A power function if degree n, is a function of the form
f ( x)  axn
where a is a real number, a  0 and n > 0 is an
integer
Examples 3x 4 (degree 4),
 x 7 (degree 7) ,
1
x
2
(degree 1)
Graphs of even power functions
y  x6
y  x4
y  x2
f ( x)  axn
The polynomial function
is even if n  2 is
even. The functions graphed above are even. Note as n
gets larger the graph becomes flatter near the origin,
between (-1, 1), but increases when x > 1 and when x <
-1. As |x| gets bigger and bigger, the graph increases
rapidly.
Properties of an even function
The domain of an even function is the set of
real numbers
Even functions are symmetric with the
y-axis
The graph of an even function contains the
points (0, 0) (1,1) (-1, 1)
Graphs of odd power functions
y  x5
y  x7
y  x3

The polynomial function f ( x)  axn is odd if n  3 is odd.
The functions graphed above are odd. Note as n gets
larger the graph becomes flatter near the origin,
-1 < x <1 but increases when x > 1 or decreases when
x < -1 . As |x| gets bigger and bigger, the graph increases
for values of x greater than 1 and decreased rapidly for
values of x less than or equal to -1.
Properties of an odd function
The domain of an odd function is the set
of real numbers
Odd functions are symmetric with the
origin
The graph of an odd function contains
the points (0, 0) (1,1) (-1,-1)
Graphs of Odd functions
y  x3
y  x3
y  ( x  2) 3  4
Graphs of Even functions
f ( x)  x2
f ( x)  ( x  2)2
f ( x)  x2  3
f ( x)   x 2  3
Zeros of a polynomial function
 A real number r is a real zero of the
polynomial f (x) if f (r) =0
 If r is a zero of the polynomial, then r is
an x – intercept.
 If r is a zero of the polynomial f (x) then
f (x) = (x – r) p (x), where p (x) is a
polynomial
The intercepts of a polynomial
 If r is an x – intercept of a polynomial x, then
 f( r ) = 0
 If r is an x – intercept then either
 1. The graph crosses the x axis at r or
 2. The graph touches the x axis at r
