

Polynomial and/or Polynomial
Functions and Equations 
Roots or zeros of
polynomial function 
Graphing
polynomial functions 
Zero polynomial 
Constant function 
Linear function 
Quadratic function
and equation 





Roots or zeros of
polynomial function 
The
zeros of a polynomial function are the values of x
for which the function equals zero. 
That
is, the solutions of the equation f
(x)
= 0,
that are called roots of the polynomial, are the zeros of the
polynomial function or the xintercepts of its
graph in a coordinate plane. 
At
these points the graph of the polynomial function cuts or
touches the xaxis. 
If
the graph of a polynomial intersects with the xaxis
at (r,
0), or x
= r
is a root or zero of a polynomial, then (x

r)
is a factor of that polynomial.

Every
polynomial of degree n
has exactly n
real and/or complex zeros. 
An
nth
degree polynomial has at most n
real zeros. 
Some
of the roots may be repeated. The number of times a root is
repeated is called multiplicity or
order of the root. 
The
number r_{i}
is a root of the
polynomial f
(x)
if and only if
f (x)
is divisible by
(x

r_{i}).

Therefore,
a polynomial and/or polynomial
function with
real coefficients can be expressed as a product of its leading
coefficient a_{n
}and
n
linear factors of the form (x
 r_{i}),
where r_{i}_{
}denotes its real root and/or complex root, 
f
(x)
= a_{n}x^{n}
+ a_{n}_{}_{1}x^{n}^{}^{1}
+
.
. .
+
a_{1}x
+ a_{0}
= a_{n}(x
 r_{1})(x
 r_{2})
.
. . (x
 r_{n}).

Thus, finding the roots of a
polynomial f(x)
is equivalent to finding its
linear divisors or is equivalent to polynomial
factorization into
linear factors. 

Graphing
polynomial functions 
Polynomial
functions are named in accordance to their degree. 
Zero polynomial 
The
constant polynomial f
(x)
= 0
is called the zero polynomial and
is graphically represented by the xaxis. 
Constant function 
A
polynomial of degree 0, f
(x)
= a_{0},
is called a constant function,
its
graph is a horizontal line with yintercept
a_{0}. 


Linear function 
The
polynomial function of the first degree, f
(x)
= a_{1}x
+ a_{0}, is called a linear function. 

Since
y
= f (x) then

y
= a_{1}x
+ a_{0}
or y
= a_{1}(x

x_{0})
or y

y_{0}
= a_{1}x, 


where 

is the slope
of the linear function, 

and
where 

are the coordinates
of translations of the linear function. 


by
setting x_{0}
= 0 or
y_{0}
= 0, we
get y
= a_{1}x,
the source linear function. 

Quadratic function
and equation 
The
polynomial function of the second degree, f
(x)
=
a_{2}x^{2}
+ a_{1}x
+ a_{0}, is called a
quadratic function. 


y
= f (x)
=
a_{2}x^{2}
+ a_{1}x
+ a_{0} or
y

y_{0}
= a_{2}(x

x_{0})^{2}, 
where 


coordinates
of translations of the quadratic function. 

By
setting x_{0}
=
y_{0}
= 0, we get 
y
=
a_{2}x^{2},
the source quadratic function.
The turning point V
(x_{0},
y_{0}). 

The
real zeros of the
quadratic function: 



y
= f (x) =
a_{2}x^{2}
+ a_{1}x
+ a_{0} = a_{2}(x
 x_{1})(x
 x_{2})
= a_{2}[x^{2}

(x_{1} +
x_{2})x
+
x_{1}x_{2}] 

The graph of a quadratic function is curve called a parabola. The parabola is symmetric with respect to a vertical line
called the axis of symmetry. 
As
the axis of symmetry passes through the vertex of the parabola
its equation is x
= x_{0}. 








College
algebra contents C




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