

Polynomial and/or Polynomial
Functions and Equations 
Zeros of
a polynomial function 
Graphing
polynomial functions 
Zero polynomial 
Constant function 
Linear function 
Quadratic function 
Transformations
of the graph of the quadratic function 





Zeros
of a 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 (a,
0), or x
= a
is a root or zero of a polynomial, then
(x

a)
is a factor of that polynomial. 
Every
polynomial of degree n
has exactly n
real 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
x_{i}
is a root of the
polynomial f
(x)
if and only if
f (x)
is divisible by
(x

x_{i}).

Thus, finding the
roots of a
polynomial f
(x)
s 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}
=
y_{0}
= 0,
follows 
y
= a_{1}x,
the source linear function. 



Quadratic function 
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,
follows 
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}. 

Transformations
of the graph of the quadratic function 
How
changes in the expression of the quadratic function affect its
graph is shown in the figures below. 




The graph of quadratic polynomial will intersect the
xaxis in two
distinct points if its leading coefficient
a_{2} and the vertical translation
y_{0}
have different signs, i.e., if a_{2}
· y_{0
}<
0. 


Example:
Find zeros and vertex of
the quadratic function y
=

x^{2}
+ 2x
+
3
and sketch its graph. 
Solution:
A quadratic function can
be rewritten into translatable form y

y_{0}
= a_{2}(x

x_{0})^{2}
by completing the square, 
y
=

x^{2}
+ 2x
+ 3 
Since a_{2}
· y_{0
}<
0 given
quadratic function must have two different real zeros. 
y
=

(x^{2}

2x)
+ 3 
To find zeros of a function, we set
y
equal to zero and solve for x.
Thus, 
y
=

[(x

1)^{2}

1]
+ 3 

4
=

(x

1)^{2} 
y 
4
=

(x

1)^{2} 
(x

1)^{2}
=
4 
y

y_{0}
= a_{2}(x

x_{0})^{2} 
x

1
=
±
sqrt(4) 
V(x_{0},
y_{0})
=> V(1,
4) 
x_{1,2}
=
1
±
2, => x_{1}
= 
1
and x_{2}
= 3. 


We
can deal with given quadratic using the property of the
polynomial explored under the title, 
'Transformations of the polynomial function
applied to the quadratic and cubic functions' above. 
Thus, 
1)
calculate the coordinates of translations of the quadratic
y
=
f (x) =

x^{2}
+ 2x
+
3 

2)
To
get the source quadratic function, plug the coordinates
of translations (with changed signs) 
into the general form
of the quadratic, i.e., 
y
+ y_{0}
= a_{2}(x
+ x_{0})^{2}
+ a_{1}(x
+ x_{0})
+ a_{0}
=>
y
+ 4
= 
(x
+ 1)^{2}
+ 2(x
+ 1)
+
3 
y
=

x^{2
}the source quadratic function 
3)
Inversely, by plugging the coordinates of translations into the source quadratic function 
y
 y_{0}
= a_{2}(x
 x_{0})^{2}
=>
y
 4
= 
(x 
1)^{2}

obtained is given quadratic in general form
y
=

x^{2}
+ 2x
+
3. 








Intermediate
algebra contents 



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