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 x-intercepts of its graph in a coordinate plane.
At these points the graph of the polynomial function cuts or touches the x-axis.
If the graph of a polynomial intersects with the x-axis 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 xi is a root of the polynomial f (x) if and only if f (x) is divisible by (x - xi).
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 x-axis.
Constant function
A polynomial of degree 0,  f (x) = a0, is called a constant function, its graph is a horizontal line with y-intercept a0.
Linear function
The polynomial function of the first degree,  f (x) = a1x + a0, is called a linear function.
Since  y = f (xthen  y = a1x + a0   or   y = a1(x - x0)   or    y - y0 = a1x,  
where is the slope of the linear function,
and where are the coordinates of translations of the linear function.   
by setting   x0 = y0 = 0,   follows y = a1x,  the source linear function.  
Quadratic function
The polynomial function of the second degree,  f (x) = a2x2 + a1x + a0 is called a quadratic function.
   y = f(x = a2x2 + a1x + a0   or   y - y0 = a2(x - x0)2,   where
 coordinates of translations of the quadratic function. 
by setting   x0 = y0 = 0,   follows y = a2x2,  the source quadratic function.  The turning point  V(x0, y0).
The real zeros of the quadratic function:  
   y = f(x= a2x2 + a1x + a0  = a2(x - x1)(x - x2) = a2[x2 - (x1 + x2)x + x1x2]  
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 = x0.
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 x-axis in two distinct points if its leading coefficient a2 and the vertical translation y0 have different signs, i.e., if   a2 y0 < 0.
Example:  Find zeros and vertex of the quadratic function  y = - x2 + 2x + 3  and sketch its graph.
Solution:  A quadratic function can be rewritten into translatable form  y - y0 = a2(x - x0)2  by completing the square,
      y = - x2 + 2x + 3   Since a2 y0 < 0 given quadratic function must have two different real zeros.
      y = - (x2 - 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 - y0 = a2(x - x0)2                      x - 1 = sqrt(4)
V(x0, y0)  =>   V(1, 4)                        x1,2 = 1 2,   =>   x1 = - 1 and  x2 = 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.
1)  calculate the coordinates of translations of the quadratic  y = f (x= - x2 + 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 + y0 = a2(x + x0)2 + a1(x + x0) + a0   =>   y + 4 = - (x + 1)2 + 2(x + 1) + 3
                                                                                              y = - x2   the source quadratic function
3)  Inversely, by plugging the coordinates of translations into the source quadratic function
y - y0 = a2(x - x0)2   =>     y - 4 = - (x - 1)2
                    obtained is given quadratic in general form     y = - x2 + 2x + 3.
Intermediate algebra contents
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