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 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 (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 ri is a root of the polynomial f (x) if and only if  f (x) is divisible by (x - ri).
Therefore, a polynomial and/or polynomial function with real coefficients can be expressed as a product of its leading coefficient an and n linear factors of the form (x - ri), where ri denotes its real root and/or complex root,
f (x) = anxn + an-1xn-1 + . . . + a1x + a0 = an(x - r1)(x - r2) . . . (x - rn).
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 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 = 0  or   y0 = 0,  we get   y = a1x,  the source linear function.
Quadratic function and equation
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,  we get 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.
College algebra contents C
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