Polynomial and/or Polynomial Functions and Equations
Polynomial functions
The source or the original polynomial function
Translating (parallel shifting) of the source polynomial function
Coordinates of translations and their role in the polynomial expression
Roots or zeros of polynomial function
Polynomial functions
The source or the original polynomial function
A polynomial  f (x) = anxn + an-1xn-1 + . . . + a1x + a0 of degree n > 1, consisting of n + 1 terms, shown
graphically, represents translation of its source (original) function in the direction of the coordinate axes.
The complete source polynomial function
 fs(x) = anxn + an-2xn-2 + . . . + a2x2 + a1x
has n - 1 terms lacking second and the constant term, since its coefficients, an-1 = 0 and a0 = 0 while the leading coefficient an, remains unchanged.
Therefore, the source polynomial function passes through the origin.
A coefficient ai of the source function is expressed by the coefficients of the general form.
A coefficient ai of the source polynomial can be calculated like coefficients of the Taylor polynomial
where  f (i ) (x0) denotes i-th derivative at x0.
Translating (parallel shifting) of the source polynomial function
Thus, to obtain the graph of a given polynomial function  f (x) we translate (parallel shift) the graph of its source function in the direction of the x-axis by x0 and in the direction of the y-axis by y0.
Inversely, to put a given graph of the polynomial function beck to the origin, we translate it in the opposite direction, by taking the values of the coordinates of translations with opposite sign.
Coordinates of translations and their role in the polynomial expression
The coordinates of translations we calculate using the formulas,
Therefore, by plugging the coordinates of translations into the source polynomial function fs(x), i.e.,
 y - y0 = an(x - x0)n + an-2(x - x0)n-2 + . . . + a2(x - x0)2 + a1(x - x0)
and by expanding above expression we get the polynomial function in the general form.
Inversely, by plugging the coordinates of translations into a given polynomial function  f (x), that is expressed in the general form, i.e.,
 y + y0 = an(x + x0)n + an-1(x + x0)n-1 + . . . + a1(x + x0) + a0
and after expanding and reducing above expression we get its source polynomial function.
Note that in the above expression the signs of the coordinates of translations are already changed.
Therefore, each polynomial missing second term (an-1 = 0), represents a source polynomial whose graph is translated in the direction of the y-axis by  y0 = a0.
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.
Pre-calculus contents E