3a
 
ALGEBRA
 Polynomial functions' properties
  Every polynomial has its initial position at the origin of the coordinate system. A polynomial function written in general form represents translation of its source (original) function
 fs(x) = anxn + an - 2 xn - 2 + an - 3 xn - 3 + . . . + a3x3 + a2x2 + a1x,
in the direction of the coordinate axes, where the coordinates of translations are
Therefore, each polynomial missing second term (an - 1 = 0), represents a source polynomial function whose graph is translated in the direction of the y-axis by  y0 = a0.
1)  By plugging the coordinates of translations with changed signs into a given polynomial  y =  f (x), expressed in the general form, i.e.,
y + y0 = an(x + x0)n + an - 1(x + x0)n - 1 + . . . + a2(x + x0)2 + a1(x + x0) + a0
and after expanding and reducing the above expression we get its source function  fs(x).
2 Inversely, by plugging the coordinates of translations into the source polynomial function, i.e.,
y - y0 = an(x - x0)n + an - 2(x - x0)n - 2 + . . . + a2(x - x0)2 + a1(x - x0),
and after expanding and reducing the above expression, we get given polynomial  f (x).
Moreover, the coefficients a of the source polynomial are related to corresponding value of the derivative of the given polynomial at x0, like coefficients of the Taylor polynomial in Taylor's or Maclaurin's formula,
where,  an = anan - 1 = 0a0 =  f (x0), and  f (n - k)(x0) denotes the (n - k)-th derivative at x0.
:: Sigma notation of the polynomial function
Sigma notation of the polynomial, where the coefficients a of its source function are given by a recursive formula,
while, for k = 0, an = an,  for k = 1, an-1 = 0   and  for k = n, a0 = f (x0) = y0.
 Graphs of polynomial functions
Polynomial functions are named in accordance to their degree.
 :: Zero polynomial   f (x) = 0
The constant polynomial  f (x) = 0 is called the zero polynomial and is graphically represented by the x-axis.
 :: Constant function  f (x) = a0
A polynomial of degree 0,  f (x) = a0, is called a constant function, its graph is a horizontal line with the y-intercept a0.
 :: Linear function, the first degree polynomial  f (x) = a1x + a0
y = a1 x + a0   or   y = m x + c
or    ya1(x - x0)   or   y - y0a1x,

where  x0 - a0 /a1  and/or   y0a0  are the x-intercept and the y-intercept respectively, and where the slope of the line  a1 m,

By setting  x0 = 0  or  y0 = 0  obtained is 
the source linear function,  ya1x.
 :: The roots of a polynomial or zero function values, x-intercepts
A zero of a function is a value of the argument (of a function) at which the value of the function is zero.
Therefore, zeros are values of the argument x that satisfy or solve the equation  f (x) = 0  or  y = 0.
The zeros (roots) of a function correspond to the x-intercepts of the graph.
An x-intercept is the point (x, 0) where the graph of the function touches or crosses the x-axis.
  :: Linear equation
Thus, the solution of the equation  f (x) = 0  or  y = 0, that is
a1 x + a0 = 0   or   m x + c = 0,
is the zero of the linear function, the x-intercept or the root of the first degree polynomial
 
 
 
 
 
 
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