This section is optional. There are no web exercises for this section.
The prior section,
Multiplicity of zeroes and graphical consequences,
mentioned that it's often difficult (or impossible) to get a single toscale graph of a polynomial
that clearly shows the interesting behavior near all the zeroes at the same time.
Also, some graphing software can exhibit puzzling behavior
when graphing polynomials,
which can leave you scratching your head!
The example of the previous section is revisited, to more deeply explore the issues that can arise.
Let $\ P(x) = (x+2)x^5(x1)^3(x4)^2\,$. Then:

Study the graphs of the polynomial $P\,$ below, paying close attention to the scales on the $y$axis.
The scales on the $x$axis are identical in all six graphs.
In (A):

(A) 
In (B):

(B) 
In (C):

(C) 
In (D):

(D) 
In (E):

(E) 
In (F):

(F) 
$P(4  6*0.045) = 6136.41$  $(3.730,6136.41)$  $P(4 + 6*0.045) = 22,687.00$  $(4.27,22,687.00)$ 
$P(4  5*0.045) = 4789.51$  $(3.775,4789.51)$  $P(4 + 5*0.045) = 14,230.70$  $(4.23,14,230.70)$ 
$P(4  4*0.045) = 3439.79$  $(3.820,3439.79)$  $P(4 + 4*0.045) = 8216.68$  $(4.18,8216.68)$ 
$P(4  3*0.045) = 2167.99$  $(3.865,2167.99)$  $P(4 + 3*0.045) = 4164.59$  $(4.135,4164.59)$ 
$P(4  2*0.045) = 1078.24$  $(3.910,1078.24)$  $P(4 + 2*0.045) = 1665.69$  $(4.090,1665.69)$ 
$P(4  1*0.045) = 301.10$  $(3.955,301.10)$  $P(4 + 1*0.045) = 374.26$  $(4.045,374.26)$ 
$P(4) = 0$ $(4,0)$ 
(G) viewing window for $y$ : $[5543,5543]$ 
(H) viewing window for $y$ : $[5542,5542]$ 
(I) viewing window for $y$ : $[27,27]$ 
(J) viewing window for $y$ : $[26,26]$ 