A root a of a polynomial
f (x) has
multiplicity k if
f (x) = (x - a)kg(x), where
g(a)≠ 0. If k > 1, then
f′(x) = k(x - a)k-1g(x) + (x - a)kg′(x) = (x - a)k-1(kg(x) + (x - a)g′(x)), and hence
gcd(f (x), f′(x)) = (x - a)k-1h(x)≠1. This observation provides a
test for multiple roots: If
gcdf (x), f′(x)
is a
constant, then f (x) has no
multiple roots; otherwise, f (x) has
at least one multiple root.
The graphs of
However,
Evaluate
gcd(f (x), f′(x)) = 791x - 2486
whereas
Evaluate
gcd(g(x), g′(x)) = 7
Thus, x = 2486/791 = 22/7 is a root of f (x) of multiplicity at least 2, whereas g(x) has no multiple roots. Solving f (x) = 0 and g(x) = 0, the real solutions are as follows.
Solve + Exact
f (x) = 0, Solution is :x =
,
x =
,
x = -
+
![]()
g(x) = 0, Solution is :x =
,
x =
,
x = -
+
,
Solve + Numeric
f (x) = 0, Solution is :x = - .68233
,
x = 3.1429
,
x = 3.1429
![]()
g(x) = 0, Solution is :x = - .68233
,
x = 3.1416
,
x = 3.1429
Thus, g has two distinct roots that are extremely close, whereas f has a root of multiplicity two.