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Properties of Quartics
Abstract: The aim of this paper is to investigate on a property of quartic polynomials. Considering a “W” shaped function with inflection points Q and R, a line was drawn through the points to meet the function again at P and S. The ratio of PQ:QR:RS is found to correspond to the Golden Ratio hence a conjecture is produced and formally proven in order to demonstrate this.
The function
€
f (x) = x 4 − 8x 3 +18x 2 −12x + 24
is a “W” shape quartic with two points of
inflection. Its first derivative
€
′ f (x) = 4x 3 − 24x 2 + 36x −12 is a cubic
function and its second derivative
€
′ ′ f (x) =12x 2 − 48x + 36 a quadratic.
Fig 1
Fig 1 shows the function
€
f (x) and its first and second derivatives. Note that the points where
€
′ f (x) = 0 are the function’s inflection points.
The quartic’s points of inflection are found by
equating the second derivative to zero.
€
′ ′ f (x) =12x 2 − 48x + 36
0 =12(x −1)(x − 3)
€
x =1 and x = 3 ,
Substituting for x,
€
f (1) = 23 and f (3) =15
hence points of inflection are Q(1,23) and
R(3,15) The equation of the straight line
joining Q and R is:
€
y − y1 = m(x − x1)
y − 23 = −4(x −1)
y = −4x + 27
Let points P and S be the intersection of the
line QR and the quartic function. They are
found by equating, factorizing using long
division and solving for x.
€
−4 x + 27 = x 4 − 8x 3 +18x 2 −12x + 24
0 = x 4 − 8x 3 +18x 2 − 8x − 3
0 = (x 2 − 4 x + 3)(x 2 − 4x +1)
x =4 ± 20
2
x = 2 + 5 and x = 2 − 5
Note that the other quadratic equation gives
the x-values of Q and R. The coordinates of
points P and S are therefore
€
P((2 − 5), f (2 − 5)) and
€
S((2 + 5), f (2 + 5)) .
The investigation must then consider the ratio
of PQ:QR:RS. This can be done using
Pythagoras Theorem, however, because the
coordinates of P and S are irrational, the
process is long and tedious. The features of
similar triangles enable us to concentrate on x
coordinates only. Since all four points lie on
the same line, the triangles they form are
similar and the ratio of their distances is the
same as the ratio of their components.
Fig 2
x
In Fig 2,points P, Q, R, and S lie on the line
€
y = −4x + 27 . The dotted lines illustrate the similar triangles they form.
The ratio PQ:QR:RS is found using the
previously calculated x-values. Let
€
xP = 2 − 5 ,
€
xS = 2 + 5 ,
€
xq =1
and
€
xR = 3
€
xP − xQ = (2 − 5) −1 = 5 −1
€
xQ − xR = 1− 3 = 2
€
xR − xS = 3 − (2 + 5) = 5 −1
PQ:QR:RS is
€
5 −1: 2 : 5 −1and dividing
through by
€
5 −1 simplifies to
€
1:1+ 5
2:1
Considering the simpler quartic function
€
g(x) = 2x 4 − 4x 3 , the points of inflection also
occur when the second derivative equals zero:
€
′ ′ g (x) = 24x 2 − 24x
€
0 = 24x(x −1)
x = 0 x =1
g(0) = 0 g(1) = −2
Hence coordinates are Q(0,0) and R(1,-2).
Since the line QR goes through the origin, its
equation is
€
y = −2x which intersects with the
quartic when:
€
2x 4 − 4x 3 = −2x
2x 4 − 4x 3 + 2x = 0
Long division using the factor
€
(x −1)
simplifies the equation into two quadratics,
which can then be solved using the formula.
€
2x 4 − 4x 3 + 2x = 0
(2x 2 − 2x)(x 2 − x −1) = 0
x =4 ± 4
4 and x =
1± 5
2
The x-coordinates of points P and S are
therefore
€
1− 5
2 and
€
1+ 5
2 respectively.
Calculating the ratio is simpler in this case
since the point of inflection Q lies on the
origin. The ratio PQ:QR:RS corresponds to
€
xP − xQ : xQ − xR : xR − xS
€
5 −1
2 :
€
1 :
€
5 −1
2
and dividing through by
€
5 −1
2 simplifies to
€
1:1+ 5
2:1
Fig 3
In both cases, we assist to the emergence of
the golden ratio defined as the positive
solution of
€
x 2 − x −1 = 0 or
€
1+ 5
2.
We therefore produce a conjecture based of
the previous investigations.
Conjecture: If
€
f (x) is a quartic polynomial
with two distinct points of inflection, Q and R,
then the straight line joining Q and R meets
the graph of
€
y = f (x) in two other points P
and S. In its simplest form, the ratio of
PQ:QR:RS is equal to
€
1:1+ 5
2:1 so that
€
QR
PQ=
1+ 5
2 or the Golden Ratio.
Let us consider the general case of a quartic
polynomial function
€
f (x) . It will have two
distinct points of inflection, Q and R, if its
second derivative has two real roots. This
means that the quadratic function
€
′ ′ f (x) must
be in the form of
€
y = bx(x − a) where a and b
are real numbers.
Fig 4
Fig 4 shows the relationship between the function and its second derivative.
The quartic function is obtained by
integrating
€
′ ′ f (x) twice.
€
bx(x − a)∫∫ dx =1
12b(x 4 − 2ax 3 + cx + d)
For simplicity, we assume that the point Q
lies on the origin (0,0). The constant d must
therefore equal zero so that
€
f (0) = 0 meaning
that the quartic is expressed as
€
f (x) =1
12b(x 4 − 2ax 3 + cx) .
Fig 3 shows the graph of
€
g(x) and
€
y = −2x . Points Q and R are inflection points and P and S intersection points.
€
y =1
12b(x 4 − ax 3 + cx)
Now if Q is the origin, the other root of the
second derivative is a hence the second point
of inflection R has coordinates
€
R(a, f (a)) .
Substituting for
€
f (a) , the coordinates are
€
R(a,1
12b(−a4 + ac)) .
The gradient of the straight line joining points
Q and R is
€
1
12bx(−a4 + ac)
a=
1
12bx(−a3 + c)
so its equation is
€
y =1
12bx(−a3 + c) because
the y-intercept is zero. The line intersects the
quartic at points P and S and the x-coordinates
of these points are the solutions of the
equation
€
1
12bx(−a3 + c) =
1
12bx(x 3 − 2ax 2 + c)
€
x(−a3) = x(x 3 − 2ax 2)
x 4 − 2ax 3 + a3x = 0
The easiest way to obtain the solutions of the
equation in terms of a is to use the quadratic
formula. This means simplifying the quartic
into a quadratic by long division using the
already known real factors
€
(x − a) and
€
(x − 0) .
Using factor
€
(x − a) first:
€
x − a x 4 − 2ax 3 + a3 x
−(x 4 − ax 3 )
− ax 3 + a3 x
−(−ax 3 + a2 x 2 )
− a2 x 2 + a3 x
−(−a2 x 2 + a3 x)
0
x 3 − ax 2 − a2 x
)
and then factor
€
(x − 0)
€
x − 0 x 3 − ax2 − a2x
−(x 3)
− ax2 − a2x
−(−ax2)
− a2x
−(−a2x)
0
x 2 − ax − a2
)
Simplifies the equation into the quadratic
€
y = x 2 − ax − a2 whose roots are found using the formula
€
x =a ± a2 − 4(−a2 )
2 hence
€
xS =a + 5a2
2,
€
xP =a − 5a2
2 ,
€
xQ = 0 and
€
xR = a.
Once again, properties of similar triangles
enable us to use x-coordinates only to find the
ratio PQ:QR:RS.
€
xP − xQ : xQ − xR : xR − xS
€
5a2 − a
2: a :
5a2 − a
2
The ratio can be greatly simplified:
€
5a2 − a
2: a :
5a2 − a
2
(dividing through by
€
5a2 − a
2)
€
1:2a
5a2 − a:1
(simplifying the fraction)
€
2a
5a2 − a =
a
a×
2
5 −1 =
1+ 5
2
Hence the ratio equals
€
1:1+ 5
2:1 or the
Golden Ratio
It is important to mention the limitations of
this conjecture that cannot be extended to
quartic functions that are not strictly of a “W”
shape. The ratio of lengths can only be found
if the function has two distinct points of
inflection meaning its second derivative must
have two real roots. There are therefore two
types of quartics that will not illustrate the
golden ratio.
The first type has no points of inflection since
its second derivative has no real roots. The
quadratic
€
′ ′ f (x) does not cross the x-axis
because its determinant
€
b2 − 4ac is negative. A
quartic with no points of inflection will have a
“U” shape as shown on fig.5
Fig 5
Fig 5 shows the graph of
€
y =x 4
12+
x 2
2 and its
second derivative
€
y = x 2 +1. Note that the quadratic has no real roots hence the quartic is “U” shaped.
The second type of quartic does not have any
points of inflection because its second
derivative has only one real solution. Indeed,
since its determinant equals zero, it touches the
x-axis at one point and does not change sign.
Quartics of such type have a flattened U shape.
Fig6
As a result, the conjecture only holds if the
determinant of the second derivative is greater
than zero. Integrating the general function
twice:
€
(ax 2 + bx + c)∫∫ dx =ax 4
12+
bx 3
6+
cx 2
2+ dx + e
Hence the conjecture only affects quartic
function with
€
b2 − 4ac > 0
Using the proof of the conjecture and the
analysis of its limitations, it is possible to
Fig6 shows the graph of
€
y = x 4 and its second derivative
€
y =12x 2 .
produce a formal theorem on this specific
property of quartic polynomials.
Theorem: Let
€
f (x) be a quartic polynomial
with two distinct points of inflection Q and R,
the straight line joining these points will meet
the function again at P and S. If points P,Q,R,S
are ordered by increasing x-
coordinates, then
€
PQ = RS and
€
QR
RS=
1+ 5
2, the Golden Ratio