Day 4. Cayley’s Discovery, The Fundamental
Theorem of Algebra and Creating Your Own
Polynomials
4.1 Basins of Attraction for Solutions of Cubics
As the previous examples have show shown, if a quadratic has real solutions a and b then
when Newton’s Method is used to solve the corresponding quadratic x2 –(a+b)x + ab = 0,
the basin of attraction for x=a is the set {x| x < (a+b)/2 } and the basin of attraction for
x = b is the set {x| x > (a+b)/2). In the mid 1800s Arthur Cayley discovered that the
basins of attraction for the solutions of cubics were far more complicated then those for
the solutions of quadratics. Here’s what he found. Consider solving x3 - x = 0 which has
solutions x = -1, 0, 1.
.
The derivative f’(x) of f(x) = ax3 + bx2 + cx + d is
f’(x) = 3ax2 + 2bx + c
The Newton function for the general cubic is
ax 3 + bx 2 + cx + d
N(x) = x 3ax 2 + 2bx + c
In this case f’(x) = 3x2 – 1 so the Newton Function for f(x) = x3 - x is
x3 - x
N(x) = x - 2
3x -1
Something interesting happens when you examine initial guesses on the interval
( - 1 , 0 , 1 ). Spreadsheet Newton Cubic allows you to do this. You can enter an interval (
a , b ) and the spreadsheet will divide the interval into 10 initial values x = a , a + d, a +
2d, a +3d, . . . a + 9d, b where d = (b-a)/10 and then calculate 10 iterations to see which
root the guess converges to. Letting a = -1.5 and b = 1.5 one gets the result shown in
Table 4.1. It looks like if an initial guess
- between -1.5 and -.6 is used, N(x) converges to -1
- between -.3 and .3 is used, N(x) converges to 0
- between .6 and 1.5 is used, N(x) converges to 1
1
Table 4.1. Iterates of Nf(x) for seeds in the interval (-1.5 , 1.5)
What Cayley discovered was that if you examined the iterates of N(x) for seeds in the
interval ( .4472, .65 ) see Table 4.2, you get a surprising result. In that interval there are
seeds that will converge to each of the three roots.
Table 4.2 Iterates of N(x) for seeds in the interval (.4472 , .65)
In fact if you examined an even smaller interval such as (-.4565, - .4435), you again find
seeds that converge to all three roots as shown in Table 4.3.
Table 4.3. Iterates of N(x) for seeds in the interval (-.4565 , -.4435)
2
What he discovered was that there were basins of attraction inside basins of attraction
inside…….It took years and the work of Fatou and Julia in the early 1900s to understand
qualitatively what was going on and it wasn’t until 1980 that people first saw the pictures
of what was going on. POLYNOMIOGRAPHY creates the pictures that Cayley, Fatou
and Julia never saw. When one uses Newton’s Method in the complex plane, the images
are often fractal like since there will be basins of attraction inside basins of attraction
inside basins of attraction……..
One can see the fractal nature of the basins of attraction for solutions of cubics using
POLYNOMIOGRAPHY. Figure 4.1 shows the image POLYNOMIOGRAPHY creates
if one enters the polynomial z3 - z.
Figure 4.1 POLYNOMIOGRAPHY x3 – x plot
Exercise 4.1 Graph x3-x in POLYNOMIOGRAPHY. Zoom in on one of the blue bulbs
and then zoom in on the red bulb at the tip of the blue bulb. Next, zoom in on the blue
bulb on the tip of the red bulb. Describe what you are seeing.
4.2 The Fundamental Theorem of Algebra
What we have seen from the above discussion is that the quadratic formula is not always
helpful in giving us a decimal approximation to the solution of a quadratic equation.
However, the existence of quadratic formula inspired scholars to solve cubic equations. A
cubic equation is
3
p(x) = ax 3 + bx 2 + cx + d, a ¹ 0
The fundamental theorem of algebra tells us that any quadratic or cubic polynomial will
always have a root, no matter what we select as coefficients. This we have already seen
in the case of quadratic and cubic polynomials with real coefficients. But the
fundamental theorem says a root of a polynomial will always exists, no matter what the
degree of the polynomial is.
It turns out that there is a formula for the solutions of a cubic equation as well. However,
the formula is complicated and not easy to keep in one's head! There is also a closed
formula in the case of quartic equation, an equation of the form
p(x) = ax 4 + bx 3 + cx 2 + dx + e, a ¹ 0
However, in the case of quintic equations - degree 5, or higher there is a famous result
found by Evariste Galois that says there is no formula that gives the solutions in terms of
radicals such as square-roots, cube-roots, etc., and the coefficients of the given
polynomial.
What this says is that there are polynomials such that the only way to approximate their
solutions is to use iterative methods. From the above we also see that if we increase the
degree of the polynomial from 4 to 5 and higher it becomes cumbersome to use the letter
of the alphabet so there is a more convenient way to denote a polynomial equation.
Formally, a polynomial of degree n  0 is an expression of the form
p(x) = an x n + an-1x n-1 +... + a1 x + a0 , an ¹ 0
Solving a polynomial equation is to find one or more roots, i.e. solutions to
p(x) = 0
When considering polynomiography of polynomials we can consider the cases when the
coefficients are real, or when they are allowed to be real and complex. However, we will
only consider polynomials with real coefficients. Of course a real number is always a
complex number because the complex numbers is a superset of the reals. However, a
complex number that has imaginary part is not a real number. So for instance for
quadratics we can first consider those with real coefficients. We saw that there are three
types: a double root, two different real roots, or a pair of conjugate roots.
For a cubic polynomial with real coefficients we can have several cases such as:
i)
ii)
iii)
A single real root.
A double real root and another real root.
A real root and a pair of complex conjugates.
4
iv)
Three distinct real roots.
Figure 4.2a A Single Real
Root
Figure 4.2c A Real Root
and A Pair of Complex
Roots
Figure 4.2b A Double Real
Root and Another Real
Root
Figure 4.2d
Three Distinct
Real Root
For a quartic (degree 4) polynomial with real coefficients we can have even more cases.
Some of these are:
i)
ii)
iii)
iv)
v)
vi)
A single real root.
Four distinct real roots.
A triple real root and a distinct second root.
Two real double roots.
A double real root and a pair of conjugate roots.
Two distinct pairs of conjugate pairs.
5
Figure 4.3. Polynomiography for these cases, i-iii (top row), iv-vi bottom
row.
4.3 Creating Designer Polynomials
If a quadratic has roots x = a and x = b then the polynomial with these roots
can be created by expanding (x – a )( x – b). Therefore, the polynomial
p(x) = x2 – (a+b)x + ab has roots a and b. For example,
p(x) = x2 – 5x + 6 has roots x = 3 and x =2.
If a quadratic polynomial has no real roots then it has complex roots of the
form x = a + bi and x = a – bi then the polynomial with theses roots can be
created by expanding (x – (a + bi) )( x – (a - bi) ). Therefore, the polynomial
p(x) = x2 – [(a+bi) + (a-bi)] x + (a + bi )( a – bi )
= x2 – 2ax + (a2 + b2)
has roots a+bi and a-bi. For example, p(x) = x2 – 8x + 25 has roots
x = 4 – 3i and x = 4 + 3i.
Since the Fundamental Theorem of Algebra states that any nth degree
polynomial with real coefficients has n roots and that the complex roots come
in conjugate pairs, the preceeding can be used to build polynomials with the
desired number of real and complex roots.
For example, to create a polynomial with 4 complex roots and two distinct
real roots do the following:
1. Select the two real roots say x = -3 and x = 4
2. Select the two conjugate pairs x = 2 +/- 3i and x = -3 +/- 5i
3. Create the quadratic polynomials that have the conjugate pairs as roots
(x – (2+3i))(x – (2-3i)) = x2 - 4x + 13
(x – (-3+2i))(x – (-3-3i)) = x2 + 6x + 34
6
4. The desired polynomial is
p(x) = (x + 3)( x – 4 )( x2 - 4x + 13)( x2 + 6x + 34 )
Using a computer algebra system, one gets p(x) in standard polynomial
form.
p(x) = x6 - 11 x5 + 69 x4 -165 x3 -196 x2 + 2126 x – 5304
Below in Figure 4.3.1 is a POLYNOMIOGRAPHY plot of
x6 - 11 x5 + 69 x4 -165 x3 -196 x2 + 2126 x – 5304 showing the 6 roots.
Figure 4.3.1 Plot of x6 - 11 x5 + 69 x4 -165 x3 -196 x2 + 2126 x – 5304
Note that given the preceding one can argue that any nth degree
polynomial can be factored into a product of linear and quadratic factors.
Exercises:
1. At the end of section 4.2, the number of different possible roots for a 4th degree
polynomials was enumerated, they were:
i)
A single real root.
ii)
Four distinct real roots.
iii)
A triple real root and a distinct second root.
iv)
Two real double roots.
v)
A double real root and a pair of conjugate roots.
vi)
Two distinct pairs of conjugate pairs.
Using the above create examples of each of the 6 possible cases.
2. Make a list of all the possible different types of roots for a 5th degree polynomial and
create an example of each possible case.
7
3. Create a polynomial with degree greater than 8 and less than 18 which has at least two
pairs of conjugate imaginary roots. Graph it using POLYNOMIOGRAPHY.
4. Split your class into groups of 4. Have each group member write down the day of the
month they were born creating a list of 4 numbers. Use these numbers to create a third
degree polynomial in standard form. Plot the polynomial solutions using
POLYNOMIOGRAPHY. Have the groups compare their results. Next, combine the
groups of 4 into groups of 8 and create seventh degree polynomials. Plot the polynomial
solutions using POLYNOMIOGRAPHY. Have the groups compare their results.
8