Constructing “nice” cubic polynomials for curve sketching.
(where
  f
   and f
  
all have integer zeros.)
(Taken from Tom Bruggeman, Xavier University and Tom Gush SUNY at Stony Brook)
Let
  be a cubic polynomial with zeros 0, a and b (note: if one of the roots is NOT zero a simple translation can make it so).
        x
3    
2  abx then (1) f
   
3 x 2

2
   ab and (2) f
   
6 x

2
  
.
Now if f
   is to have nice integer roots (say c and d), then
(3) f
     
( x d )

3 x
2 
3
 c
  
3 cd
Comparing (1) and (3) yields
(4) 2
 a b
  c d

and (5) ab

3 cd .
From (5), one of a or b is divisible by 3 and because of symmetry suppose it is a.
From (4) (a + b) is divisible by 3, and so b must be divisible by 3 also.
Let a = 3A and b = 3B and (4) becomes
(6) 2( A + B ) = c + d and (5) becomes
(7) 3AB = cd
Eliminating A from the system (6) (7) yields..
2


Cd
B

B



3 C
 d or
(8) 2 B 2 

3 C



2 Cd

0
If B is to be an integer, the discriminant in the quadratic (8) above must be a perfect square

3 C
 d

2 
16 Cd

H
2 which can be rearranged to
(9) H
2 
4 C
 d

5 C

2
.
We know how to find all of the Pythagorean triples satisfying (9)
(10) 4 C

2 kst , H


2  t
2

, d

5 C


2  t
2

for integers s, t and any k.

From (8) B


3 C
 
H

Taking the plus sign (it turns out that choosing the minus sign gives the same result
4 with s and t interchanged) (10) gives… B


3 C

5 C


2  t
2
  

4
4 kst

2 ks 2
4

.
Putting k = 2r we get
 
From (10) C
 rst and d

5 rst

2

2  t
2

 r

2
  s

2 t

Finally, substituting back we get a

3 rt


2 s
 t b

3 rs s

2 t

 c

3 rst d
 r

2 s
 t
 s

2 t

The root of the second derivative is
 
3


2 
4 st
 t 2

The table below gives some examples of these “nice” cubics (I just used the third one down on a quiz I gave today)
Polynomial Roots of f Roots of f

Roots of f
 x
3 
3 x
2 
4

1, 2, 2 0, 2 1 x
3 
33 x
2 
216 x
0,9,24 4,18 11 x
3 
6 x
2 
135 x

9, 0,15

5, 9 2 x 3 
147 x

286

13, 2,11

7, 7 0 x
3 
3 x
2 
144 x

140 x
3 
3 x
2 
144 x

432

10, 1,14

12, 3,12

6,8

6,8
1
1
One thing worth mentioning…..don’t expect the function values at the relative extrema and P.O.I to be close to the x – axis.
For x
3 
6 x
2 
135 x , the relative max is at
 
5, 400

the relative min is at
 
and the P.O.I is at
 
I guess you can’t ask for everything to be “nice”.
Example: Let r = 1, t = 2 and s = 3. Then a = 48, b = 63, c = 18 and d = 56. This will yield
 
 x 3 
111 x

3024 x f
  

3 x 2 
222 x

3024 f
  

6 x

222 roots of f

0, 48, 63 roots of f
 
18,56 and the root of f
 
37