Rationalizing a Quadratic Polynomial

Rationalizing a Quadratic Polynomial

Problem

Rewrite the Quadratic Polynomial 1 + A*x + B*x^2 as a rational function of polynomials.  A and B are real numbers (however, this should work if A and B were complex numbers).  

1 + A*x + B*x^2 → p(x) / q(x)

Attempt 1:  1 + A*x + B*x^2 → (1 + C*x) / (1 + D*x)

1 + A*x + B*x^2 = (1 + C*x) / (1 + D*x)

(1 + A*x + B*x^2) * (1 + D*x) = (1 + C*x) / (1 + D*x) * (1 + D*x)

A*x + B*x^2 + D*x + A*D*x^2 + D*B*x^3 = C*x

Comparing the powers of x:

constant:  0 = 0

x:  A + D = C

x^2:  B + A*D = 0

x^3:  B*D = 0

This leads to either B=0 or D=0

Assume B=0. 

Then B + A*D = 0

A*D = 0

If A=0, then D = C, which leads to:

1 + A*x + B*x^2 = (1 + C*x) / (1 + D*x)

1 = (1 + C*x) / (1 + C*x)

1 = 1

If D=0, then A = C

1 + A*x + B*x^2 = (1 + C*x) / (1 + D*x)

1 + C*x = (1 + C*x) 

If D=0, then B=0, and we get the same results as above.   

Ultimately this transformation to (1 + C*x) / (1 + D*x) leads to nothing useful.

Attempt 2:  1 + A*x + B*x^2 → (1 + C*x^2) / (1 + D*x)

 

1 + A*x + B*x^2 = (1 + C*x^2) / (1 + D*x)

(1 + A*x + B*x^2) * (1 + D*x) = 1 + C*x^2 

(A + D)*x + (B + A*D)*x^2 + B*D*x^3 = C*x^2

Comparing the powers of x:

constant:  0 = 0

x:  A + D = 0

x^2:  B + A*D = C

x^3:  B*D = 0

A + D = 0 implies that A = -D or D = -A

Also either B = 0 or D = 0.

Assume B = 0. Then with A = -D:

A*D = C

-D*D = C

C = -D^2

1 + A*x + B*x^2 = (1 + C*x^2) / (1 + D*x)

1 -  D*x = (1 - D*x^2) / (1 + D*x)

1 -  D*x = ((1 - D*x) * (1 + D*x))/ (1 + D*x)

1 - D*x = 1 - D*x

If we assume that D = -A, then:

C = -A^2 and

1 + A*x = (1 - A*x^2) / (1 - A*x)

1 + A*x = ((1 - A*x)  * (1 + A*x)) / (1 - A*x)

1 + A*x = 1 + A*x

Assume D = 0.

Then A = 0 and B = C:

1 + B*x^2 = 1 + B*x^2

1 + C*x^2 = 1 + C*x^2

Again, we have transformations that are trivial.

Attempt 3:  1 + A*x + B*x^2 → (1 + C*x^3) / (1 + D*x)

1 + A*x + B*x^2 = (1 + C*x^3) / (1 + D*x)

(1 + A*x + B*x^2) * (1 + D*x) = 1 + C*x^3

(A + D)*x + (B + A*D)*x^2 + B*D*x^3 = 1 + C*x^3

Comparing the powers of x:

constant:  0 = 0

x:  A + D = 0

x^2:  B + A*D = 0

x^3:  B*D = C

This implies that:

A + D = 0

D = -A

B + A*D = 0

B+ A*-A = 0

B = A^2   (this restricts A and B)

B * D = C

(A^2)*(-A) = C

C = -A^3

We can conclude that B = A^2, C = -A^3, D = -A

The relationship between A, B, C, and D are all connected in this case.

Examples:

A = 2 ⇒ B = 4, C = -8, D = -2 and

1 + 2*x + 4*x^2 = (1 - 8*x^3) / (1 - 2*x)

A = -2 ⇒ B = 4, C = 8, D = 2 and

1 - 2*x + 4*x^2 = (1 + 8*x^3) / (1 + 2*x)

Note:  Casio fx-991EX Week - September 5, 2022 to September 9, 2022 

Eddie

All original content copyright, © 2011-2022.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.