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
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