Is it 2nd order polynomials, or 2nd order quadratics?

[Mol_Bolom]

I thought this was rather odd, and wanted to just show it to see what you all thought of it. Well, also, if anyone knows what I should read to exactly understand what I did.

1: Let's define each answer of a polynomial such as (ax + b)(cx + d) as x1 = (-b/a), x2 = (-d/c).

2: The polynomial written out is qx^2 + rx + s

3: Change the polynomial from qx^2 + rx + s to where x = s, and rewrite the problem as qS^2 + rS = S.

4: Devide S out, which leaves qS + r = 1

5: subtract r, qS + r - r = 1 - r. qS = 1 - r

6: Devide q, qS/q = (1 - r)/q. S = (1-r)/q

7: Now if the equation (q x1 + 1)/q is subtracted from S will solve for x2. Or if x1 is replaced with x2 the equation then solves for x1.
(1-r)/q - (q x1 + 1)/q = x2
(1-r)/q - (q x2 + 1)/q = x1

I did simplify this, but I still think it is rather interesting as to 'how' I did this, rather than the 'simplicity' of it.
(Darn it, I forgot what these kinds of polynomials are called. Is it 2nd order polynomials, or 2nd order quadratics?)

Thanks

 

[Mol_Bolom]

I just ran it through a couple of problems but it seems that in the third order...

qx³ + rx²+sx+t
(ax+b)(cx+d)(ex+f)
x₁ = -b/a
x₂ = -d/c
x₃ = -f/e

(1-r)/q - (q x₁ + 1)/q = x₂ + x₃
(1-r)/q - (q x₂ + 1)/q = x₁ + x₃
(1-r)/q - (q x₃ + 1)/q = x₁ + x₂

 

[mathman]

3: Change the polynomial from qx^2 + rx + s to where x = s, and rewrite the problem as qS^2 + rS = S.

The original polynomial is 0 only at its roots. x=s is NOT a root, so your rewrite is wrong. Also you put in a sign change in the S term.

 

[HallsofIvy]

I thought this was rather odd, and wanted to just show it to see what you all thought of it. Well, also, if anyone knows what I should read to exactly understand what I did.

1: Let's define each answer of a polynomial such as (ax + b)(cx + d) as x1 = (-b/a), x2 = (-d/c).

2: The polynomial written out is qx^2 + rx + s

3: Change the polynomial from qx^2 + rx + s to

To what?? Isn't that a crucial part of the problem?

where x = s, and rewrite the problem as qS^2 + rS = S.

4: Devide S out, which leaves qS + r = 1

5: subtract r, qS + r - r = 1 - r. qS = 1 - r

6: Devide q, qS/q = (1 - r)/q. S = (1-r)/q

7: Now if the equation (q x1 + 1)/q is subtracted from S will solve for x2. Or if x1 is replaced with x2 the equation then solves for x1.
(1-r)/q - (q x1 + 1)/q = x2
(1-r)/q - (q x2 + 1)/q = x1

I did simplify this, but I still think it is rather interesting as to 'how' I did this, rather than the 'simplicity' of it.
(Darn it, I forgot what these kinds of polynomials are called. Is it 2nd order polynomials, or 2nd order quadratics?)

Thanks

 

[Mol_Bolom]

Yikes...I had thought I did a better job at explaining it, I guess not...Well back to the drawing board...

I just thought it was interesting how I came up with it...

I'll try one more time, though...


qx² + rx = s, and 'Falsely' implying s as the value of x (Actually, I consider this change as 'False' and call the new variable formed as 'False x', however, I use s instead here)

qx²+rx = s...Making S = x (Yes, I know there is a change in the variable, that is what I find interesting)

qS²+rS = S...Deviding S out then leaves qS + r = 1.

qS = 1 - r, then divide q out.

Thus S is the 'False' x.

(Which is the reason that I started using the capital letter S instead of the lowercase s when I was expressing that in the first post. Perhaps I should have mentioned that...)

Then using the 'False' x in the equation

(FALSE X) - (qx₂ + 1)/q = x₁