my roots are (1+isq.root2)(1-isq.root2)
I then used long division:
X^4-4X^3+4X^2-9/X^2-X-4
I got the answer X^2-8X+32 with a remander of 119. Where did I go wrong?
I am uncertain what you are trying to do.
i am trying to solve the equation. I have to find 2 more roots of the X^4 equation
To determine where you went wrong in your long division, let's retrace the steps and perform the division correctly.
Given that your roots are (1 + √2i) and (1 - √2i), we have:
(x - (1 + √2i))(x - (1 - √2i)) = 0
Next, let's multiply the factors together to find the quadratic equation:
(x - 1 - √2i)(x - 1 + √2i) = 0
(x - 1)^2 - (√2i)^2 = 0
(x - 1)^2 - 2i^2 = 0
(x - 1)^2 + 2 = 0
Expanding the above expression, we get:
x^2 - 2x + 1 + 2 = 0
x^2 - 2x + 3 = 0
Therefore, the correct quadratic equation with the given roots is x^2 - 2x + 3 = 0.
Now, let's verify the long division you performed for the polynomial (x^4 - 4x^3 + 4x^2 - 9) divided by (x^2 - x - 4).
____________________
x^2 - x - 4 | x^4 - 4x^3 + 4x^2 - 9
-(x^4 - x^3 - 4x^2)
____________________
-3x^3 + 8x^2
(3x^3 - 3x^2 - 12x)
____________________
11x^2 - 9x - 9
(11x^2 - 11x - 44)
____________________
2x - 53
Based on the long division, the quotient obtained so far is x^2 - x - 4, with a remainder of 2x - 53. Therefore, your previous result of x^2 - 8x + 32 with a remainder of 119 must have been an error.
To correctly divide (x^4 - 4x^3 + 4x^2 - 9) by (x^2 - x - 4), the resulting quotient is x^2 - x - 4, with a remainder of 2x - 53.