solve (x + 3)/(x - 1) = 3/(x + 1) + (x ^ 2)/(x ^ 2 - 1)
To solve the equation, let's start by simplifying it:
(x + 3)/(x - 1) = 3/(x + 1) + (x^2)/(x^2 - 1)
First, let's get rid of the denominators by cross-multiplying:
(x + 3)(x + 1) = 3(x - 1) + (x^2)(x + 1)/(x^2 - 1)
Expanding each term:
(x^2 + x + 3x + 3) = 3x - 3 + (x^3 + x^2 + x + x^2)/(x^2 - 1)
Combining like terms:
(x^2 + 4x + 3) = 3x - 3 + (x^3 + 2x^2 + x)/(x^2 - 1)
Getting rid of the fraction:
(x^2 + 4x + 3)(x^2 - 1) = (3x - 3)(x^2 - 1) + (x^3 + 2x^2 + x)
Expanding each term:
x^4 - x^2 + 4x^3 - 4x + 3x^2 + 3 = 3x^3 - 3x - 3 + x^3 + 2x^2 + x
Combining like terms:
x^4 + 4x^3 + 2x^3 - x^2 + 3x^2 - 4x + x - 3 = 4x^3 - 2x^2 - 3x - 3
Simplifying the equation:
x^4 + 6x^3 + 2x^3 + 2x^2 + 4x^2 - 4x + x - 3 = 4x^3 - 2x^2 - 3x - 3
x^4 + 8x^3 + 6x^2 - 4x - 3 = 4x^3 - 2x^2 - 3x - 3
Rearranging the equation:
x^4 + 8x^3 + 6x^2 - 4x - 3 - 4x^3 + 2x^2 - 3x - 3 = 0
x^4 + 4x^3 + 8x^3 + 6x^2 + 2x^2 - 4x - 3x - 3 = 0
x^4 + 12x^3 + 8x^2 - 7x - 3 = 0
Now, we need to find the roots of this fourth-degree polynomial. Unfortunately, there is no general formula to find the roots of a polynomial of degree four or higher. We must either approximate the roots or use specialized algorithms or software.