How do you divide #(x^4-9x^2-2) / (x^2+3x-1) #?

One method involves separating out multiples of the denominator from the numerator, starting with the highest degree term. This is equivalent to polynomial long division.

We find:

To divide (x^4-9x^2-2) by (x^2+3x-1), you can use long division or synthetic division. Here is the step-by-step process using long division:

1. Divide the first term of the numerator (x^4) by the first term of the denominator (x^2). The result is x^2.
2. Multiply the entire denominator (x^2+3x-1) by x^2, and subtract the result from the numerator (x^4-9x^2-2). (x^4-9x^2-2) - (x^2+3x-1) * x^2 = -12x^2 - 3x - 1
3. Bring down the next term from the numerator (-12x^2), and divide it by the first term of the denominator (x^2). The result is -12x.
4. Multiply the entire denominator (x^2+3x-1) by -12x, and subtract the result from the previous step (-12x^2 - 3x - 1). (-12x^2 - 3x - 1) - (x^2+3x-1) * -12x = -9x - 1
5. Bring down the next term from the numerator (-9x), and divide it by the first term of the denominator (x^2). The result is -9.
6. Multiply the entire denominator (x^2+3x-1) by -9, and subtract the result from the previous step (-9x - 1). (-9x - 1) - (x^2+3x-1) * -9 = 8x - 8
7. Bring down the last term from the numerator (-2), and divide it by the first term of the denominator (x^2). The result is 0.
8. Multiply the entire denominator (x^2+3x-1) by 0, and subtract the result from the previous step (8x - 8). (8x - 8) - (x^2+3x-1) * 0 = 8x - 8

The final result of the division is x^2 - 12x - 9, with a remainder of 8x - 8.