How do you divide #\frac{x^2+2x-5}{x}#?

To divide \frac{x^2+2x-5}{x}, you can use long division or synthetic division.

Using long division:

1. Divide x into x^2 to get x.
2. Multiply x by x to get x^2, and multiply x by 2 to get 2x.
3. Subtract x^2 + 2x from x^2 + 2x - 5 to get -5.
4. Bring down the -5.
5. Divide x into -5 to get -5.
6. Multiply -5 by x to get -5x.
7. Subtract -5x from -5 to get 0.

The quotient is x - 5, and there is no remainder.

Using synthetic division:

1. Set up the synthetic division table with the divisor x on the left and the coefficients of the dividend x^2 + 2x - 5 on the right.
2. Bring down the coefficient of x^2, which is 1.
3. Multiply the divisor x by 1 to get x, and write it below the next coefficient.
4. Add the two numbers in the second column to get 2x.
5. Multiply the divisor x by 2x to get 2x^2, and write it below the next coefficient.
6. Add the two numbers in the third column to get 2x - 5.
7. Multiply the divisor x by 2x - 5 to get 2x^2 - 5x, and write it below the next coefficient.
8. Add the two numbers in the fourth column to get -5x.
9. The last number in the synthetic division table is the remainder, which is 0.

The quotient is x - 5, and there is no remainder.

To divide (\frac{x^2+2x-5}{x}), you can use polynomial long division or synthetic division. Here's how to do it using polynomial long division:

1. Divide the highest degree term of the numerator by the highest degree term of the denominator.
2. Multiply the divisor (the result from step 1) by the denominator, and subtract the result from the numerator.
3. Repeat steps 1 and 2 until the degree of the remainder is less than the degree of the denominator, or until the remainder is zero.

Here's how it works:

[ \begin{array}{r|ll} x & x & +2 \ \hline x & x^2 & +2x & -5 \ & -(x^2 & & ) \ \hline & & 2x & -5 \ & & -(2x & ) \ \hline & & & -5 \ \end{array} ]

So, (\frac{x^2+2x-5}{x} = x + 2 - \frac{5}{x}).