x(2nd power)+ 2x - 3
________________________ (Divided by)
x (2nd power) + 3x
Factor numerator: (x+3)(x-1)
Factor denominator: x(x+3)
It appears something will divide out.
To simplify the given expression, let's perform polynomial long division. Here's how to do it step-by-step:
Step 1: Arrange the terms in descending order, considering the powers of x.
Denominator: x^2 + 3x
Numerator: x^2 + 2x - 3
Step 2: Divide the first term of the numerator by the first term of the denominator.
x^2 / x^2 = 1
Step 3: Multiply the result (1) by the denominator (x^2 + 3x) and write it under the numerator.
1 * (x^2 + 3x) = x^2 + 3x
Step 4: Subtract the obtained product from the numerator.
(x^2 + 2x - 3) - (x^2 + 3x) = -x - 3
Step 5: Bring down the next term from the numerator.
- x - 3
Step 6: Repeat steps 2-5 with the result from step 4.
- x / x^2 = -1/x
-1/x * (x^2 + 3x) = -x - 3
(-x - 3) - (-x - 3) = 0
Step 7: There are no terms left in the numerator, and the remainder is 0. Therefore, the expression simplifies to:
1 - 1/x
Hence, x^2 + 2x - 3 / x^2 + 3x = 1 - 1/x