Hello,
Thanks for all of the help guys. I really appreciate it.
I was woudnering if you could show me how to do this step by step and the thought process that goes through your head while you do this. This way I will be able to try to solve some of these problems on my own becasue I really have no idea how to do them.
Perform the indicated operation and simplify the ersult. Leave your anser in factored form.
1/x - 2/(x^2 + x) + 3/(x^3 - x^2)
THANK YOU!!!
remember to add/subtract fractions we need a common denominator.
let's write our expression in factored form
1/x - 2/(x^2 + x) + 3/(x^3 - x^2)
= 1/x - 2/[x(x+1)] + 3/[x^2(x-1)]
so the common denominator is x^2(x+1)(x-1)
first term: x(x+1)(x-1)/[x^2(x+1)(x-1)]
notice if we cancel we get 1/x
second term: -2x(x-1)/[x^2(x+1)(x-1)]
again, see what you get when you cancel
third term: 3(x+1)/[x^2(x+1)(x-1)]
so we get
[x(x+1)(x-1) - 2x(x-1) + 3(x+1)]/[x^2(x+1)(x-1)]
I will let you finish it,
(I got (x^3 - 2x^2 + 4x + 3)/[x^2(x^2 - 1)]
I bet you have a sign wrong but will do as shown.
1/x
-2/[x(x+1)]
+3/[x^2(x-1)]
we need a common denominator
need LCD = x^2 (x-1)(x+1) = x^2(x^2-1)
so
1[(x)(x+1)(x-1)]/LCD
-2 [ (x)(x-1) ] /LCD
+3 [ (x+1 ] / LCD
then
[1 (x(x^2-1) -2 (x^2-x) +3 (x+1) ]/LCD
[ x^3-x -2x^2 +2x +3x+3 ]/LCD
(x^3 -2x^2 +4x +3)/[x^2(x^2-1)]
I could not simplify, suspct mistake but that is the idea.
Oh, I see Reiny got the same answer. If you typed it right the answer is right.
Sure, I'll be happy to explain the step-by-step process for solving this problem and share the thought process involved. Let's break it down:
Step 1: Find a common denominator for the fractions involved. In this case, the least common denominator (LCD) is the product of all the denominators, which are x, (x^2 + x), and (x^3 - x^2).
Step 2: Rewrite each fraction with the LCD as the denominator. To do this, we need to multiply the numerator and denominator of each fraction by any missing factors.
The first fraction, 1/x, already has the x as the denominator, so we don't need to do anything with it.
For the second fraction, 2/(x^2 + x), we need to multiply both the numerator and denominator by (x - 1) to get rid of the binomial in the denominator. This gives us 2(x - 1)/(x(x - 1)).
For the third fraction, 3/(x^3 - x^2), we need to multiply both the numerator and denominator by (x^2) to get rid of the cubic expression in the denominator. This gives us 3(x^2)/(x^2(x - 1)).
Step 3: Combine the fractions. Now that we have a common denominator for all fractions, we can add or subtract the numerators.
Thus, the expression becomes: 1/x - 2(x - 1)/(x(x - 1)) + 3x^2/(x^2(x - 1)).
Step 4: Simplify the expression. We can simplify further by combining like terms and factoring out any common factors.
For the second fraction, by distributing the -2, we get (-2x + 2)/(x(x - 1)).
Since 3x^2 is already in its simplest form, we don't need to do anything with it.
Step 5: Combine like terms. We can do this by adding or subtracting the numerators of the fractions.
The expression becomes: (1 - 2x + 2 + 3x^2)/(x(x - 1)).
Step 6: Further simplify if possible. In this case, we cannot factor or simplify the numerator any further.
Therefore, the final simplified expression in factored form is: (3x^2 - 2x + 3)/(x(x - 1)).
And that's the step-by-step process for solving the given problem! Remember, practice is key in improving your problem-solving skills, so try solving similar problems on your own to gain confidence and mastery.