Fully simplify.
start fraction, 2, plus, start fraction, 2, divided by, x, end fraction, divided by, start fraction, 1, divided by, x, squared, end fraction, minus, 1, end fraction
x
2
1
−1
2+
x
2
To fully simplify the expression, we can begin by simplifying the division of fractions:
2 + (2/x) / (1/x^2) - 1
Next, we can simplify the expression with the common denominator of x^2:
2 + (2/x) * (x^2/1) - 1
Simplifying further, we have:
2 + 2x - 1
Combining like terms, we get:
2x + 1
To fully simplify the expression:
Start by simplifying the fractions separately:
2 + 2/x / 1/x^2 - 1
To add the first two fractions, we need a common denominator which is x. So, multiplying the first fraction's numerator and denominator by x:
2x/x + 2/x / 1/x^2 - 1
Simplifying the numerators:
(2x + 2)/x / 1/x^2 - 1
Now, we simplify the second fraction:
(2x + 2)/x / (1 - x^2)/x^2
To divide by a fraction, we can multiply by its reciprocal:
(2x + 2)/x * x^2/(1 - x^2)
Multiplying the numerators and denominators:
(2x^3 + 2x^2)/(x - x^3)
Now, we can factor out a common factor in the numerator:
2x^2(x + 1)/(x - x^3)
And finally, we can cancel out common factors in the numerator and denominator:
2x^2(x + 1)/(-x^2(x - 1))
So, the expression is fully simplified as:
2x(x + 1)/(-x(x - 1))
To fully simplify the expression, let's break it down step by step.
First, we can simplify the expression within each fraction separately.
2 + (2/x) can be rewritten as (2x + 2)/x since the common denominator is x.
1/(x^2) can be left as it is since there are no like terms to simplify further.
Now let's simplify the overall expression.
We have ((2x + 2)/x) - 1/(x^2).
To combine the two terms, we need to find a common denominator. The common denominator in this case is x^2.
So, ((2x + 2)/x) can be multiplied by (x/x) to get (2x^2 + 2x)/x^2.
Therefore, the expression becomes ((2x^2 + 2x)/x^2) - 1/(x^2).
Now that we have a common denominator, we can subtract the fractions.
The numerator becomes (2x^2 + 2x - 1). The denominator remains x^2.
Hence, the fully simplified expression is (2x^2 + 2x - 1)/x^2.