Simplify the difference and type the answer in factored form. State any restrictions on the variable.
3x/x^2+x-2 - 5x/x^2-3x+2
To simplify the expression, we need to find a common denominator for both fractions.
First, let's find the factors of the denominators:
x² + x - 2 = (x - 1)(x + 2)
x² - 3x + 2 = (x - 1)(x - 2)
Now, let's rewrite each fraction with the common denominators:
3x/(x - 1)(x + 2) - 5x/(x - 1)(x - 2)
Since the denominators are the same, we can combine the numerators:
(3x - 5x)/(x - 1)(x + 2)
Simplifying the numerator:
-2x/(x - 1)(x + 2)
The final simplified expression, in factored form, is -2x/(x - 1)(x + 2). The restriction is that the denominator cannot be equal to zero, so x ≠ 1 and x ≠ -2.
To simplify the difference:
The first step is to factor both denominators:
For the first fraction, x^2 + x - 2, we need to find two numbers that multiply to -2 and add up to +1 (the coefficient of x). The numbers are -1 and +2, so we can factor it as (x - 1)(x + 2).
For the second fraction, x^2 - 3x + 2, we need to find two numbers that multiply to +2 and add up to -3 (the coefficient of x). The numbers are -1 and -2, so we can factor it as (x - 1)(x - 2).
Now, we can rewrite the expression:
3x/(x - 1)(x + 2) - 5x/(x - 1)(x - 2)
To subtract fractions, we need a common denominator, which in this case is (x - 1)(x + 2)(x - 2).
Next, we will multiply the first fraction by (x - 2)/(x - 2) and the second fraction by (x + 2)/(x + 2) to obtain a common denominator:
[3x(x - 2)]/[(x - 1)(x + 2)(x - 2)] - [5x(x + 2)]/[(x - 1)(x + 2)(x - 2)]
Expanding the numerators, we get:
[3x^2 - 6x - 5x^2 - 10x]/[(x - 1)(x + 2)(x - 2)]
Combining like terms in the numerator, we have:
[-2x^2 - 16x]/[(x - 1)(x + 2)(x - 2)]
Now, we can factor out the common factor of -2x:
-2x(x + 8)/[(x - 1)(x + 2)(x - 2)]
Therefore, the simplified difference in factored form is:
-2x(x + 8)/[(x - 1)(x + 2)(x - 2)]
Restrictions on the variable:
The restrictions on the variable are x ≠ 1, -2, and 2.
To simplify the given expression and express the answer in factored form, let's start by factoring the denominators.
For the first fraction, x^2 + x - 2, we need to find two numbers that multiply to -2 and add up to +1 (the coefficient of the middle term, which is 1). These numbers are +2 and -1. Therefore, we can factor the denominator as (x + 2)(x - 1).
For the second fraction, x^2 - 3x + 2, we need to find two numbers that multiply to +2 and add up to -3 (the coefficient of the middle term, which is -3). These numbers are -2 and -1. Therefore, we can factor the denominator as (x - 2)(x - 1).
Now that we have factored the denominators, the given expression becomes:
3x/(x + 2)(x - 1) - 5x/(x - 2)(x - 1)
To combine the fractions, we need a common denominator, which is (x + 2)(x - 1)(x - 2). We can multiply each fraction by the appropriate factors of the common denominator as follows:
(3x * (x - 2))/(x + 2)(x - 1)(x - 2) - (5x * (x + 2))/(x - 2)(x - 1)(x + 2)
Simplifying further:
(3x^2 - 6x - 5x^2 - 10x)/(x + 2)(x - 1)(x - 2)
Combining like terms:
(-2x^2 - 16x)/(x + 2)(x - 1)(x - 2)
Now, to express this answer in factored form, we can factor out the common factor in the numerator, which is -2x:
(-2x(x + 8))/(x + 2)(x - 1)(x - 2)
Therefore, the simplified difference in factored form is:
(-2x(x + 8))/(x + 2)(x - 1)(x - 2)
As for restrictions on the variable, we need to consider any values of x that would make the denominator(s) equal to zero, as division by zero is undefined. In this case, we have three denominators: x + 2, x - 1, and x - 2.
Therefore, the restrictions on the variable (x) are:
x ≠ -2, x ≠ 1, x ≠ 2