Simplify the difference and type the answer in factored form. State any restrictions on the variable.
2x/x^2-2x-15 - 6x/x^2-8x+15
To simplify the difference, we first need to factor the denominators of each rational expression.
For the first rational expression, we have x^2 - 2x - 15. This can be factored as (x - 5)(x + 3).
For the second rational expression, we have x^2 - 8x + 15. This can be factored as (x - 3)(x - 5).
Now, we can rewrite the original expression:
2x/(x - 5)(x + 3) - 6x/(x - 3)(x - 5)
Next, we need to find a common denominator. The two denominators have common factors of (x - 5), so we only need to consider the other factors.
The least common denominator (LCD) is (x - 3)(x - 5)(x + 3).
To rewrite the expression with the common denominator, we need to multiply the numerator and denominator of each rational expression by the missing factors from the LCD.
For the first rational expression, we need to multiply the numerator and denominator by (x - 3):
(2x * (x - 3))/((x - 5)(x + 3)(x - 3)) = (2x^2 - 6x)/((x - 5)(x + 3)(x - 3))
Similarly, for the second rational expression, we need to multiply the numerator and denominator by (x + 3):
(6x * (x + 3))/((x - 3)(x - 5)(x + 3)) = (6x^2 + 18x)/((x - 3)(x - 5)(x + 3))
Now, we can subtract the two rational expressions:
(2x^2 - 6x)/((x - 5)(x + 3)(x - 3)) - (6x^2 + 18x)/((x - 3)(x - 5)(x + 3))
To combine the two terms, we need a common denominator. Since the denominators are already the same, we can write the difference as a single rational expression:
(2x^2 - 6x - (6x^2 + 18x))/((x - 3)(x - 5)(x + 3))
Now, let's simplify the numerator:
2x^2 - 6x - 6x^2 - 18x = -4x^2 - 24x
Thus, the simplified difference is:
(-4x^2 - 24x)/((x - 3)(x - 5)(x + 3))
Restrictions on the variable: The restrictions on the variable occur when the denominator becomes zero. In this case, the restrictions are x = 3, x = 5, and x = -3.
To simplify the difference and express it in factored form, we first need to factor both denominators and find the common denominator.
For the first fraction, the denominator is x^2 - 2x - 15. Factoring this quadratic, we have:
x^2 - 2x - 15 = (x - 5)(x + 3)
For the second fraction, the denominator is x^2 - 8x + 15. Factoring this quadratic, we have:
x^2 - 8x + 15 = (x - 3)(x - 5)
Now we can rewrite the original expression using the common denominator:
2x / (x - 5)(x + 3) - 6x / (x - 3)(x - 5)
The common denominator is (x - 5)(x + 3)(x - 3).
Now, let's combine the numerators over the common denominator:
(2x(x - 3) - 6x(x + 3)) / (x - 5)(x + 3)(x - 3)
Simplifying the expression in the numerator:
2x^2 - 6x(x + 3) - 6x(x + 3) = 2x^2 - 6x^2 - 18x - 6x^2 - 18x = -10x^2 - 36x
So, the simplified expression is:
(-10x^2 - 36x) / (x - 5)(x + 3)(x - 3)
Restrictions on the variable:
Since there are denominators involving (x - 5), (x + 3), and (x - 3), we have the following restrictions:
x ≠ 5
x ≠ -3
x ≠ 3
These values would cause the denominators to be equal to zero, which is undefined.
To simplify the given expression, let's first factor the denominators.
For the first fraction, x^2-2x-15 can be factored as (x-5)(x+3).
For the second fraction, x^2-8x+15 can be factored as (x-5)(x-3).
Now, let's rewrite the expression with the factored denominators:
2x / [(x-5)(x+3)] - 6x / [(x-5)(x-3)]
To combine the fractions, we need a common denominator. In this case, the common denominator is (x-5)(x+3)(x-3).
Now, we can rewrite the expression with the common denominator:
(2x * (x-3)) / [(x-5)(x+3)(x-3)] - (6x * (x+3)) / [(x-5)(x+3)(x-3)]
Next, we can simplify the numerators:
2x * (x - 3) = 2x^2 - 6x
-6x * (x + 3) = -6x^2 - 18x
Now, we can rewrite the simplified expression:
(2x^2 - 6x) / [(x-5)(x+3)(x-3)] - (-6x^2 - 18x) / [(x-5)(x+3)(x-3)]
To subtract the fractions, we need to find a common denominator. In this case, the common denominator is already present.
Therefore, we can combine the fractions:
(2x^2 - 6x + 6x^2 + 18x) / [(x-5)(x+3)(x-3)]
Simplifying the numerator:
2x^2 - 6x + 6x^2 + 18x = 8x^2 + 12x
Finally, the simplified expression in factored form is:
(8x^2 + 12x) / [(x-5)(x+3)(x-3)]
Restrictions on the variable:
From the factored denominators, we can see that the expressions (x-5), (x+3), and (x-3) cannot equal zero to avoid division by zero. Hence, the variable has restrictions at x = 5, x = -3, and x = 3.