Simplify each rational expression. State all restrictions on the variables.
10𝑥+30 / 𝑥^2−2𝑥−15 × 𝑥^2−5𝑥+6 / 5(𝑥−3)2
To simplify the rational expression, we need to cancel out any common factors in the numerator and denominator.
First, let's factor each quadratic expression:
x^2 - 2x - 15 = (x - 5)(x + 3)
x^2 - 5x + 6 = (x - 2)(x - 3)
Now, we can simplify the rational expression:
(10𝑥 + 30) / (𝑥^2 − 2𝑥 − 15) × (𝑥^2 − 5𝑥 + 6) / (5(𝑥 − 3)^2)
= (10(x + 3)) / ((x - 5)(x + 3)) × ((x - 2)(x - 3)) / (5(𝑥 − 3)^2)
= (10(x + 3)(x - 2)(x - 3)) / (5(x - 5)(𝑥 − 3)^2)
Now, we can cancel out the common factors:
= (2(x - 2)) / (x - 5)
= (2x - 4) / (x - 5)
The restriction in this expression is that x cannot be equal to 5 or 3, as this would make some denominators zero, resulting in undefined values.