Write each rational expression in simplest form. State all restrictions on the variables.
4𝑥^2+5𝑥−6/𝑥^2−6𝑥−16
To simplify the rational expression 4𝑥^2 + 5𝑥 - 6 / 𝑥^2 - 6𝑥 - 16, we can factor both the numerator and denominator.
The numerator factors as: (4𝑥 - 3)(𝑥 + 2)
The denominator factors as: (𝑥 - 8)(𝑥 + 2)
Cancel out the common factor of (𝑥 + 2) from both the numerator and denominator:
(4𝑥 - 3)(𝑥 + 2) / (𝑥 - 8)(𝑥 + 2) = (4𝑥 - 3) / (𝑥 - 8)
The expression simplifies to (4𝑥 - 3) / (𝑥 - 8) in its simplest form.
Restrictions on the variable: The expression is undefined when the denominator (𝑥 - 8) is equal to zero. Therefore, 𝑥 cannot be equal to 8.