Questions: Understand equivalent rational expressions: Rational Expressions Writing equivalent rational expressions with polynomial denominators Fill in the blank to make equivalent rational expressions. 4/(x+5) = [ ]/((x-9)(x+5))

Solution Steps

To make the rational expressions equivalent, we need to multiply the numerator and the denominator of the original expression by the same factor. In this case, the denominator of the second expression is \((x-9)(x+5)\), so we need to multiply the numerator of the first expression by \((x-9)\) to maintain equivalence.

The original rational expression is given by: \[ \frac{4}{x + 5} \]

We want to express this in terms of a new denominator: \[ \frac{\square}{(x - 9)(x + 5)} \] The new denominator is \((x - 9)(x + 5)\).

To maintain equivalence, we multiply the original numerator by \((x - 9)\): \[ \text{New Numerator} = 4 \cdot (x - 9) = 4x - 36 \]

Final Answer