Write each rational expression in lowest terms. See Example 2. 3 (3 - t) / ((t + 5) (t - 3))

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Start by writing the given rational expression clearly: \(\frac{3(3 - t)}{(t + 5)(t - 3)}\).

Look for opportunities to factor or simplify the numerator and denominator. Notice that \(3 - t\) can be rewritten by factoring out a negative sign: \(3 - t = -(t - 3)\).

Substitute this back into the expression to get \(\frac{3 \cdot (-(t - 3))}{(t + 5)(t - 3)} = \frac{-3(t - 3)}{(t + 5)(t - 3)}\).

Now, observe that \((t - 3)\) appears in both numerator and denominator, so you can cancel these common factors, keeping in mind the domain restrictions where \(t \neq 3\) to avoid division by zero.

After canceling, the simplified expression is \(\frac{-3}{t + 5}\). This is the rational expression in lowest terms.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Factoring Polynomials

Factoring involves rewriting polynomials as products of simpler expressions. Recognizing common factors or special products like difference of squares helps simplify rational expressions by canceling common terms in numerator and denominator.

Simplifying Rational Expressions

A rational expression is simplified by factoring numerator and denominator and canceling out common factors. This process reduces the expression to its lowest terms, making it easier to work with or evaluate.

Domain Restrictions in Rational Expressions

The domain of a rational expression excludes values that make the denominator zero. Identifying these restrictions is essential to avoid undefined expressions and to correctly state the simplified form's valid input values.

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