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

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

Look for any factors in the numerator and denominator that can be simplified or canceled. Notice that \(3 - t\) can be rewritten to relate to \(t - 3\) by factoring out a negative sign: \(3 - t = -(t - 3)\).

Rewrite the numerator using this relationship: \(3(3 - t) = 3[-(t - 3)] = -3(t - 3)\).

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

Now, cancel the common factor \((t - 3)\) from numerator and denominator, remembering to keep the negative sign, resulting in \(\frac{-3}{t + 5}\), which is the expression in lowest terms.

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주요 개념

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Rational Expressions

A rational expression is a fraction where both the numerator and denominator are polynomials. Simplifying rational expressions involves factoring and reducing common factors, similar to simplifying numerical fractions.

Factoring Polynomials

Factoring is the process of breaking down a polynomial into simpler polynomials that multiply to give the original. Recognizing common factors or special products helps simplify expressions and is essential for reducing rational expressions.

Simplifying Rational Expressions

To simplify a rational expression, factor both numerator and denominator completely, then cancel out any common factors. This reduces the expression to its lowest terms, making it easier to work with or interpret.