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

Verified step by step guidance

1. Identify the numerator and the denominator of the rational expression. The numerator is $3(3-t)$ and the denominator is $(t+5)(t-3)$.
2. Factor the numerator if possible. Notice that $3(3-t)$ can be rewritten as $3(-1)(t-3)$ by factoring out $-1$ from $(3-t)$, resulting in $-3(t-3)$.
3. Rewrite the expression using the factored form of the numerator: $-3(t-3)/((t+5)(t-3))$.
4. Cancel out the common factor $(t-3)$ from the numerator and the denominator, as long as $t\ne 3$ to avoid division by zero.
5. The simplified expression is $-3/(t+5)$.

Key Concepts

Rational Expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. To work with rational expressions, it is essential to understand how to manipulate polynomials, including factoring and simplifying them. This concept is foundational for performing operations such as addition, subtraction, multiplication, and division of rational expressions.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its factors. This is crucial for simplifying rational expressions, as it allows us to identify common factors in the numerator and denominator. Techniques such as finding the greatest common factor (GCF) or using special factoring formulas (like difference of squares) are commonly employed in this process.

Lowest Terms

A rational expression is said to be in lowest terms when the numerator and denominator have no common factors other than 1. To reduce a rational expression to its lowest terms, one must factor both the numerator and denominator and then cancel out any common factors. This simplification is important for clarity and accuracy in mathematical expressions.