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Chapter 6: Problem 85

Write each rational expression in lowest terms. $$ \frac{6-t}{t-6} $$

Short Answer

Expert verified
The expression simplifies to -1.

Step by step solution

01

Identify the Rational Expression

The given rational expression is \( \frac{6 - t}{t - 6} \). Identify the numerator and the denominator.
02

Factor the Expression

Notice that the numerator and the denominator are very similar. In fact, \( 6 - t \) can be rewritten as \( -(t - 6) \).
03

Simplify the Expression

Rewrite \( \frac{6 - t}{t - 6} \) as \( \frac{-(t - 6)}{t - 6} \). The \( t - 6 \) terms cancel out, leaving \( -1 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simplifying Rational Expressions
Simplifying rational expressions is a fundamental concept that can help make complex problems manageable. A rational expression is simply a fraction where both the numerator and the denominator are polynomials. For example, in the exercise we just saw, we had $$ \frac{6 - t}{t - 6} $$ which is itself a rational expression. Simplifying such expressions often involves factoring polynomials and canceling out common factors. By simplifying, we convert the given expression into its simplest form, making further calculations easier.
Factoring Polynomials
Factoring polynomials is a critical step in simplifying rational expressions. In simple terms, factoring means breaking down a polynomial into simpler 'factors' that, when multiplied together, will give you the original polynomial. In our exercise, we noticed that the numerator and the denominator are close in form: $$6-t$$ and $$t-6$$. We can factor the numerator by recognizing that it can be written as the product of -1 and $$t-6$$. So,
  • $$6-t$$ can be rewritten as $$-(t-6)$$
By factoring numerator and rewriting it, the rational expression simplifies significantly.
Numerator and Denominator
A key part of understanding rational expressions involves knowing the differences between the numerator and the denominator.
  • The numerator is the top part of the fraction.
  • The denominator is the bottom part.
In the expression $$\frac{6 - t}{t - 6}$$, the numerator is 6 - t, and the denominator is t - 6.
  • Notice how they are nearly identical but have opposite signs.
By understanding this, you can easily convert one into another to simplify the expression. Once the numerator was factored as $$-(t-6)$$ instead of $$6-t$$, simplifying the fraction became straightforward. By cancellation of $$t-6$$ terms, we arrived at the simplest form: -1, showing how understanding the parts of the expression can ease simplification.

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Most popular questions from this chapter

involve factoring by grouping (Section 5.1) and factoring sums and differences of cubes (Section 5.4). Write each rational expression in lowest terms $$ \frac{1+p^{3}}{1+p} $$

Multiply or divide as indicated. Write each answer in lowest terms. $$ \frac{7}{12} \div \frac{15}{4} $$

Find the greatest common factor of each group of terms. $$ 84 q^{3}, 90 q^{6} $$

Write each rational expression in lowest terms. $$ \frac{q^{2}-4 q}{4 q-q^{2}} $$

Rewrite each rational expression with the indicated denominator. $$ \frac{36 r}{r^{2}-r-6}=\frac{?}{(r-3)(r+2)(r+1)} $$
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