Question

Perform the indicated operations and simplify.\(\frac{t^{2}-t-6}{t^{2}+6 t+9} \cdot \frac{t+3}{t^{2}-4}\)

Short answer

The simplified expression is \(\frac{1}{(t-2)}\)

Step-by-step solution

Factorize the given polynomials

Factorize the given polynomials: \(t^{2}-t-6 = (t-3)(t+2)\), \(t^{2}+6t+9 = (t+3)^2\), \(t^2 - 4 = (t-2)(t+2)\). Then the expression can be rewritten as: \(\frac{(t-3)(t+2)}{(t+3)^2} \cdot \frac{(t+3)}{(t-2)(t+2)}\)

Cancel out similar factors

Let's cancel out the similar factors in the numerator and the denominator. We keep in mind that this can only be done if the same factor appears in both the numerator and the denominator. What can be canceled is (t+2) and (t+3) as: \(\frac{(t-3)}{(t+3)}\cdot\frac{1}{(t-2)}\)

Simplify

Lastly, simplify the equation to get the answer as: \(\frac{(t-3)}{(t-3)(t-2)}\)

Key concepts

Factorization

Factorization is a key component in simplifying polynomial expressions and involves breaking down a complex expression into simpler, multipliable components, known as factors. Each factor is a simpler polynomial, sometimes a linear term. In this exercise, we began by factorizing each polynomial from the given rational expression:

• The quadratic polynomial \(t^{2} - t - 6\) was factorized to \((t-3)(t+2)\).
• The trinomial \(t^{2} + 6t + 9\) factorized to a perfect square: \((t+3)^2\).
• The polynomial \(t^{2} - 4\) is a difference of squares, and its factors are \((t-2)(t+2)\).

By understanding factorization, you can transform polynomials into a product of simpler terms, making subsequent operations like simplification more manageable.
This initial step is crucial because it sets the stage for canceling terms in the rational expression, thereby simplifying the problem.

Rational Expressions

Rational expressions are fractions that involve polynomials in the numerator and the denominator. The operations performed on these expressions are similar to those for numerical fractions, but they incorporate algebraic expressions.
In this problem, after the initial factorization, the rational expression was reconfigured into \[\frac{(t-3)(t+2)}{(t+3)^2} \cdot \frac{(t+3)}{(t-2)(t+2)}\]This form is essential because it allows for the cancellation of common terms, provided those terms are not zero. By interpreting and working with rational expressions correctly, you can simplify the entire algebraic expression effectively.
It's similarly fundamental to be cautious of implicated restrictions, especially when simplifying these expressions, such as avoiding division by zero. Always ensure that none of the values of \(t\) that make the denominator zero are included in solutions, as they are undefined in math.

Simplification

Simplification is the process of reducing an expression to its simplest form, which often makes it more comprehensible and easier to work with. In the exercise, after factorization, we proceeded to simplify by canceling out common factors from both the numerator and the denominator.

• The term \((t+2)\) was canceled out, as it appeared both in the numerator and denominator.
• Similarly, \((t+3)\) was canceled, since it appeared in both the numerator and denominator.

After canceling, the expression simplifies further to\[\frac{(t-3)}{(t-2)}\]From \[\frac{t-3}{t+3}\cdot\frac{1}{(t-2)}\]
Simplifying rational expressions helps you arrive at the most reduced form of the expression, making it better suited for further operations or evaluations.
The concept of simplification is essential because it leads you to the most elegant and condensed expression possible, while retaining all original relationships and values, except those eliminated due to division by zero.