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Factoring polynomials is a crucial step in simplifying rational expressions. To 'factor' means to break down a polynomial into simpler components that, when multiplied together, give back the original polynomial. For example, in the exercise, the numerator is given as \(2t + 6\). The greatest common factor (GCF) here is 2, so we can factor the numerator to get \(2(t + 3)\). This simplification makes it much easier to work with the expression.
Polynomials can take many forms, but factoring often involves recognizing patterns such as:

• Common factors (as shown above).
• Difference of squares.
• Perfect square trinomials.

Understanding these patterns will help you break down more complex polynomials into manageable parts.

The difference of squares is a special type of polynomial that can be factored easily. It follows the form \(a^2 - b^2\), which can be factored into \((a + b)(a - b)\). In the given exercise, the denominator is \(t^2 - 9\), which is a difference of squares since \(t^2\) is a perfect square and 9 is \(3^2\). Thus, the expression \(t^2 - 9\) can be rewritten as \((t + 3)(t - 3)\).
Recognizing a difference of squares allows us to factor quickly and simplify the expressions efficiently. Always look for this pattern when you see a subtraction between two squared terms.

Writing a rational expression in its lowest terms means simplifying it so that the numerator and the denominator have no common factors other than 1. After factoring, you can often cancel out these common factors. In the exercise, after factoring both the numerator and the denominator, we get the expression \(\frac{2(t + 3)}{(t + 3)(t - 3)}\). Because \(t + 3\) appears in both the numerator and the denominator, these terms cancel each other out, leaving us with \(\frac{2}{t - 3}\).
However, it's essential to note any restrictions. In this case, \(t\) cannot equal 3 because it would make the original denominator zero, leading to an undefined expression.
By always aiming for the lowest terms, you simplify calculations and reveal the core structure of the problem.

• Factor both the numerator and denominator first.
• Identify and cancel common factors.
• State any restrictions that apply to the variables.