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Polynomial factorization involves breaking down a polynomial into simpler factors that, when multiplied together, give back the original polynomial. This is similar to how we can factor numbers, like 12 into 2 and 6. For polynomials, the process often involves finding numbers (or expressions) that can be multiplied to reach the respective terms in the polynomial.

In our exercise, we dealt with the polynomial in the numerator:

• \( t^{2} - 3t + 2 \), which can be factored as \((t-1)(t-2)\).

This is achieved by finding two numbers that multiply to the constant term (2 in this case), and add up to the coefficient of the linear term (-3 here).

Similarly, the polynomial in the denominator,

• \( t^2 - 4 \), is factored as \((t-2)(t+2)\).

This uses the difference of squares formula which is particularly useful for any expression in the form of \(a^2 - b^2\) resulting in \((a-b)(a+b)\). Learning to recognize these patterns will make factorization quicker and easier.

Simplification of rational expressions is the process of making these expressions as simple as possible. Rational expressions, like the one in our exercise, are fractions where the numerator and the denominator are polynomials.

After factoring both the numerator and the denominator, you can cancel out any common factors. In our exercise, both the numerator and the denominator had \(t-2\) as a factor:

• Original form: \( \frac{(t-1)(t-2)}{(t-2)(t+2)} \).

Since \(t-2\) appears in both parts of the fraction, it can be canceled out, simplifying the expression to:

• \(\frac{t-1}{t+2}\).

Simplifying helps in evaluating the functions more easily and also reveals more about their behaviors and properties. Remember to not cancel factors directly without ensuring they are indeed factors of both parts of the fraction.

The domain of a function consists of all the possible values of the variable that make the function work without any issues. In rational functions, we particularly focus on the values that make the denominator zero, as these are the points where the function becomes undefined.

In our example, the denominator \(t+2\) becomes zero when \(t = -2\), making our function undefined at this value. Thus, \(t = -2\) is not included in the domain of the function.

The domain of \(f(t)\), then, is all real numbers except \(t = -2\). Checking the domain is crucial since it indicates where the function can be smoothly evaluated without running into mathematical problems. Always look for values that might make any part of the expression undefined and exclude those from the domain.