91Ó°ÊÓ

To master factoring polynomials, it's crucial to understand the concept of the Greatest Common Factor (GCF). The GCF is the largest factor that can divide each term in a polynomial without leaving a remainder. Finding the GCF is an essential step in simplifying and factoring expressions.
For the polynomial in our exercise, we first grouped the terms into two pairs: \(t^3 + 8t^2 - t - 8\) becomes \( (t^3 + 8t^2) + (-t - 8) \).
Next, we need to find the GCF of each group. For the first group, \(t^3 + 8t^2 \), the GCF is \(t^2 \), because \(t^2 \) is the highest power of t common to both terms. So, we factor out \(t^2 \), leaving us with \[ t^2(t + 8) \].
For the second group, \(-t - 8 \), the GCF is \(-1 \), because \(-1 \) is the simplest common factor. Factoring out \(-1 \) gives us \[-1(t + 8) \].
With the GCF factored out in each group, the expression simplifies to \[ t^2(t + 8) - 1(t + 8) \].

Factoring by grouping is a technique used to factor polynomials with four or more terms. The idea is to group terms in such a way that each group has a common factor, which then can be factored out, leading to a simpler expression.
In our exercise, after grouping and factoring out the GCF from each group, our expression is \[ t^2(t + 8) - 1(t + 8) \].
We notice that \(t + 8 \) is a common factor in both terms of the simplified expression. We can factor out \(t + 8 \), leading to: \[ (t^2 - 1)(t + 8) \].
Factoring by grouping thus simplifies the polynomial to an easier expression where further factoring techniques, like the difference of squares, can be applied.

The difference of squares is a special factorization method used when a polynomial takes the form \(a^2 - b^2 \). This can be factored into \[(a + b)(a - b) \].
In our exercise, we now have \[ (t^2 - 1)(t + 8) \]. The term \(t^2 - 1 \) fits the difference of squares pattern: \[ t^2 - 1 = (t + 1)(t - 1) \].
Using this pattern, we factor the expression further to \[ (t + 1)(t - 1)(t + 8) \].
This method is very powerful for simplifying expressions and solving equations involving squares, making complex polynomials more manageable.