When factoring a trinomial, why is it necessary to write the trinomials in four terms

When factoring a trinomial, it is not always necessary to write it in four terms. The four-term expression is most commonly used in a specific case known as "factoring by grouping." Factoring by grouping involves dividing the trinomial into two groups of two terms each, and then factoring out a common factor from each group.

To factor a trinomial using the four-term approach, follow these steps:

1. Write the trinomial in expanded form, with the highest power term first, the middle term second, and the constant term last.
2. Break down the middle term into two terms whose sum matches the coefficient of the middle term and whose product equals the constant term.
3. Group the terms in pairs, where the first pair consists of the highest power term and one of the terms obtained in step 2, and the second pair consists of the remaining term and the other term obtained in step 2.
4. Factor out the greatest common factor (if any) from each pair.
5. Look for common factors between the two grouped pairs.
6. Factor out the common factors from each pair.
7. Write the factored form of the trinomial using the factored expressions from step 4 and step 6.

In other cases where factoring a trinomial does not involve grouping, you may not need to write it in four terms. For example, when factoring a perfect square trinomial or a difference of squares trinomial, you can directly apply specific factoring patterns without the need to manipulate the trinomial into four terms.