Factoring trinomials: use product and sum, not random guessing

The monic case has two clean conditions

When the leading coefficient is one, list factor pairs of c and choose the pair whose sum is b. Signs follow from the product and sum.

A positive c means the pair has matching signs; a negative c means opposite signs.

Use ac to handle a leading coefficient

For ax² + bx + c, find numbers whose product is ac and whose sum is b. Split bx into those two terms, then factor by grouping.

This method is systematic and makes every coefficient visible, which is safer than guessing binomial fronts and backs.

Not every trinomial factors over the integers

If no integer pair works, the trinomial may be prime over the integers even though it factors over the reals or complex numbers.

The discriminant b² − 4ac predicts the root structure and can explain why integer factoring fails.

Worked example

Common mistakes

• Looking for a product of c instead of ac when a ≠ 1.
• Choosing a pair with the right product but wrong sum.
• Declaring failure after one factor pair.

Keep these ideas

• Factoring reverses multiplication.
• Use product ac and sum b.
• Some trinomials are prime over the integers.