Factoring trinomials
Multiplying two binomials gives a trinomial. Handed only the trinomial, how do you find the two binomials, and how do you know you have finished?
Where you are 3 inside
What this area is about
Factoring a trinomial runs multiplication backwards: x^2 + 13x + 22 becomes (x + 2)(x + 11), an equivalent expression written as a product instead of a sum. This area works with integer coefficients, where a trinomial may also turn out to be prime, with no such factoring at all.
How these fit together
All three children are procedures.
Factoring when the leading coefficient is one handles x^2 + bx + c: find two integers that multiply to c and add to b.
Factoring when the leading coefficient is not one handles ax^2 + bx + c with a \ne 1: split the middle term using two numbers with product ac and sum b, then factor by grouping.
Factoring completely and choosing a strategy puts every method together: take out the greatest common factor first, factor what is left, and stop only when each factor is prime over the integers.
A path through it
Leading coefficient one first, then any leading coefficient, then factoring completely, which draws on both.
Factoring is what makes the zero product property solve quadratics, and it is the first step with rational expressions and limits found by factoring.