Basic Concepts Involved in Factoring Trinomials

(where the coefficient of the squared term is \(1\))

To factor a trinomial of the form \(x^{2}+bx+c\), start by finding two numbers, \(f\) and \(g\), that

• Add together to give \(b\) (the coefficient of the x term); and
• Multiply together to give \(c\) (the constant term).
  

Then:

\(x^{2}+bx+c=x^{2}+\overset{=b}{\overbrace{(f+g))}}x+\overset{=c}{\overbrace{fg}}=(x+f)(x+g)\)

For example, to factor \(x^{2}+5x+6\), we must find two numbers that add to \(5\) and multiply to \(6\).

The numbers \(2\) and \(3\) work, since \(2+3=5\) and \(2\cdot 3=6\).

Thus:

\(x^{2}+5x+6=x^{2}+(2+3)x+(2\cdot 3)=(x+2)(x+3)\)

(FOIL it out to check!)

When everything in sight is positive and coefficients are small, then it may be easy to come up with the numbers that work. For example, it may not be too hard for you to find numbers that add to \(5\) and multiply to \(6\).

However, bring some negative numbers into the picture and make coefficients bigger, and things can get considerably trickier.

Fortunately, there are some key ideas that will help you find the numbers that work (if they exist), and the purpose of this web exercise is to give you practice with these ideas.