Factoring Quadratic Expressions

A few quick points

• A quadratic expression is a polynomial that can be written in the form \(ax^2+bx+c\) where \(a,b,c\in \mathbb{R}\).
  • That is, the coefficients in front of the \(x\)-stuff and at the end are any real numbers.
  • Ex: \(3.124x^2+\frac{1}{3}x-5\) is a quadratic expression.
• Quadratic expressions also take the form \((ax+b)(cx+d)\). After FOILing, you will get an expression of the same form as the above. Notice that the highest power that you get (after FOILing) is \(2\). This comes from multiplying the \(x\)’s as part of the FOILing process. This maximum power of \(2\) is what makes the expression quadratic.
  • To see this, try FOILing \((2x+4)(5x-2)\)
• Quadratic expressions can also have the form \((ax+b)^2+c\). Again, fully FOILing will give you an expression that has at most a power of \(2\), making the expression quadratic.
  • To see this, try expanding \((2x+5)^2+3\) and see what you get!

The following topics will talk about how we can factor many quadratic expressions. Remember throughout these topics that factoring is a means of “undoing” FOILing.