Polynomial Inequalities

What is a polynomial?

A polynomial, informally speaking, is a chunk of algebraic terms (things with x) with variable powers and constants in front.

2x+6 is a polynomial.

3x2-2x+9 is a polynomial.

1679x4+81x2-9x is also a polynomial even though it may look crazy complicated.

What are polynomial inequalities?

2x+6>0 is a polynomial inequality. A polynomial is placed beside a greater than sign (>), not an equal sign, therefore making it an inequality.

3x2-2x+9<6 is also a polynomial inequality since there is a lesser than sign (<).

x2-2x+9>=-10x-3 is also a polynomial inequality since there is a greater or equal than sign (>=).

We encounter polynomial inequalities everywhere. So how do we solve them?

For example we will solve 2x+6>0.

There are some general steps:

Here's anther example:

(You might have been thinking that simple algebraic manipulation could have gotten the answer x<-3, instead of doing all this polynomial stuff. You’re right, but I wanted to show you the essence of the method. Here’s a harder example that shows you how useful this approach is to solving general polynomials.)