Polynomial Inequalities

Hello, again.

That was cute.
Anyways, I'm here to make music and explain polynomial inequalities, and I'm all out of music. Seriously, I'm not in band this tri. It's awful. I haven't had 5 academic classes in a row for an extended period of time since 4th grade. But enough about the real world. Let's talk of polynomial inequalities.

One of the first things we did to start class this year was define some things:

Variable-A letter or symbol that represents a certain quantity.

Numeric Expression-Any set of numbers and/or variables and/or operations

Equation-Two expressions with one side equaling another.

Solution-A number or set of numbers (interval, perhaps?) that a variable may be to make an equation true.

So what is an inequality? I propose that an Inequality is an expression that shows the relative value of two other expressions. Examples:
X>Y

X is greater than Y

How about we add an operation or two?

2X>Y+9

How about we have fun with this?

6(X^2+12X+32)>Y/5

Oh my, it seems we have stumbled upon quite a brain buster. Well, you're supposed to treat inequalities the same as equations when it comes to solving, so let's do that. We'll try to find the zeros, so let's set Y=0 for simplicity's sake.

5*6(X^2+12X+32)>0

We'll, we can cancel some numbers here. We do, after all, have the zero product property, don't we? Let's cancel the 5*6

X^2+12X+32>0
Wait, this totally looks like it factors.

(X+4)(X+8)>0

Well, we know from the zero product property that either X+4>0 or X+8 is greater than zero, right? so X>-4 or X>-8.

Wait a minute...

If X>-4 or X>-8, doesn't that mean that X could be any number above -4 or -8, not just zero? So they could multiply into something other than 0? There are infinitely more things they could multiply into! Therefore, THE ZERO PRODUCT PROPERTY DOES NOT APPLY TO INEQUALITIES!!!!!!

What it can do for us, however, is give us test values by setting the in equality into an equation.
That in mind, let's change X=-4 and X=-8 into test values.

We have essentially three segments of a number line here to plot the solutions of this inequality.
We have everything below -8, everything between -8 and -4, and everything above negative 4. So, what we need to do is pick some arbitrary points on the number line and plug them in to our in equality that is set for 0. All that matters, since 0 is our reference, is whether the number is positive or negative. Let's do -9, -5, and 0

-9^2+12(-9)+32=+

-5^2+12(-5)+32=-

0+0+32=+

So, we see that the values of X are positive (>0) at all numbers below -8 and above -4.

So our number line would end up looking something like this...


And the graph would look something like this...


And that is how we deal with polynomial inequalities. It's the same with cubes and all other degrees of polynomials: get zero on one side of the inequality and find your test values, then test them.

A brief refresher is always nice. Honors' Precalc is tough. Chin up, eyes foreward. We're all in this together. Let's make this one heck of a learning experience.
-Shane McPartlin