I'll explain.
First of all, when people say 'find the zeros' they basically mean 'find the solutions.' The reason for calling them 'zeroes' is ACTUALLY important. It tells you WHAT a solution is.
Many people, even people who are all the way up in pre-calculus don't know this...
The solutions of a function are the points at which it crosses the x-axis.
With that being said, let's find the zeros of this function:
y=x^4-8x^2+16
First of all, we must factor it.
y=x^4-8x^2+16
Becomes...
y=(x+2)(x-2)(x+2)(x-2)
We can find the solutions by setting these equal to 0!
x+2=0, x=-2
x-2=0, x=2
x+2=0, x=-2
x-2=0, x=2
Now, those are your answers. Continue if you want to learn something interesting.
Look at the above picture of the graph. I want you to look at where the graph touches the x-axis. These are the solutions.
Now I will point out something interesting.
There are 3 ways in which a polynomial function touches the x-axis. I like to call them the...
Cross through: this is when the graph goes strait through the x-axis like a line.
Bounce: this is when the graph touches the x-axis at one point and then bounces back.
Wiggle through: this is when the graph makes a little 'wiggle' as it crosses the x-axis.
WHY am I pointing this out?
Well, the way in which the graph crosses the x-axis will tell you how many of the same factor the equation has.
Cross through: This indicates that the graph only has one factor that equals the value of the point it crosses through
Example: y=(x-2)(x-2)(x+1) this means at -1, the graph crosses strait through the x-axis.
Bounce: like in our example, a bounce indicates that there are two factors that equal the value of the point it touches.
Example: y=(x+2)(x-2)(x+2)(x-2) this means that at -2 and 2, the graph bounces off the x-axis
Wiggle: this indicates that 3 factors that equal the value of the point it crosses through
Example: y=(x+1)(x+1)(x+1)(x-2) the graph wiggles through the x-axis at -1
I hope this helps! Ty for reading:)
