Question 701443: How do you find the number of possible x-intercepts and the number of changes in direction?
Answer by MathLover1(20849)   (Show Source):
You can put this solution on YOUR website! The  of a polynomial are where the polynomial intersects the  on the real coordinate plane. Mathematically speaking, these only occur when  is equal to  .
Polynomials can have   because of the way they curve. The number of a certain polynomial can have is the  of the polynomial.
A   polynomial can only have   . A  degree can have  to  , but it doesn't have to have four. For even degree polynomials, it is possible that there are no .
 degree polynomials must have    .
The are key to graphing a polynomial. They are points that you can connect that lie on the . These are also known as  to the  .
So, how do we find these ? Simply, these points are where  . So, you simply solve the polynomial for when or  is .
Usually, cross the straight through. However, there is more than one way that the polynomial can intercept the . There are actually    that the graph intercepts the . In the , it passes straight through no problem. In the  , it goes down and touches the and then rebounds off it. In the  , the graph sort of lingers around the interception point before crossing.
Why are there three types of intercepts? This is governed by a mathematical thing called  . Multiplicity is the number of times a particular  or solution appears. What if you ended up with the same twice? That means that that particular has a multiplicity of . It  twice, and it therefore has a multiplicity of  . An that  times has a of  .
Let us look at the following example:
We could commence the normal procedure for finding by setting the equal to  and solving. However, we end up with  twice, which means we get the intercept, ( ,), twice. The , (,) has a multiplicity of .
here are three cases how graph might cross
1. Normally, an has a multiplicity of , or it   . When this happens, the graph simply passes straight through the . It occurs once, so it passes through and continues along with the normal path that it takes.
2. If the intercept has an  multiplicity, meaning it occurs  , times,  times, etc., then the graph appears to touch the x-axis and then bounces off in the same direction it came from. The graph never passes through the x-axis, it simply touches it and goes back. As multiplicity increases, the valley will become flatter and flatter.
3.If the has an  multiplicity, meaning it   ,  times, etc., then the graph kind of lingers around the interception point before passing through. The graph does actually pass through, but it is sort of delayed before actually passing through, like in the image. As multiplicity increases, the deflection becomes closer and closer to the .
The graph of a polynomials of degree ,  for some constant  is a  line or  is a  line . It has no turning points and its tails are flat.
The graph of a polynomial of degree  ,  ,with  or  is a slant line with ,  points, and tails in opposite direction.
The graph of a polynomial of degree .  with  (leading coefficient) is a parabola that opens if and  if . The graph has turning point. It can have , , or  , those that are obtained by shifts, stretching or shrinking and reflections in the  negative (or negative) axis from  . The function that is not shifted, like  , have graphs with tails in opposite directions, one and no turning points. They are  if and  if .
-relative  or  minimum are values where the curve  
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