When P(x) touches but does not cross the x-axis at (r, 0), what must be true of P(x)? Give an example of a polynomial for which this is true.

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Paplow

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When P(x) touches but does not cross the x-axis at (r, 0), what must be true of P(x)? Give an example of a polynomial for which this is true.
 
Surely you can come up with an example. Think one of the simplest polynomials there is. Translate it so it's vertex appears at [MATH]x=r[/MATH], rather than [MATH]x=0[/MATH]
 
Romsek said:
Surely you can come up with an example. Think one of the simplest polynomials there is. Translate it so it's vertex appears at [MATH]x=r[/MATH], rather than [MATH]x=0[/MATH]
Click to expand...
How about y=x^2+2x+1
 
Paplow said:
How about y=x^2+2x+1
Click to expand...

that only touches the x-axis at [MATH]r[/MATH] if [MATH]r=-1[/MATH]
move the parabola [MATH]y=x^2[/MATH] over so the vertex is at [MATH]x=r[/MATH]what do you get?
 
Romsek said:
that only touches the x-axis at [MATH]r[/MATH] if [MATH]r=-1[/MATH]
move the parabola [MATH]y=x^2[/MATH] over so the vertex is at [MATH]x=r[/MATH]what do you get?
Click to expand...
P(x)=r^2+2x+1? I'm sorry this one question is just so complicated to me, forgive my brain.
 
Let's bring this back from the abstract to the concrete.
For now let r=3, say. We can generalise later.
If the graph passes through (3, 0) then 3 must be an x intercept. Agree?
So (x-3) must be a factor.
BUT the question says it TOUCHES the x axis at 3, but doesn't cross it.
May I suggest you look at the graphs of
y=x-3
y =(x-3)^2
y =(x-3)^3
y =(x-3)^4 etc and see if that helps you.
 
If x=r is a root of P(x), then (x-r) must be a factor of P(x).
 
Paplow said:
P(x)=r^2+2x+1? I'm sorry this one question is just so complicated to me, forgive my brain.
Click to expand...

You really should know that if [MATH]P(x)=x^2[/MATH] has a vertex at [MATH]x=0[/MATH]then [MATH]P(x) = (x-r)^2[/MATH] has a vertex at [MATH]x=r[/MATH]
The function [MATH]P(x)=(x-r)^2[/MATH] touches but does not cross the x-axis at [MATH]x=r[/MATH]
As far as what must be necessary of [MATH]P(x)[/MATH] I imagine it's that if you were to write
[MATH]P(x) = (x-r)^N (x-\rho_1)(x-\rho_2) \dots (x-\rho_M)[/MATH], where the [MATH]\rho[/MATH]'s are all the roots
of [MATH]P(x)[/MATH] other than [MATH]r[/MATH], then [MATH]N[/MATH] must be an even number 2 or greater.

In the cast of the simple parabola I used as an example, [MATH]N=2[/MATH] and there are no roots other than [MATH]r[/MATH]That's why that example is the simplest one possible.
 
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