Polynomial Functions, what am I doing wrong?

[L3]

I'm trying to learn about Polynomial Funtions and am being rather confused. So it says they have the form,

f(x) = a[sub:2nfsv4p3]n[/sub:2nfsv4p3] x[sup:2nfsv4p3]n[/sup:2nfsv4p3] + a[sub:2nfsv4p3]n – 1[/sub:2nfsv4p3] x[sup:2nfsv4p3]n – 1[/sup:2nfsv4p3] + ... + a[sub:2nfsv4p3]2[/sub:2nfsv4p3] x[sup:2nfsv4p3]2[/sup:2nfsv4p3] + a[sub:2nfsv4p3]1[/sub:2nfsv4p3] x + a[sub:2nfsv4p3]0[/sub:2nfsv4p3]

I thought I understood that, but then got confused when they started showing examples.

So they said if, n = 4 then it's a quartic polynomials: f(x) = a[sub:2nfsv4p3]4[/sub:2nfsv4p3] x[sup:2nfsv4p3]4[/sup:2nfsv4p3] + a[sub:2nfsv4p3]3[/sub:2nfsv4p3] x[sup:2nfsv4p3]3[/sup:2nfsv4p3] + a[sub:2nfsv4p3]2[/sub:2nfsv4p3] x[sup:2nfsv4p3]2[/sup:2nfsv4p3] + a[sub:2nfsv4p3]1[/sub:2nfsv4p3] x + a[sub:2nfsv4p3]0[/sub:2nfsv4p3]

And they gave as an example,
f(x) = -x[sup:2nfsv4p3]4[/sup:2nfsv4p3] - 2 * x[sup:2nfsv4p3]3[/sup:2nfsv4p3] - (0.5)

I don't understand at all. Why do we go from x[sup:2nfsv4p3]4[/sup:2nfsv4p3] to x[sup:2nfsv4p3]3[/sup:2nfsv4p3] and then skip the other x's? Or, as in another example,

f(x) = x[sup:2nfsv4p3]4[/sup:2nfsv4p3]- 2 * x[sup:2nfsv4p3]2[/sup:2nfsv4p3],

Do we skip x[sup:2nfsv4p3]3[/sup:2nfsv4p3]?

Also, why does in first example does a appear to be -1, then -2 and then 0.5? Does it's value change depending on whether it's a[sub:2nfsv4p3]n[/sub:2nfsv4p3] or a[sub:2nfsv4p3]n-1[/sub:2nfsv4p3].

I realize there's clearly some concept here I'm not grasping but I don't know what it is, and I tried looking up Polynomial Functions on other sites and still was left confused.

 

[galactus]

A polynomial does not have to have all the consecutive x terms. i.e. if we have \(\displaystyle x^{3}+x-1\)

There is no x^2 term. That is OK. It is still a polynomial.

The coefficients(numbers in front of the x terms) can be any real value.


The Fundamental Theorem of Algebra states that a polynomial has the same number of roots as the highest power.

So, if we have \(\displaystyle x^{5}+3x^{4}-2x-5=0\). It has 5 roots. A root is that value of x that makes the polynomial equal 0.

In this case, the 5 solutions are \(\displaystyle -.01875+1.14359i, \;\ -.01875-1.14359i, \;\ -3.0124, \;\ -1.1017, \;\ 1.15166\)

5 roots. Two are complex and three are real.

 

[fasteddie65]

The "missing terms" simply have a coefficient of 0. Not all terms have to be present for a polynomial.

 

[L3]

Thank-you very very much for the help, math tends to confuse me.