Polynomial function - different degrees don't understand

[zeion]

Homework Statement

Hello.
I don't understand this:

Let f(x) be a polynomial function of degree k+1, then f(x) has the form
a_k+1x^k+1 + ... + a₁x + a₀

Now the polynomial function has degree h(x) = f(x) - a_k+1(x-a) has degree <= k
How?

Homework Equations

The Attempt at a Solution

I can understand if (x-a) = x^k+1 then the k+1 th term will be gone. But what does (x-a) do?

 

[Mentallic]

Just think of (x-a) as (x-2) or (x-10), a being any number, just as a_k and a₂ etc. are any constants.

So $$h(x)=f(x)-a_{k+1}(x-a)$$, now we can just expand out and simplify:

$$h(x)=f(x)-a_{k+1}x+a\cdot a_{k+1}$$

Now obviously at this point the $$a_{k+1}x$$ will cancel with that from the polynomial f(x) and $$a\cdot a_{k+1}$$ is just some constant we can add to that of a₀ from f(x). See now that the degree of h(x) will be $$\leq k$$ simply because we have eliminated the (k+1)^th degree and the k^th degree could have a coefficient of zero, just like the cubic polynomial

$$2x^3-3x+1=0$$ has a coefficient of 0 for its 2^nd degree.

 

[zeion]

Now obviously at this point the $$a_{k+1}x$$ will cancel with that from the polynomial f(x)

So x has to equal x^k+1 ?

 

[Mark44]

Homework Statement

Hello.
I don't understand this:

Let f(x) be a polynomial function of degree k+1, then f(x) has the form
a_k+1x^k+1 + ... + a₁x + a₀

Now the polynomial function has degree h(x) = f(x) - a_k+1(x-a) has degree <= k
How?

As stated above, the degree of h(x) is still k + 1. I think you have a typo or there is is typo in what you're getting this from.

You have f(x) = a_k+1x^k+1 + ... + a₁x + a₀, and h(x) = f(x) - a_k+1(x - a). This means that h(x) = a_k+1x^k+1 + ... + a₁x - a_k+1x + a*a_k+1+ a₀.

I believe that h(x) should be defined this way: h(x) = f(x) - a_k+1(x-a)^{k + 1}[/color]. If so, then the two x^{k + 1} terms cancel, and you're left with a polynomial whose highest-degree term is x^k, making its degree <= k.

Homework Equations

The Attempt at a Solution

I can understand if (x-a) = x^k+1 then the k+1 th term will be gone. But what does (x-a) do?

 

[SpeedOfDark]

I'd like to make things simple for you the degree of a polynomial is it's highest exponent. Generally Trinomial + is considered a higher degree.

x^2 = Binomial
x^3 = Trinomial

Also another hint about degrees
1. They tell you the overall shape of the graph
2. They tell you how many x values there is

 

[zeion]

I'd like to make things simple for you

Thanks I never knew things could be so simple.

 

[Mentallic]

Oh man I should really avoid posting when I'm sleepy... zeion as you defined h(x) in your OP, it isn't true that it will be of degree less than or equal to k. Mark44 fixed it up from there.
Sorry about that.