Polynomial function - different degrees don't understand

In summary, the conversation discusses the degree of a polynomial function and how it can be determined by looking at the highest exponent. The original poster is confused about the degree of a polynomial function h(x) and how it can be less than or equal to k. The expert explains that the degree is determined by the highest exponent and provides examples to clarify the concept.
  • #1
466
1

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
ak+1xk+1 + ... + a1x + a0

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



Homework Equations





The Attempt at a Solution



I can understand if (x-a) = xk+1 then the k+1 th term will be gone. But what does (x-a) do?
 
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  • #2
Just think of (x-a) as (x-2) or (x-10), a being any number, just as ak and a2 etc. are any constants.

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

[tex]h(x)=f(x)-a_{k+1}x+a\cdot a_{k+1}[/tex]

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

[tex]2x^3-3x+1=0[/tex] has a coefficient of 0 for its 2nd degree.
 
  • #3
Mentallic said:
Now obviously at this point the [tex]a_{k+1}x[/tex] will cancel with that from the polynomial f(x)

So x has to equal xk+1 ?
 
  • #4
zeion said:

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
ak+1xk+1 + ... + a1x + a0

Now the polynomial function has degree h(x) = f(x) - ak+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) = ak+1xk+1 + ... + a1x + a0, and h(x) = f(x) - ak+1(x - a). This means that h(x) = ak+1xk+1 + ... + a1x - ak+1x + a*ak+1+ a0.

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

zeion said:

Homework Equations





The Attempt at a Solution



I can understand if (x-a) = xk+1 then the k+1 th term will be gone. But what does (x-a) do?
 
  • #5
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
 
  • #6
SpeedOfDark said:
I'd like to make things simple for you

Thanks I never knew things could be so simple.
 
  • #7
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.
 

What is a polynomial function?

A polynomial function is a mathematical function that consists of one or more terms, each of which is a constant multiplied by one or more variables raised to non-negative integer exponents. It can have different degrees, which indicate the highest exponent in the function.

What does it mean for a polynomial function to have different degrees?

The degree of a polynomial function is the highest exponent in the function. For example, a polynomial function with a degree of 2 would have an exponent of 2 in its highest term. A polynomial function can have different degrees for different terms within the function.

Can a polynomial function have more than one degree?

No, a polynomial function can only have one degree. However, it can have different degrees for different terms within the function, which can make it appear to have multiple degrees.

How do I determine the degree of a polynomial function?

To determine the degree of a polynomial function, you need to look at the highest exponent in the function. This will be the degree of the polynomial function.

Why is it important to understand different degrees in polynomial functions?

Understanding different degrees in polynomial functions is important because it allows us to accurately describe and analyze the behavior of the function. It also helps us to determine the number of roots and the shape of the graph of the function.

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