Where g is a polynomial function of degree n-1

In summary, if f(a)=0 for some a \in \mathbb{R}, the either f is constant or f(x)=(x-a)g(x), where g is a polynomial function of degree n-1.
  • #1
17
0
Let
Code:
 [ tex ]f(x)=\sum_{i=0}^n c_i x^i[ / tex ]
be an arbitrary polynomial function of degree n

Show that if f(0)=0 then either f is constant or f(x)=xg(x), where g is a polynomial function of degree n-1

I don't know how to start. Please help

Thank you in advance
 
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  • #2


So, 0 is a root of this polynomial. What does that tell you? Factoring?
 
  • #3


Since 0 is a root of f, the polynomial x divides the polynomial f. Therefore, there exists g such that xg(x)=f(x). However, if we write p the degree of g, degree of f= p+1 (since xg(x)=f(x)). Therefore if f equals zero, g equals zero. If not p=n-1
 
  • #4


is that correct ?
 
  • #5


jawad1 said:
is that correct ?

It seems you are making the argument much more obscure than it needs to be. Just put x = 0 in f(x) = c[0] + c[1]*x + c[2]*x^2 + ... + c[n]*x^n.

RGV
 
  • #6


For the first part, I understood: we will get f(0)=c_0=0 which is a constant

However show that f(x)=xg(x) doesn't seem so obvious to me
 
  • #7


jawad1 said:
For the first part, I understood: we will get f(0)=c_0=0 which is a constant

However show that f(x)=xg(x) doesn't seem so obvious to me

You don't see why
$$c_1x + c_2x^2 + \cdots + c_n x^n = x \cdot (\text{something})?$$

RGV
 
  • #8


ok I see. I do the same thing then for question 2 right ?
 
  • #9


jawad1 said:
ok I see. I do the same thing then for question 2 right ?

I do not see any question 2.

RGV
 
  • #10


Oh I am so sorry. I forgot to post it.

I need to show that if f(a)=0 for some a \in \mathbb{R}, the either f is constant or f(x)=(x-a)g(x), where g is a polynomial function of degree n-1
 

1. What is a polynomial function?

A polynomial function is a mathematical expression made up of constants and variables raised to non-negative integer powers, with the variables being multiplied together. It can also have coefficients, which are constants multiplied by the variables. Examples of polynomial functions include x^2 + 3x + 5 and 2y^3 - 6y + 1.

2. What is the degree of a polynomial function?

The degree of a polynomial function is the highest exponent of the variable in the expression. For example, in the polynomial function 2x^3 + 5x^2 + 3x + 1, the degree is 3 because the variable x is raised to the highest power of 3.

3. How do you determine the degree of a polynomial function?

To determine the degree of a polynomial function, you can look at the term with the highest exponent of the variable. If there are multiple terms with the same highest exponent, then the degree is equal to that exponent. If there are no exponents in the expression, then the degree is 0.

4. What does it mean for a polynomial function to have degree n-1?

When a polynomial function is described as having degree n-1, it means that the highest exponent of the variable in the expression is one less than the value of n. For example, if n=5, then a polynomial function of degree n-1 would have a highest exponent of 4.

5. How does the degree of a polynomial function affect its graph?

The degree of a polynomial function affects its graph in several ways. The degree determines the number of possible x-intercepts and turning points of the function. It also affects the end behavior of the graph, as well as the overall shape and symmetry of the curve. Generally, the higher the degree of a polynomial function, the more complex and varied its graph will be.

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