Zero Polynomial Function

[mathdad]

In section 5.1, Sullivan introduces polynomial functions. On page 320, he made the following statement without giving THE WHY of his comment.

Here it is:

"The zero polynomial function f(x) = 0 + 0x + 0x^2 + • • • + 0x^(n) is not assigned a degree."

Can anyone explain, in simple terms, why the above statement is true?

 

[Harry_the_cat]

The degree of a polynomial in x is the highest power of x with a non-zero coefficient. Thus f(x) =0 has no degree.

 

[mathdad]

The degree of a polynomial in x is the highest power of x with a non-zero coefficient. Thus f(x) =0 has no degree.

Are you saying that x^(1) is the highest power of x when the coefficient is not zero?
When you say f(x) = 0, do you also mean y = 0?

 

[Harry_the_cat]

y=f(x) yes to your second question

I don't understand your first question "Are you saying that x^(1) is the highest power of x when the coefficient is not zero?"

Here are some examples which might help you understand "degree of a polynomial".

\(\displaystyle y=3x^4-5x^2 + 2x +3\) …..degree = 4

\(\displaystyle y=0.5x^2-3x +7\) ….. degree = 2

\(\displaystyle y = 2x +3\) ….. degree = 1

\(\displaystyle y = 7\) ….degree = 0 (since \(\displaystyle 7=7x^0\))

\(\displaystyle y=0\) ….. not assigned a degree, as Sullivan pointed out.

 

[mathdad]

y=f(x) yes to your second question

I don't understand your first question "Are you saying that x^(1) is the highest power of x when the coefficient is not zero?"

Here are some examples which might help you understand "degree of a polynomial".

\(\displaystyle y=3x^4-5x^2 + 2x +3\) …..degree = 4

\(\displaystyle y=0.5x^2-3x +7\) ….. degree = 2

\(\displaystyle y = 2x +3\) ….. degree = 1

\(\displaystyle y = 7\) ….degree = 0 (since \(\displaystyle 7=7x^0\))

\(\displaystyle y=0\) ….. not assigned a degree, as Sullivan pointed out.

When you said "The degree of a polynomial in x is the highest power of x with a non-zero coefficient" I thought you meant x to the first power. I now understand what you mean by the examples provided.