MHB What Determines the Degree and Coefficients of Polynomials?

[mathdad]

Specify the degree and the (nonzero) coefficients of each polynomial.

(A) 4

(B) 0

Solution:

The number 4 can be expressed as 4x^0. Is this correct?
If this is right, then the nonzero coefficient must be 4 itself. Is this right? The degree is 0.

The whole number 0 can be expressed as 0x^0. The degree is 0. What is the nonzero coefficient of 0?

Why is 4 a polynomial?

Why is 0 a polynomial?

 

[joypav]

(A) The degree of a constant is always 0. Any constant c can be written as cx^0.
(B) The degree of 0 is technically undefined. This is a polynomial but has no nonzero terms (obviously) and therefore has no degree.

These are certainly polynomials! More specifically, monomials, meaning they only have one term. A polynomial is a collection of constants and variables with exponents, but you cannot divide by a variable. Both 4 and 0 are then polynomials, because they do not break this rule.

 

[mathdad]

(A) The degree of a constant is always 0. Any constant c can be written as cx^0.
(B) The degree of 0 is technically undefined. This is a polynomial but has no nonzero terms (obviously) and therefore has no degree.

These are certainly polynomials! More specifically, monomials, meaning they only have one term. A polynomial is a collection of constants and variables with exponents, but you cannot divide by a variable. Both 4 and 0 are then polynomials, because they do not break this rule.

You said that we cannot divide a variable. Say, for example, x. Is x/2 not considered x divided by 2?

 

[joypav]

You said that we cannot divide a variable. Say, for example, x. Is x/2 not considered x divided by 2?

Not quite, if I understand what you're asking.

x/2 would be a polynomial. In this case, x is in the numerator. You CAN divide a variable by a constant. That is not an issue.

What I meant was, you CANNOT divide by a variable. Meaning, 2/x would not be a monomial. In this case, you have a variable in the denominator.

 

[mathdad]

Not quite, if I understand what you're asking.

x/2 would be a polynomial. In this case, x is in the numerator. You CAN divide a variable by a constant. That is not an issue.

What I meant was, you CANNOT divide by a variable. Meaning, 2/x would not be a monomial. In this case, you have a variable in the denominator.

I get it now.