# What is the number of zeros of the ‘zero polynomial’?

(a). 0

(b). 1

(c). 2

(d). Infinite

Answer

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Hint: Recall the definition for zero polynomial and zeros of a polynomial and determine the number of roots of the zero polynomial.

Complete step-by-step answer:

A polynomial is an expression consisting of variables raised to some exponents along with their coefficients. It involves only the operations of addition, subtraction, multiplication, and not negative integer exponentiation. Examples of a polynomial are 5x, \[6{x^2} + 5\] and so on.

Zero polynomial is the constant polynomial whose coefficients are all equal to 0, hence, it’s name. It is a constant function with value zero and it takes all values of x to a single value, that is, zero. It is also the additive identity of the group of polynomials, since any polynomial added to zero polynomial gives back the same polynomial.

The zero polynomial is represented as follows:

P(x) = 0

We know that the zeros of any polynomial is the value of x for which the value of the polynomial becomes zero. For example, the zero of the polynomial p(x) = x – 1 is x = 1, since, for the value of x equal to 1, the value of the polynomial becomes 0.

We see that for the zero polynomial, P(x) = 0, any value of x, will give the value of P(x) to be zero.

Hence, the number of zeros of the polynomial is infinite.

Hence, option (d) is the correct answer.

Note: You might wrongly select option (a), thinking that the degree of the polynomial is zero, hence, it has zero solution. Yes, it is true the number of zeros is equal to the degree but zero polynomial is special and its value itself is equal to zero.

Complete step-by-step answer:

A polynomial is an expression consisting of variables raised to some exponents along with their coefficients. It involves only the operations of addition, subtraction, multiplication, and not negative integer exponentiation. Examples of a polynomial are 5x, \[6{x^2} + 5\] and so on.

Zero polynomial is the constant polynomial whose coefficients are all equal to 0, hence, it’s name. It is a constant function with value zero and it takes all values of x to a single value, that is, zero. It is also the additive identity of the group of polynomials, since any polynomial added to zero polynomial gives back the same polynomial.

The zero polynomial is represented as follows:

P(x) = 0

We know that the zeros of any polynomial is the value of x for which the value of the polynomial becomes zero. For example, the zero of the polynomial p(x) = x – 1 is x = 1, since, for the value of x equal to 1, the value of the polynomial becomes 0.

We see that for the zero polynomial, P(x) = 0, any value of x, will give the value of P(x) to be zero.

Hence, the number of zeros of the polynomial is infinite.

Hence, option (d) is the correct answer.

Note: You might wrongly select option (a), thinking that the degree of the polynomial is zero, hence, it has zero solution. Yes, it is true the number of zeros is equal to the degree but zero polynomial is special and its value itself is equal to zero.

Last updated date: 30th Sep 2023

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