Question

Determine whether each of the following is a polynomial expression or not. Give your reasons.

1) Expression 1 is a polynomial because it only contains a single term with a variable raised to the power of 1.
2) Expression 1 is not a polynomial because it does not contain any variable.
3) Expression 1 is not a polynomial because it contains a variable raised to a non-integer power.
4) Expression 1 is not a polynomial because it contains a square root symbol.

Answer 1

Determining if an expression is a polynomial involves checking for variables raised to non-negative integer powers, no variables in denominators, and no radical signs; statements 1 and 2 could represent polynomials, while statements 3 and 4 cannot.

Step-by-step explanation:

To determine whether each statement about a polynomial expression is true, we need to understand the definition of a polynomial. A polynomial is an expression that can have constants, variables and exponents, but never division by a variable, variables in the denominator, negative exponents, or variables under a radical. It consists of terms which are simply the product of a coefficient and a variable raised to a non-negative integer power.

1) The first statement can be true for a polynomial if the term has a variable raised to the power of 1. This fits the definition of a polynomial term and can be part of a polynomial expression.

2) The second statement is false for identifying a polynomial because the lack of variables means it is simply a constant term, which can still be part of a polynomial expression.

3) The third statement is true for identifying a non-polynomial because a variable raised to a non-integer power does not fit the definition of a polynomial term.

4) The fourth statement is true for identifying a non-polynomial because a square root symbol implies a variable raised to a fractional power, which is not allowed in a polynomial expression.