Question

Which of the following is a polynomial?

Options

• `x^2 - 5x + 4sqrt(x) + 3`

• `x^(3//2) - x + x^(1//2) + 1`

• `sqrt(x) + 1/sqrt(x)`

• `sqrt(2)x^2 - 3sqrt(3)x + sqrt(6)`

Answer 1

`bb(sqrt(2)x^2 - 3sqrt(3)x + sqrt(6))`

Explanation:

A polynomial is an algebraic expression where the powers (exponents) of the variables must be non-negative integers (i.e., 0, 1, 2, 3,...).

Let’s evaluate each option by looking at the exponents of x:

a. `x^2 - 5x + 4sqrt(x) + 3`: The term `4sqrt(x)` can be written as `4x^(1//2)`. Since the exponent `1/2` is a fraction (not an integer), this is not a polynomial.

b. `x^(3//2) - x + x^(1//2) + 1`: The exponents `3/2` and `1/2` are fractions, so this is not a polynomial.

c. `sqrt(x) + 1/sqrt(x)`: This can be written as `x^(1//2) + x^(-1//2)`. The exponents are a fraction and a negative number, so this is not a polynomial.

d. `sqrt(2)x^2 - 3sqrt(3)x + sqrt(6)`: The exponents of x are 2 and 1, which are both non-negative integers. The square roots are only on the coefficients (numbers), which is completely valid. Therefore, this is a polynomial.