A rational expression is a fraction where the numerator and the denominator are polynomials. We will determine whether each of the given expressions is rational or not. To do so, we will check if the numerator and the denominator of each expression is a polynomial. Let's start by checking the numerators.
Fraction
Numerator
Is the Numerator a Polynomial?
1/x-1
1
Yes ✓
x^2+3x+2/x-5
x^2+3x+2
Yes ✓
x+1/sqrt(x+3)
x+1
Yes ✓
x^2+1/3
x^2+1
Yes ✓
We can see that all numerators of the given expressions are polynomials. So far, we cannot identify the expression that does not belong. Now let's take a look at the denominators.
Fraction
Denominator
Is the Denominator a Polynomial?
1/x-1
x-1
Yes ✓
x^2+3x+2/x-5
x-5
Yes ✓
x+1/sqrt(x+3)
sqrt(x+3)
No *
x^2+1/3
3
Yes ✓
The denominator of the expression x+1sqrt(x+3) is not a polynomial. Therefore, this expression is not a rational expression. At the same time the other three expressions are all rational. Thus, the expression x+1sqrt(x+3) is the one that does not belong with the other three.
