To determine which of the given expressions is NOT a rational expression, we need to first understand the definition. A rational expression is a ratio of two polynomials. Therefore, if any part of the expression involves a non-polynomial element (such as roots, trigonometric functions, etc.), it will not be considered a rational expression.
Let's analyze each given expression one by one:
### Expression 1: [tex]\(\frac{2 x^2 - 3 x^3 + 5}{5 x}\)[/tex]
- Numerator: [tex]\(2 x^2 - 3 x^3 + 5\)[/tex] is a polynomial.
- Denominator: [tex]\(5 x\)[/tex] is also a polynomial.
- Since both the numerator and the denominator are polynomials, this is a rational expression.
### Expression 2: [tex]\(\frac{3 x^2 + 3 x}{4 x + 5}\)[/tex]
- Numerator: [tex]\(3 x^2 + 3 x\)[/tex] is a polynomial.
- Denominator: [tex]\(4 x + 5\)[/tex] is a polynomial.
- Since both the numerator and the denominator are polynomials, this is a rational expression.
### Expression 3: [tex]\(\frac{(x + 3)(2 x - 1)}{x + 3}\)[/tex]
- Numerator: When expanded, [tex]\((x + 3)(2 x - 1)\)[/tex] is a polynomial.
- Denominator: [tex]\(x + 3\)[/tex] is a polynomial.
- Since both the numerator and the denominator are polynomials, this is a rational expression.
### Expression 4: [tex]\(\frac{3 x + 4 \sqrt{x} - 7}{2 x + 2}\)[/tex]
- Numerator: [tex]\(3 x + 4 \sqrt{x} - 7\)[/tex] includes a square root term ([tex]\(\sqrt{x}\)[/tex]), which is not a polynomial.
- Denominator: [tex]\(2 x + 2\)[/tex] is a polynomial.
- Since the numerator contains a square root, it is not a polynomial. Thus, this expression is NOT a rational expression.
### Conclusion:
The expression that is NOT a rational expression is:
[tex]\[
\frac{3 x + 4 \sqrt{x} - 7}{2 x + 2}
\][/tex]
So the answer is the fourth expression.
