Alright, let's analyze each given expression to determine if they are rational expressions. A rational expression is defined as a ratio of two polynomials. For this, both the numerator and the denominator must be polynomials, except in the case of a single polynomial (which can be considered a ratio where the denominator is `1`).
a. [tex]\(\frac{x+3}{x^3+x-9}\)[/tex]
- Numerator: [tex]\(x + 3\)[/tex] is a polynomial (degree 1).
- Denominator: [tex]\(x^3 + x - 9\)[/tex] is also a polynomial (degree 3).
Since both the numerator and the denominator are polynomials, this expression is a rational expression.
Result: Rational
b. [tex]\(\frac{1}{x^3+\sqrt{x}+1}\)[/tex]
- Numerator: [tex]\(1\)[/tex] is a polynomial (degree 0).
- Denominator: [tex]\(x^3 + \sqrt{x} + 1\)[/tex] includes [tex]\(\sqrt{x}\)[/tex], which is not a polynomial term (as it involves a fractional exponent).
Since the denominator is not entirely a polynomial due to the [tex]\(\sqrt{x}\)[/tex] term, this expression is not a rational expression.
Result: Not Rational
c. [tex]\(\frac{x^2-x}{\sqrt{x^4+4x^2+4}}\)[/tex]
- Numerator: [tex]\(x^2 - x\)[/tex] is a polynomial (degree 2).
- Denominator: [tex]\(\sqrt{x^4+4x^2+4}\)[/tex] involves a square root, which makes it not a polynomial unless the expression inside the square root is a perfect square. Even then, the square root function itself makes it non-polynomial.
Since the denominator involves a square root, this expression is not a rational expression.
Result: Not Rational
d. [tex]\(\frac{x^{1/2} + 2x - 1}{6x^3 + 4x - 7}\)[/tex]
- Numerator: [tex]\(x^{1/2} + 2x - 1\)[/tex] includes [tex]\(x^{1/2}\)[/tex], which is not a polynomial (as it involves a fractional exponent).
- Denominator: [tex]\(6x^3 + 4x - 7\)[/tex] is a polynomial (degree 3).
Since the numerator contains a term with a fractional exponent, this expression is not a rational expression.
Result: Not Rational
e. [tex]\(2x^4 - 3x^3 + 2x^2 + 3x - 1\)[/tex]
This is a single polynomial expression (degree 4). A polynomial by itself can be considered a rational expression with the denominator [tex]\(1\)[/tex].
Since this is a single polynomial, it's considered a rational expression.
Result: Rational
Therefore, the rationality of the given expressions are:
- (a): Rational
- (b): Not Rational
- (c): Not Rational
- (d): Not Rational
- (e): Rational
