Welcome to Westonci.ca, the ultimate question and answer platform. Get expert answers to your questions quickly and accurately. Ask your questions and receive precise answers from experienced professionals across different disciplines. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
Sagot :
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.
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.
Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.