Answer :
To determine which expressions are polynomials, let's first define a polynomial. A polynomial is an algebraic expression made up of variables and coefficients, where the variables are raised to whole number (non-negative integer) exponents. The coefficients can be real or complex numbers.
Now, let's examine each option:
A. [tex]\(2x^2 + 5x - 3\)[/tex]
- This expression has terms with exponents as whole numbers: [tex]\(x^2\)[/tex] (exponent is 2), [tex]\(x\)[/tex] (exponent is 1), and a constant [tex]\(-3\)[/tex] (exponent is 0).
- The coefficients (2, 5, and -3) are real numbers.
- Conclusion: This is a polynomial.
B. [tex]\(3x^3 - 19\)[/tex]
- The term [tex]\(x^3\)[/tex] has a whole number exponent (3), and [tex]\(-19\)[/tex] is a constant term with an exponent of 0.
- The coefficients (3 and -19) are real numbers.
- Conclusion: This is a polynomial.
C. [tex]\(-x^3 + 5x^2 + 7x - 1\)[/tex]
- The terms have whole number exponents: [tex]\(x^3\)[/tex] (exponent is 3), [tex]\(x^2\)[/tex] (exponent is 2), [tex]\(x\)[/tex] (exponent is 1), and a constant [tex]\(-1\)[/tex] (exponent is 0).
- The coefficients (-1, 5, 7, and -1) are real numbers.
- Conclusion: This is a polynomial.
D. [tex]\(\frac{3}{5}x^4 - 18x^3 + x^2 - 10x + 3.5\)[/tex]
- The terms are [tex]\(x^4\)[/tex] (exponent is 4), [tex]\(x^3\)[/tex] (exponent is 3), [tex]\(x^2\)[/tex] (exponent is 2), [tex]\(x\)[/tex] (exponent is 1), and the constant 3.5 (exponent is 0), all with whole number exponents.
- The coefficients ([tex]\(\frac{3}{5}\)[/tex], -18, 1, -10, and 3.5) are real and rational numbers.
- Conclusion: This is a polynomial.
E. [tex]\(-x^3 + \sqrt{-x}\)[/tex]
- [tex]\(x^3\)[/tex] has a whole number exponent (3), but [tex]\(\sqrt{-x}\)[/tex] is not typical for a polynomial. The square root indicates a power of [tex]\(1/2\)[/tex] due to the square root of [tex]\(x\)[/tex], violating the polynomial condition.
- Additionally, [tex]\(-x\)[/tex] under a square root can introduce complex numbers, depending on the context.
- Conclusion: This is not a polynomial.
Based on the criteria for polynomials, options A, B, C, and D are polynomials. Option E is not a polynomial because it involves a variable under a square root, introducing fractional and possibly imaginary components.
Now, let's examine each option:
A. [tex]\(2x^2 + 5x - 3\)[/tex]
- This expression has terms with exponents as whole numbers: [tex]\(x^2\)[/tex] (exponent is 2), [tex]\(x\)[/tex] (exponent is 1), and a constant [tex]\(-3\)[/tex] (exponent is 0).
- The coefficients (2, 5, and -3) are real numbers.
- Conclusion: This is a polynomial.
B. [tex]\(3x^3 - 19\)[/tex]
- The term [tex]\(x^3\)[/tex] has a whole number exponent (3), and [tex]\(-19\)[/tex] is a constant term with an exponent of 0.
- The coefficients (3 and -19) are real numbers.
- Conclusion: This is a polynomial.
C. [tex]\(-x^3 + 5x^2 + 7x - 1\)[/tex]
- The terms have whole number exponents: [tex]\(x^3\)[/tex] (exponent is 3), [tex]\(x^2\)[/tex] (exponent is 2), [tex]\(x\)[/tex] (exponent is 1), and a constant [tex]\(-1\)[/tex] (exponent is 0).
- The coefficients (-1, 5, 7, and -1) are real numbers.
- Conclusion: This is a polynomial.
D. [tex]\(\frac{3}{5}x^4 - 18x^3 + x^2 - 10x + 3.5\)[/tex]
- The terms are [tex]\(x^4\)[/tex] (exponent is 4), [tex]\(x^3\)[/tex] (exponent is 3), [tex]\(x^2\)[/tex] (exponent is 2), [tex]\(x\)[/tex] (exponent is 1), and the constant 3.5 (exponent is 0), all with whole number exponents.
- The coefficients ([tex]\(\frac{3}{5}\)[/tex], -18, 1, -10, and 3.5) are real and rational numbers.
- Conclusion: This is a polynomial.
E. [tex]\(-x^3 + \sqrt{-x}\)[/tex]
- [tex]\(x^3\)[/tex] has a whole number exponent (3), but [tex]\(\sqrt{-x}\)[/tex] is not typical for a polynomial. The square root indicates a power of [tex]\(1/2\)[/tex] due to the square root of [tex]\(x\)[/tex], violating the polynomial condition.
- Additionally, [tex]\(-x\)[/tex] under a square root can introduce complex numbers, depending on the context.
- Conclusion: This is not a polynomial.
Based on the criteria for polynomials, options A, B, C, and D are polynomials. Option E is not a polynomial because it involves a variable under a square root, introducing fractional and possibly imaginary components.
