Answer :
To determine which of the given expressions are polynomials, we need to understand what a polynomial is. A polynomial is a mathematical expression that consists of variables (also known as indeterminates) raised to whole number powers and multiplied by coefficients. The coefficients can be any real numbers, including fractions and negative numbers. However, the powers of the variables must be non-negative integers (whole numbers).
Let's analyze each option:
A. [tex]\(3x^3 - 19\)[/tex]
- This expression consists of a term [tex]\(3x^3\)[/tex] with the variable [tex]\(x\)[/tex] raised to the power of 3, which is a whole number, and a constant term [tex]\(-19\)[/tex].
- Since both parts fit the rules of a polynomial, this expression is a polynomial.
B. [tex]\(-x^3 + 5x^2 + 7x - 1\)[/tex]
- This expression has terms [tex]\(-x^3\)[/tex], [tex]\(5x^2\)[/tex], [tex]\(7x\)[/tex], and [tex]\(-1\)[/tex], all with whole number exponents: 3, 2, 1, and 0, respectively.
- Therefore, this expression is a polynomial.
C. [tex]\(-x^3 + \sqrt{-x}\)[/tex]
- This expression has a term [tex]\(-x^3\)[/tex] with a whole number exponent, which is fine. However, the term [tex]\(\sqrt{-x}\)[/tex] is equivalent to [tex]\((-x)^{1/2}\)[/tex], which involves a fractional exponent.
- Since polynomials cannot have fractional powers, this expression is not a polynomial.
D. [tex]\(2x^2 + 5x - 3\)[/tex]
- This expression consists of terms [tex]\(2x^2\)[/tex], [tex]\(5x\)[/tex], and [tex]\(-3\)[/tex], where the variable [tex]\(x\)[/tex] is raised to the whole number exponents of 2, 1, and 0, respectively.
- Thus, this expression is a polynomial.
E. [tex]\(\frac{3}{5}x^4 - 18x^3 + x^2 - 10x + 3.5\)[/tex]
- This expression includes terms with exponents 4, 3, 2, 1, and 0, all of which are whole numbers. The coefficients include a fraction [tex]\(\frac{3}{5}\)[/tex] and a decimal [tex]\(3.5\)[/tex], but these are allowable in polynomials.
- Hence, this expression is a polynomial.
In conclusion, options A, B, D, and E are polynomials.
Let's analyze each option:
A. [tex]\(3x^3 - 19\)[/tex]
- This expression consists of a term [tex]\(3x^3\)[/tex] with the variable [tex]\(x\)[/tex] raised to the power of 3, which is a whole number, and a constant term [tex]\(-19\)[/tex].
- Since both parts fit the rules of a polynomial, this expression is a polynomial.
B. [tex]\(-x^3 + 5x^2 + 7x - 1\)[/tex]
- This expression has terms [tex]\(-x^3\)[/tex], [tex]\(5x^2\)[/tex], [tex]\(7x\)[/tex], and [tex]\(-1\)[/tex], all with whole number exponents: 3, 2, 1, and 0, respectively.
- Therefore, this expression is a polynomial.
C. [tex]\(-x^3 + \sqrt{-x}\)[/tex]
- This expression has a term [tex]\(-x^3\)[/tex] with a whole number exponent, which is fine. However, the term [tex]\(\sqrt{-x}\)[/tex] is equivalent to [tex]\((-x)^{1/2}\)[/tex], which involves a fractional exponent.
- Since polynomials cannot have fractional powers, this expression is not a polynomial.
D. [tex]\(2x^2 + 5x - 3\)[/tex]
- This expression consists of terms [tex]\(2x^2\)[/tex], [tex]\(5x\)[/tex], and [tex]\(-3\)[/tex], where the variable [tex]\(x\)[/tex] is raised to the whole number exponents of 2, 1, and 0, respectively.
- Thus, this expression is a polynomial.
E. [tex]\(\frac{3}{5}x^4 - 18x^3 + x^2 - 10x + 3.5\)[/tex]
- This expression includes terms with exponents 4, 3, 2, 1, and 0, all of which are whole numbers. The coefficients include a fraction [tex]\(\frac{3}{5}\)[/tex] and a decimal [tex]\(3.5\)[/tex], but these are allowable in polynomials.
- Hence, this expression is a polynomial.
In conclusion, options A, B, D, and E are polynomials.
