Answer :
To determine which of the given expressions are polynomials, we need to understand what defines a polynomial. A polynomial is an expression that consists of variables raised to non-negative integer powers and coefficients that are typically real numbers.
Let's go through each option:
A. [tex]\(-x^3 + 5x^2 + 7x - 1\)[/tex]
- This expression is a polynomial because each term is a power of [tex]\(x\)[/tex] that is a non-negative integer ([tex]\(3, 2, 1, 0\)[/tex]), and the coefficients [tex]\(-1, 5, 7, -1\)[/tex] are real numbers.
B. [tex]\(3x^3 - 19\)[/tex]
- This is a polynomial. Here, [tex]\(3x^3\)[/tex] has a non-negative integer power, and the constant [tex]\(-19\)[/tex] can be considered as a term with [tex]\(x^0\)[/tex].
C. [tex]\(-x^3 + \sqrt{-x}\)[/tex]
- This is NOT a polynomial. The term [tex]\(\sqrt{-x}\)[/tex] is problematic because taking the square root of a negative involves imaginary numbers and also [tex]\(\sqrt{x}\)[/tex] implies a power of [tex]\(\frac{1}{2}\)[/tex], which is not an integer.
D. [tex]\(5x^4 - 18x^3 + x^2 - 10x + 3.5\)[/tex]
- This expression is a polynomial. All terms are powers of [tex]\(x\)[/tex] with non-negative integer exponents, and [tex]\(3.5\)[/tex] is a real number constant.
E. [tex]\(2x^2 + 5x - 3\)[/tex]
- This is a polynomial. The powers of [tex]\(x\)[/tex] are non-negative integers, and the coefficients [tex]\(2, 5, -3\)[/tex] are real numbers.
In summary, the expressions that are polynomials are: A, B, D, and E.
Let's go through each option:
A. [tex]\(-x^3 + 5x^2 + 7x - 1\)[/tex]
- This expression is a polynomial because each term is a power of [tex]\(x\)[/tex] that is a non-negative integer ([tex]\(3, 2, 1, 0\)[/tex]), and the coefficients [tex]\(-1, 5, 7, -1\)[/tex] are real numbers.
B. [tex]\(3x^3 - 19\)[/tex]
- This is a polynomial. Here, [tex]\(3x^3\)[/tex] has a non-negative integer power, and the constant [tex]\(-19\)[/tex] can be considered as a term with [tex]\(x^0\)[/tex].
C. [tex]\(-x^3 + \sqrt{-x}\)[/tex]
- This is NOT a polynomial. The term [tex]\(\sqrt{-x}\)[/tex] is problematic because taking the square root of a negative involves imaginary numbers and also [tex]\(\sqrt{x}\)[/tex] implies a power of [tex]\(\frac{1}{2}\)[/tex], which is not an integer.
D. [tex]\(5x^4 - 18x^3 + x^2 - 10x + 3.5\)[/tex]
- This expression is a polynomial. All terms are powers of [tex]\(x\)[/tex] with non-negative integer exponents, and [tex]\(3.5\)[/tex] is a real number constant.
E. [tex]\(2x^2 + 5x - 3\)[/tex]
- This is a polynomial. The powers of [tex]\(x\)[/tex] are non-negative integers, and the coefficients [tex]\(2, 5, -3\)[/tex] are real numbers.
In summary, the expressions that are polynomials are: A, B, D, and E.
