Answer :
To determine which expressions are polynomials, we need to understand the definition of a polynomial. A polynomial is a mathematical expression involving a sum of powers of a variable, where the exponents are whole numbers (non-negative integers) and the coefficients can be real numbers. Now, let's evaluate each expression:
A. [tex]\(2x^2 + 5x - 3\)[/tex]
- This expression consists of powers of [tex]\(x\)[/tex] that are non-negative integers: [tex]\(x^2\)[/tex], [tex]\(x^1\)[/tex], and [tex]\(x^0\)[/tex].
- Therefore, it is a polynomial.
B. [tex]\(3x^3 - 19\)[/tex]
- This expression includes the term [tex]\(x^3\)[/tex], with [tex]\(x\)[/tex] raised to a non-negative integer power, and [tex]\(-19\)[/tex] which can be considered as [tex]\(x^0\)[/tex].
- Therefore, it is a polynomial.
C. [tex]\(-x^3 + \sqrt{-x}\)[/tex]
- This expression contains a term [tex]\(\sqrt{-x}\)[/tex], which involves a square root. Powers must be non-negative integers in polynomials, and the square root does not fit this requirement.
- Therefore, it is not a polynomial.
D. [tex]\(\frac{3}{5}x^4 - 18x^3 + x^2 - 10x + 3.5\)[/tex]
- All the powers of [tex]\(x\)[/tex] in this expression are non-negative integers: [tex]\(x^4\)[/tex], [tex]\(x^3\)[/tex], [tex]\(x^2\)[/tex], [tex]\(x^1\)[/tex], and [tex]\(x^0\)[/tex].
- Therefore, it is a polynomial.
E. [tex]\(-x^3 + 5x^2 + 7x - 1\)[/tex]
- This expression has terms with non-negative integer powers: [tex]\(x^3\)[/tex], [tex]\(x^2\)[/tex], [tex]\(x^1\)[/tex], and [tex]\(x^0\)[/tex].
- Therefore, it is a polynomial.
In conclusion, the expressions that are polynomials are: A, B, D, and E.
A. [tex]\(2x^2 + 5x - 3\)[/tex]
- This expression consists of powers of [tex]\(x\)[/tex] that are non-negative integers: [tex]\(x^2\)[/tex], [tex]\(x^1\)[/tex], and [tex]\(x^0\)[/tex].
- Therefore, it is a polynomial.
B. [tex]\(3x^3 - 19\)[/tex]
- This expression includes the term [tex]\(x^3\)[/tex], with [tex]\(x\)[/tex] raised to a non-negative integer power, and [tex]\(-19\)[/tex] which can be considered as [tex]\(x^0\)[/tex].
- Therefore, it is a polynomial.
C. [tex]\(-x^3 + \sqrt{-x}\)[/tex]
- This expression contains a term [tex]\(\sqrt{-x}\)[/tex], which involves a square root. Powers must be non-negative integers in polynomials, and the square root does not fit this requirement.
- Therefore, it is not a polynomial.
D. [tex]\(\frac{3}{5}x^4 - 18x^3 + x^2 - 10x + 3.5\)[/tex]
- All the powers of [tex]\(x\)[/tex] in this expression are non-negative integers: [tex]\(x^4\)[/tex], [tex]\(x^3\)[/tex], [tex]\(x^2\)[/tex], [tex]\(x^1\)[/tex], and [tex]\(x^0\)[/tex].
- Therefore, it is a polynomial.
E. [tex]\(-x^3 + 5x^2 + 7x - 1\)[/tex]
- This expression has terms with non-negative integer powers: [tex]\(x^3\)[/tex], [tex]\(x^2\)[/tex], [tex]\(x^1\)[/tex], and [tex]\(x^0\)[/tex].
- Therefore, it is a polynomial.
In conclusion, the expressions that are polynomials are: A, B, D, and E.
