To determine which of the given algebraic expressions are polynomials, we need to recall the definition of a polynomial. A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The exponents of these variables must be non-negative integers.
Let's analyze each expression one by one:
1. [tex]\(\pi x - \sqrt{3} + 5y\)[/tex]
In this expression, [tex]\(\pi\)[/tex] and [tex]\(\sqrt{3}\)[/tex] are constants, and both [tex]\(x\)[/tex] and [tex]\(y\)[/tex] appear to the first power, which is a non-negative integer. Therefore, [tex]\(\pi x - \sqrt{3} + 5y\)[/tex] is a polynomial.
2. [tex]\(x^2 y^2 - 4 x^3 + 12 y\)[/tex]
Here, [tex]\(x\)[/tex] and [tex]\(y\)[/tex] appear to integer powers: [tex]\(x\)[/tex] and [tex]\(y\)[/tex] both to the power of 2, [tex]\(x\)[/tex] to the power of 3, and [tex]\(y\)[/tex] to the power of 1. All powers are non-negative integers. Therefore, [tex]\(x^2 y^2 - 4 x^3 + 12 y\)[/tex] is a polynomial.
3. [tex]\(\frac{4}{x} - x^2\)[/tex]
In this expression, [tex]\(\frac{4}{x}\)[/tex] can be rewritten as [tex]\(4x^{-1}\)[/tex]. The exponent [tex]\(-1\)[/tex] is a negative integer, which does not satisfy the definition of a polynomial. Therefore, [tex]\(\frac{4}{x} - x^2\)[/tex] is not a polynomial.
4. [tex]\(\sqrt{x} - 16\)[/tex]
The term [tex]\(\sqrt{x}\)[/tex] can be written as [tex]\(x^{1/2}\)[/tex]. The exponent [tex]\(1/2\)[/tex] is not an integer, which does not satisfy the definition of a polynomial. Therefore, [tex]\(\sqrt{x} - 16\)[/tex] is not a polynomial.
5. [tex]\(3.9 x^3 - 4.1 x^2 + 7.3\)[/tex]
In this expression, [tex]\(x\)[/tex] appears to the powers of 3 and 2, both of which are non-negative integers. The constant term [tex]\(7.3\)[/tex] is also valid in a polynomial. Therefore, [tex]\(3.9 x^3 - 4.1 x^2 + 7.3\)[/tex] is a polynomial.
Summary:
- [tex]\(\pi x - \sqrt{3} + 5y\)[/tex] is a polynomial
- [tex]\(x^2 y^2 - 4 x^3 + 12 y\)[/tex] is a polynomial
- [tex]\(\frac{4}{x} - x^2\)[/tex] is not a polynomial
- [tex]\(\sqrt{x} - 16\)[/tex] is not a polynomial
- [tex]\(3.9 x^3 - 4.1 x^2 + 7.3\)[/tex] is a polynomial
