To determine which algebraic expressions are polynomials, we must understand the definition of a polynomial. A polynomial is a mathematical expression involving a sum of powers of one or more variables with non-negative integer exponents and real coefficients. Let's examine each given expression in detail:
1. [tex]\(\pi x - \sqrt{3} + 5y\)[/tex]
- This expression is a sum of terms involving the variables [tex]\(x\)[/tex] and [tex]\(y\)[/tex], and constants.
- [tex]\(\pi x\)[/tex] is a linear term in [tex]\(x\)[/tex], and [tex]\(\pi\)[/tex] is a real number.
- [tex]\(-\sqrt{3}\)[/tex] is a constant term.
- [tex]\(5y\)[/tex] is a linear term in [tex]\(y\)[/tex], and [tex]\(5\)[/tex] is a real number.
All terms have non-negative integer exponents and real coefficients, so this is a polynomial.
2. [tex]\(x^2 y^2 - 4x^3 + 12y\)[/tex]
- This expression includes terms [tex]\(x^2 y^2\)[/tex], [tex]\(-4x^3\)[/tex], and [tex]\(12y\)[/tex].
- [tex]\(x^2 y^2\)[/tex] is a product of terms with non-negative integer exponents (degree 4 in total).
- [tex]\(-4x^3\)[/tex] is a polynomial term with the coefficient [tex]\(-4\)[/tex].
- [tex]\(12y\)[/tex] is a linear term in [tex]\(y\)[/tex].
All terms have non-negative integer exponents and real coefficients, so this is a polynomial.
3. [tex]\(\frac{4}{x} - x^2\)[/tex]
- This expression includes the terms [tex]\(\frac{4}{x}\)[/tex] and [tex]\(-x^2\)[/tex].
- [tex]\(\frac{4}{x}\)[/tex] can be rewritten as [tex]\(4x^{-1}\)[/tex], where the exponent of [tex]\(x\)[/tex] is [tex]\(-1\)[/tex], a negative integer.
- [tex]\(-x^2\)[/tex] is a polynomial term with the exponent [tex]\(2\)[/tex].
Since [tex]\(\frac{4}{x}\)[/tex] (or [tex]\(4x^{-1}\)[/tex]) has a negative exponent, this is not a polynomial.
4. [tex]\(\sqrt{x} - 16\)[/tex]
- This expression includes the terms [tex]\(\sqrt{x}\)[/tex] and [tex]\(-16\)[/tex].
- [tex]\(\sqrt{x}\)[/tex] can be rewritten as [tex]\(x^{\frac{1}{2}}\)[/tex], where the exponent of [tex]\(x\)[/tex] is [tex]\(\frac{1}{2}\)[/tex], a non-integer.
- [tex]\(-16\)[/tex] is a constant term.
Since [tex]\(\sqrt{x}\)[/tex] (or [tex]\(x^{\frac{1}{2}}\)[/tex]) has a non-integer exponent, this is not a polynomial.
5. [tex]\(3.9 x^3 - 4.1 x^2 + 7.3\)[/tex]
- This expression includes terms [tex]\(3.9x^3\)[/tex], [tex]\(-4.1x^2\)[/tex], and [tex]\(7.3\)[/tex].
- [tex]\(3.9 x^3\)[/tex] is a cubic term in [tex]\(x\)[/tex] with a real coefficient [tex]\(3.9\)[/tex].
- [tex]\(-4.1 x^2\)[/tex] is a quadratic term in [tex]\(x\)[/tex] with a real coefficient [tex]\(-4.1\)[/tex].
- [tex]\(7.3\)[/tex] is a constant term.
All terms have non-negative integer exponents and real coefficients, so this is a polynomial.
### Summary:
The following expressions are polynomials:
1. [tex]\(\pi x - \sqrt{3} + 5y\)[/tex]
2. [tex]\(x^2 y^2 - 4x^3 + 12y\)[/tex]
5. [tex]\(3.9 x^3 - 4.1 x^2 + 7.3\)[/tex]
The following expressions are NOT polynomials:
3. [tex]\(\frac{4}{x} - x^2\)[/tex]
4. [tex]\(\sqrt{x} - 16\)[/tex]
