To determine which algebraic expressions are polynomials, we need to understand the basic definition of a polynomial. A polynomial in variables \(x\) and \(y\) is an expression that is formed using non-negative integer powers of \(x\) and \(y\) and involves only sums, differences, and constant multiples.
Let's analyze each of the given expressions to see if they meet this criterion:
1. Expression: \(\pi x - \sqrt{3} + 5 y\)
- Terms: \(\pi x\), \(- \sqrt{3}\), \(5 y\)
- Analysis:
- \(\pi x\) has the variable \(x\) with exponent 1.
- \(\sqrt{3}\) is a constant term and is valid in a polynomial.
- \(5 y\) has the variable \(y\) with exponent 1.
- None of these terms have negative or fractional exponents, and they are combined through addition and subtraction.
- Result: This is a polynomial.
2. Expression: \(x^2 y^2 - 4 x^3 + 12 y\)
- Terms: \(x^2 y^2\), \(-4 x^3\), \(12 y\)
- Analysis:
- \(x^2 y^2\) involves \(x\) and \(y\) with non-negative integer exponents.
- \(-4 x^3\) involves \(x\) with a non-negative integer exponent.
- \(12 y\) involves \(y\) with a non-negative integer exponent.
- All terms fit the definition of a polynomial.
- Result: This is a polynomial.
3. Expression: \(\frac{4}{x} - x^2\)
- Terms: \(\frac{4}{x}\), \(-x^2\)
- Analysis:
- \(\frac{4}{x}\) is equivalent to \(4x^{-1}\), which involves a negative exponent.
- \(-x^2\) involves \(x\) with a non-negative integer exponent.
- The presence of \(4x^{-1}\) makes the expression not a polynomial.
- Result: This is not a polynomial.
4. Expression: \(\sqrt{x} - 16\)
- Terms: \(\sqrt{x}\), \(-16\)
- Analysis:
- \(\sqrt{x}\) is equivalent to \(x^{1/2}\), which involves a fractional exponent.
- \(-16\) is a constant term and is valid in a polynomial.
- The presence of the term \(x^{1/2}\) (a fractional exponent) disqualifies this from being a polynomial.
- Result: This is not a polynomial.
5. Expression: \(3.9 x^3 - 4.1 x^2 + 7.3\)
- Terms: \(3.9 x^3\), \(-4.1 x^2\), \(7.3\)
- Analysis:
- \(3.9 x^3\) involves \(x\) with a non-negative integer exponent.
- \(-4.1 x^2\) involves \(x\) with a non-negative integer exponent.
- \(7.3\) is a constant term and is valid in a polynomial.
- All terms fit the definition of a polynomial.
- Result: This is a polynomial.
Summary:
The expressions that are polynomials are:
1. \(\pi x - \sqrt{3} + 5 y\)
2. \(x^2 y^2 - 4 x^3 + 12 y\)
5. \(3.9 x^3 - 4.1 x^2 + 7.3\)
The ones that are not polynomials are:
3. \(\frac{4}{x} - x^2\)
4. [tex]\(\sqrt{x} - 16\)[/tex]
