Answer :
To determine which algebraic expressions are polynomials, we must define what constitutes a polynomial. A polynomial is an algebraic expression consisting of variables (also known as indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
Let's analyze each given expression:
1. [tex]\(\pi x - \sqrt{3} + 5 y\)[/tex]
- This expression involves only the variables [tex]\(x\)[/tex] and [tex]\(y\)[/tex] with coefficients [tex]\(\pi\)[/tex], [tex]\(\sqrt{3}\)[/tex] (as a constant term), and 5.
- All exponents of the variables [tex]\(x\)[/tex] and [tex]\(y\)[/tex] are 1, which are non-negative integers.
- This expression is a polynomial.
2. [tex]\(x^2 y^2 - 4 x^3 + 12 y\)[/tex]
- This expression involves the variables [tex]\(x\)[/tex] and [tex]\(y\)[/tex] raised to non-negative integer powers: 2, 2, and 1.
- The coefficients are 1, -4, and 12.
- This expression is a polynomial.
3. [tex]\(\frac{4}{x} - x^2\)[/tex]
- The first term, [tex]\(\frac{4}{x}\)[/tex], can be rewritten as [tex]\(4 x^{-1}\)[/tex], which involves a negative exponent of [tex]\(x\)[/tex].
- The second term, [tex]\(-x^2\)[/tex], is valid as it involves a non-negative integer exponent.
- Since a polynomial cannot have negative exponents, this expression is not a polynomial.
4. [tex]\(\sqrt{x} - 16\)[/tex]
- The first term, [tex]\(\sqrt{x}\)[/tex], can be rewritten as [tex]\(x^{1/2}\)[/tex], which involves a fractional (non-integer) exponent.
- The second term, [tex]\(-16\)[/tex], is simply a constant term.
- Since a polynomial cannot have fractional exponents, this expression is not a polynomial.
5. [tex]\(3.9 x^3 - 4.1 x^2 + 7.3\)[/tex]
- This expression involves the variable [tex]\(x\)[/tex] raised to non-negative integer powers: 3 and 2.
- The coefficients are 3.9, -4.1, and 7.3.
- This expression is a polynomial.
Based on the analysis:
- The expressions [tex]\(\pi x - \sqrt{3} + 5 y\)[/tex], [tex]\(x^2 y^2 - 4 x^3 + 12 y\)[/tex], and [tex]\(3.9 x^3 - 4.1 x^2 + 7.3\)[/tex] are polynomials.
- The expressions [tex]\(\frac{4}{x} - x^2\)[/tex] and [tex]\(\sqrt{x} - 16\)[/tex] are not polynomials.
Therefore, the polynomials among the given expressions are:
- [tex]\(\pi x - \sqrt{3} + 5 y\)[/tex]
- [tex]\(x^2 y^2 - 4 x^3 + 12 y\)[/tex]
- [tex]\(3.9 x^3 - 4.1 x^2 + 7.3\)[/tex]
Let's analyze each given expression:
1. [tex]\(\pi x - \sqrt{3} + 5 y\)[/tex]
- This expression involves only the variables [tex]\(x\)[/tex] and [tex]\(y\)[/tex] with coefficients [tex]\(\pi\)[/tex], [tex]\(\sqrt{3}\)[/tex] (as a constant term), and 5.
- All exponents of the variables [tex]\(x\)[/tex] and [tex]\(y\)[/tex] are 1, which are non-negative integers.
- This expression is a polynomial.
2. [tex]\(x^2 y^2 - 4 x^3 + 12 y\)[/tex]
- This expression involves the variables [tex]\(x\)[/tex] and [tex]\(y\)[/tex] raised to non-negative integer powers: 2, 2, and 1.
- The coefficients are 1, -4, and 12.
- This expression is a polynomial.
3. [tex]\(\frac{4}{x} - x^2\)[/tex]
- The first term, [tex]\(\frac{4}{x}\)[/tex], can be rewritten as [tex]\(4 x^{-1}\)[/tex], which involves a negative exponent of [tex]\(x\)[/tex].
- The second term, [tex]\(-x^2\)[/tex], is valid as it involves a non-negative integer exponent.
- Since a polynomial cannot have negative exponents, this expression is not a polynomial.
4. [tex]\(\sqrt{x} - 16\)[/tex]
- The first term, [tex]\(\sqrt{x}\)[/tex], can be rewritten as [tex]\(x^{1/2}\)[/tex], which involves a fractional (non-integer) exponent.
- The second term, [tex]\(-16\)[/tex], is simply a constant term.
- Since a polynomial cannot have fractional exponents, this expression is not a polynomial.
5. [tex]\(3.9 x^3 - 4.1 x^2 + 7.3\)[/tex]
- This expression involves the variable [tex]\(x\)[/tex] raised to non-negative integer powers: 3 and 2.
- The coefficients are 3.9, -4.1, and 7.3.
- This expression is a polynomial.
Based on the analysis:
- The expressions [tex]\(\pi x - \sqrt{3} + 5 y\)[/tex], [tex]\(x^2 y^2 - 4 x^3 + 12 y\)[/tex], and [tex]\(3.9 x^3 - 4.1 x^2 + 7.3\)[/tex] are polynomials.
- The expressions [tex]\(\frac{4}{x} - x^2\)[/tex] and [tex]\(\sqrt{x} - 16\)[/tex] are not polynomials.
Therefore, the polynomials among the given expressions are:
- [tex]\(\pi x - \sqrt{3} + 5 y\)[/tex]
- [tex]\(x^2 y^2 - 4 x^3 + 12 y\)[/tex]
- [tex]\(3.9 x^3 - 4.1 x^2 + 7.3\)[/tex]
