Westonci.ca offers quick and accurate answers to your questions. Join our community and get the insights you need today. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
Sagot :
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]
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]
Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.