Identify if the given is polynomial or not. If not, explain why. If it is a polynomial, classify according to number of terms: 1) -7x + y/4x 2) y 3) 8mn^(-1) 4) x^2 - 5xy + 7y^2 5)... Identify if the given is polynomial or not. If not, explain why. If it is a polynomial, classify according to number of terms: 1) -7x + y/4x 2) y 3) 8mn^(-1) 4) x^2 - 5xy + 7y^2 5) sqrt(m) - n^4 6) sqrt(6)uv^3.
Understand the Problem
The question is asking to determine whether each provided expression is a polynomial and to explain the reasoning. If an expression is a polynomial, it also requires classification based on the number of terms. This involves checking the definitions of polynomials and how terms are categorized.
Answer
1) Not a polynomial 2) Polynomial (1 term) 3) Not a polynomial 4) Polynomial (3 terms) 5) Not a polynomial 6) Polynomial (1 term)
Answer for screen readers
- Not a polynomial
- Polynomial (1 term)
- Not a polynomial
- Polynomial (3 terms)
- Not a polynomial
- Polynomial (1 term)
Steps to Solve
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Identify Polynomial Criteria A polynomial is an expression involving a sum of powers in one or more variables multiplied by coefficients, where all exponents are non-negative integers.
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Analyze Expression 1: (-7x + \frac{y}{4x}) The term (-7x) is a polynomial term (degree 1), but (\frac{y}{4x}) can be rewritten as (\frac{1}{4}y \cdot x^{-1}), which has a negative exponent. Therefore, this expression is not a polynomial.
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Analyze Expression 2: (y) This expression is a single term with variable (y) and can be considered a polynomial of degree 1. It has 1 term.
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Analyze Expression 3: (8mn^{-1}) The term (8mn^{-1}) can be rewritten as (8m \cdot n^{-1}) which has a negative exponent. Therefore, this expression is not a polynomial.
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Analyze Expression 4: (x^2 - 5xy + 7y^2) All terms have non-negative integer exponents. This is a polynomial of degree 2 (highest exponent is 2). It has 3 terms.
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Analyze Expression 5: (\sqrt{m} - n^4) The term (\sqrt{m}) can be rewritten as (m^{1/2}), which has a fractional exponent, so this is not a polynomial.
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Analyze Expression 6: (\sqrt{6}uv^3) This expression consists of constants and variables with non-negative integer exponents. Thus, this is a polynomial with 1 term.
- Not a polynomial
- Polynomial (1 term)
- Not a polynomial
- Polynomial (3 terms)
- Not a polynomial
- Polynomial (1 term)
More Information
A polynomial must consist only of terms with non-negative integer exponents. Polynomials can be classified by their number of terms: a single term is a monomial, two terms is a binomial, and three or more terms is a polynomial.
Tips
- Confusing negative or fractional exponents as valid terms in a polynomial.
- Not properly counting the number of terms in polynomial expressions.
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