State why the following expression is not a polynomial. $\newline$ $4 \mathrm{x}^{-1}+6 \mathrm{x}$ $\newline$ Choose the correct answer below. $\newline$ A. The exponent of the variable is a rational number. $\newline$ B. The exponent of the variable is a negative integer. $\newline$ C. The expression is a quotient of polynomials, which cannot be simplified to a polynomial. $\newline$ D. The exponent of the variable is an irrational number.

Full solution

Q. State why the following expression is not a polynomial.

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4 \mathrm{x}^{-1}+6 \mathrm{x}

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Choose the correct answer below.

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A. The exponent of the variable is a rational number.

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B. The exponent of the variable is a negative integer.

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C. The expression is a quotient of polynomials, which cannot be simplified to a polynomial.

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D. The exponent of the variable is an irrational number.

Identify Characteristics of Polynomial: Identify the characteristics of a polynomial. A polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
Examine Given Expression: Examine the given expression $4x^{-1}+6x$ to see if it meets the criteria of a polynomial. We need to check if all the exponents of the variable $x$ are non-negative integers.
Identify Exponents of Variable: Identify the exponents of the variable $x$ in the expression. The first term has an exponent of $-1$ ( $4x^{-1}$ ) and the second term has an exponent of $1$ ( $6x$ ).
Determine Exponents for Polynomial: Determine if the exponents of the variable $x$ meet the criteria for a polynomial. The exponent $-1$ is a negative integer, which does not satisfy the condition of having non-negative integer exponents.
Choose Correct Answer: Choose the correct answer based on the analysis. Since the expression contains a term with a negative exponent, it does not qualify as a polynomial. The correct answer is that the exponent of the variable is a negative integer.