Is the algebraic expression a polynomial? If it is, write the polynomial in standard form. (2x+3)/x

Verified step by step guidance

1. Identify the given expression: \(\frac{2x + 3}{x}\).
2. Recall that a polynomial is an expression consisting of variables raised to non-negative integer powers, combined using addition, subtraction, and multiplication only. Division by a variable (like \(x\)) is not allowed in polynomials.
3. Rewrite the expression by dividing each term in the numerator by \(x\): \(\frac{2x}{x} + \frac{3}{x}\).
4. Simplify the terms: \(\frac{2x}{x} = 2\) and \(\frac{3}{x} = 3x^{-1}\).
5. Since \$3x^{-1}$ has a negative exponent, the expression is not a polynomial. Therefore, it cannot be written in standard polynomial form.

Key Concepts

Definition of a Polynomial

A polynomial is an algebraic expression consisting of variables and coefficients, involving only non-negative integer exponents of variables combined using addition, subtraction, and multiplication. Expressions with variables in denominators or with negative or fractional exponents are not polynomials.

Standard Form of a Polynomial

The standard form of a polynomial arranges terms in descending order of their exponents, starting with the highest degree term. This format helps in easily identifying the degree and leading coefficient of the polynomial.

Simplifying Rational Expressions

A rational expression is a ratio of two polynomials. To determine if it is a polynomial, simplify the expression by performing division or factoring. If the variable remains in the denominator after simplification, the expression is not a polynomial.