Identify each expression as a polynomial or not a polynomial. For each polynomial, give the degree and identify it as a monomial, binomial, trinomial, or none of these. 6x+3x⁴

Verified step by step guidance

1. First, write down the given expression clearly: \$6x + 3x^4$.
2. Recall that a polynomial is an expression consisting of variables raised to non-negative integer exponents, combined using addition, subtraction, and multiplication by constants. Check if each term fits this definition.
3. Identify the terms in the expression: \$6x\( and \)3x^4$. Both have variables with whole number exponents (1 and 4 respectively), so both are valid polynomial terms.
4. Count the number of terms: there are two terms, so this is a binomial.
5. Determine the degree of the polynomial by finding the highest exponent of the variable, which is 4 in \$3x^4$.

Key Concepts

Definition of a Polynomial

A polynomial is an algebraic expression consisting of variables and coefficients, involving only non-negative integer exponents and operations of addition, subtraction, and multiplication. Expressions with variables raised to negative or fractional powers, or involving division by variables, are not polynomials.

Degree of a Polynomial

The degree of a polynomial is the highest exponent of the variable in the expression. For example, in 6x + 3x^4, the term 3x^4 has the highest exponent 4, so the polynomial's degree is 4. The degree helps classify and understand the behavior of the polynomial.

Classification by Number of Terms

Polynomials are classified based on the number of terms: a monomial has one term, a binomial has two terms, and a trinomial has three terms. Expressions with more than three terms are simply called polynomials without a special name. For example, 6x + 3x^4 has two terms, so it is a binomial.