To determine whether the expression [tex]\(2x + 8x^5\)[/tex] is a polynomial and, if so, its degree and type, follow these steps:
1. Identify if the Expression is a Polynomial:
A polynomial is an expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. The given expression is:
[tex]\[
2x + 8x^5
\][/tex]
This expression involves two terms:
- [tex]\(2x\)[/tex], where the variable [tex]\(x\)[/tex] is raised to the power of 1.
- [tex]\(8x^5\)[/tex], where the variable [tex]\(x\)[/tex] is raised to the power of 5.
Both terms involve non-negative integer exponents and valid coefficients. Therefore, this expression is a polynomial.
2. Determine the Degree of the Polynomial:
The degree of a polynomial is the highest power of the variable within the polynomial. In this expression, the terms have the following exponents:
- [tex]\(2x\)[/tex] has an exponent of 1.
- [tex]\(8x^5\)[/tex] has an exponent of 5.
The highest power of [tex]\(x\)[/tex] in [tex]\(2x + 8x^5\)[/tex] is 5. Hence, the degree of the polynomial is 5.
3. Identify the Type of the Polynomial:
- Monomial: A polynomial with exactly one term.
- Binomial: A polynomial with exactly two terms.
- Trinomial: A polynomial with exactly three terms.
The given polynomial [tex]\(2x + 8x^5\)[/tex] consists of two terms. Therefore, it is classified as a binomial.
To summarize:
- The expression [tex]\(2x + 8x^5\)[/tex] is a polynomial.
- The degree of the polynomial is 5.
- The polynomial is a binomial.
