Question

Complete the following table by stating whether the polynomial is a monomial, binomial, or trinomial; then list its terms and state its degree.

Polynomial	Type	Term	Degree
$x-x^2+x^3-x^4$

Short answer

Polynomial	Type	Term	Degree
$x-x^2+x^3-x^4$	None	$x,-x^2,x^3,-x^4$	4

Step-by-step solution

Step 1. Apply the concept of polynomials.

Apolynomial in the variable x is an expression of the form$a_n{x}^n+a_{n-1}{x}^{n-1}+a_{n-2}{x}^{n-2}+\mathrm{...}+a_{1}x+a_0$

$a_n,a_{n-1},a_{n-2},\mathrm{...},a_1,a_0$are real numbers, and n is a nonnegative integer. If $a_n\ne 0$, then the polynomial hasdegree n. The monomials $a_k{x}^k$that make up the polynomial are called theterms of the polynomial.

Step 2. Find the degree of polynomial.

Given$x-x^2+x^3-x^4$

We arrange the terms of the polynomial such the exponents are in descending order.

$-x^4+x^3-x^2+x$

Since the highest exponent of variable x is 4, the degree of polynomial is 4.

Step 3. Find the terms of polynomial.

Clearly, there are 4 terms in the polynomial$x-x^2+x^3-x^4$

The terms are$x,-x^2,x^3,-x^4$

Step 4. Find the type of polynomial.

Since the number of terms in the polynomial is 4,$x-x^2+x^3-x^4$ is none of monomial, binomial or trinomial.

Step 5. Complete the table.

Polynomial	Type	Term	Degree
$x-x^2+x^3-x^4$	None	$x,-x^2,x^3,-x^4$	4