Question

How do you write a polynomial in standard form, then classify it by degree and number of terms $4{x^4} + 6{x^3} - 2 - x4$ ?

Answer 1

In this question, we have to find out the polynomial in standard form, also need to find the degree and the number of terms.
To write any polynomial in a standard form we need to start writing the term with the highest degree, or exponent, then in decreasing order.
To find the degree, we need to find the highest component of the polynomial and to find the number of terms we just need to count the number of terms.

Complete step-by-step answer:
It is given that, $4{x^4} + 6{x^3} - 2 - x4$ .
We need to write the polynomial in standard form, then classify it by degree and number of terms.
To write any polynomial in a standard form we need to write starting with the term with the highest degree, or exponent (in this case, the ${x^4}$term), then in decreasing order.
Since the ${x^4}$term is the highest term in the given polynomial, we will write in decreasing order ${x^3}$ term then x term and lastly end with the constant term $\left( {{x^0}} \right)$ .
Thus, we get the given polynomial in standard form as:
$4{x^4} + 6{x^3} - 4x - 2$ .
To classify a polynomial by degree, we need to look at the highest component, or degree.
Since $4$ is the highest exponent ($4{x^4}$), it is a quartic equation. Thus, we can say that the polynomial is of degree $4$ .
Here are the degrees of the polynomial with their respective definition.

Degree	Equation
$1$	Linear
$2$	Quadratic
$3$	Cubic
$4$	Quartic
$5$	Quintic

To classify a polynomial by the number of terms, count how many terms are there in the polynomial.
In this polynomial there are four terms, $4{x^4},6{x^3},4x,2$,

$1$ term	Monomial
$2$ terms	Binomial
$3$ terms	Trinomial
$4 +$ terms	Polynomial

Therefore, it is a polynomial.
Hence, the polynomial in standard form is $4{x^4} + 6{x^3} - 4x - 2$ .

The polynomial is of degree $4$ and the number of terms is also four.

Note:
Degree of a term:
In mathematics, the degree of a polynomial is the highest of the degrees of the polynomial’s monomials (individual terms) with non-zero coefficients. The degree of a term is the sum of the exponents of the variables that appear in it, and thus is a non-negative integer.
Constant term:
In mathematics, a constant term is a term in an algebraic expression that has a value that is constant or cannot change, because it does not contain any modifiable variables.