Question

First: State the degree and tell which monomial (term) you determined to be the one as the greatest degree. Second: Classify each of the 2 polynomials. If it is NOT a polynomial, explain why you think it is not a polynomial. For example you may say: Polynomial # 4. has degree of 5 and is classified as a quintic polynomial of 4 terms. We covered this in class (see below). Explain in sentences how you determined degree and "name" or why it is NOT a polynomial: 1. 7 x 4 + 5 x 2 + x − 9 2. 7 x 6 − 4 x 3 + 1 /x

Answer 1

Answer:

$1.\text{ }7x^4+5x^2+x-9$

• The term with the greatest degree is $7x^4$

• The polynomial has degree 4 and is classified as a quadratic polynomial of 4 terms.

$2.\text{ }7x^6-4x^3+1/x$

• It is not a polynomial.

Explanation:

Polynomials are expressions with terms composed of monomials.

A monomial is an expression formed by the product of a coefficient (constant numbers, like 3, 4/5, or even a root), and a literal part.

The literal part may consist of one or more letters and each letter may be raised to a non-negative integer number (positive or zero).

Hence, for instance, (2/9)x²y⁵z⁴³ is a monomial.

The given expressions are:

       $1.\text{ }7x^4+5x^2+x-9$

       $2.\text{ }7x^6-4x^3+1/x$

$1.\text{ }7x^4+5x^2+x-9$

This expression is a polynomial formed by 4 terms or monomials.

The monomials are:

            $7x^4,\\ \\ 5x^2,\\ \\ x,\text{ }and\\ \\ 9$

The degrees of each monomial are:

         $7x^4,\text{ }degree\text{ }4\\ \\ 5x^2,\text{ }degree\text{ }2\\ \\ x,\text{ }degree\text{ }1,\text{ }and\\ \\ 9\text{ }degree\text{ }0$

Thus, the one with the greatest degree is $7x^4$ . It defines the degree of the of the polynomial; hence the polynomial has degree four and is classified as quartic polynomial of four terms.

$2.\text{ }7x^6-4x^3+1/x$

This is not a  polynomial, because the letter in the last term, i.e. 1/x has a negative power:

                                $1/x=x^{-1}$

As stated above, the powers have to be positive integers or zero (for the constant term).

Some special names used to classify the polynomials are:

•     Degree 1 – linear

•    Degree 2 – quadratic

•    Degree 3 – cubic

•    Degree 4 – quartic

•    Degree 5 – quintic

 

You can find other classifications in your book or in the internet.