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. 7x4 + 5x2 + x − 92. 7x6 − 4x3 + 1x

A monomial is basically an individual term.
A polynomial is the sum of monomials. Any monomial (term) which contains the highest power of the variable would be considered as the degree of a polynomial. For example, [tex]2x^{2} + 3x + 1[/tex] is a polynomial of degree 3, as the first monomial (term) has the highest power (degree) of the variable x.

Now, lets analyze the given algebraic expressions

Examining the first polynomial

Considering the first polynomial

[tex]7x^4\:+\:5x^2\:+\:x\:-\:9[/tex]

Carefully notice that the first term [tex]7x^{4}[/tex] which is also called a monomial does have the highest exponent or power of a variable x, which is 4. Thus, it can be determined that the degree of [tex]7x^4\:+\:5x^2\:+\:x\:-\:9[/tex] will be 4.

Here is the summary of the classification of the first polynomial [tex]7x^4\:+\:5x^2\:+\:x\:-\:9[/tex].

[tex]7x^4\:+\:5x^2\:+\:x\:-\:9[/tex] is a polynomial of 4 terms.

Its degree is 4.

It can also be named as 'quintic' because its degree is 4

Examining the second polynomial

Considering the expression

[tex]7x^6\:-\:4x^3\:+\:1/x[/tex]

It is easy to figure out that [tex]7x^6\:-\:4x^3\:+\:1/x[/tex] is not a polynomial as it does not fulfill the criteria to be a polynomial as its last term is [tex]\frac{1}{x}[/tex] has the variable x in the denominator.

Here are some important things that need to be fulfilled for any expression to be a polynomial.

The polynomial can not contain any variable having fractional or negative powers. For example, any expression containing [tex]2v^{-5}[/tex] in any of its terms can not be a polynomial as the variable v has a negative power of exponent.
Similarly, the polynomial can not contain any square root of variables. For example, any algebraic expression with the term [tex]2\sqrt{v}[/tex] can not be a polynomial as the variable v is right inside the radical symbol.
The polynomial can not have any variable in the denominator. For example, any expression that contains the term [tex]\frac{7}{v}[/tex] would not be treated as the polynomial as the variable v is in the denominator.

Therefore, from the above discussion we can safely say that the expression [tex]7x^6\:-\:4x^3\:+\:1/x[/tex] can not be treated as as a Polynomial as its last term [tex]\frac{1}{x}[/tex] contains the variable x which is in the denominator.

Keywords: matrimonial, polynomial, exponent, degree

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